UNIVERSITY OF CALIFORNIA.
Class
THE
RESISTANCE AND PROPULSION
OF
SHIPS.
BY
WILLIAM F. DURAND,
Professor of Mechanical Engineering, Leland Stanford* Junior University.
SECOND EDITION, THOROUGHLY REVISED.
FIRST THOUSAND.
NEW YORK:
JOHN WILEY & SONS.
LONDON : CHAPMAN & HALL, LIMITED.
1909.
GENERAL
Copyright, 1898, 1909,
I BY
WILLIAM F DURAND.
frbr g-rintttfir
Sohrrl Srimunnnii anD Company
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PREFACE TO SECOND EDITION.
DURING the ten years since the first edition of this book was
prepared, the chief contributions to the material available for
the discussion of the resistance and propulsion of ships consist,
of the results of various experimental researches, notably in
regard to the operation of the screw propeller. In the prepa-
ration of the present revision no attempt has been made to
change the general character of the theoretical and descriptive
treatment of the subject. References have been given here and
there, however, to some of the more important recent contri-
butions to general theory.
Regarding the results of experience, an effort has been made
to summarize some of the more important researches in this
field, and especially in relation 'to the problem of design. Some
matter of relatively minor importance, and certain discussions
which could well be superseded by new matter based on more
recent experience have been omitted, thus permitting the inser-
tion of a considerable amount of new matter without material
change in the size of the volume. It is hoped that the present
revision will serve as a reasonably comprehensive introduction
;o the principal problems of ship resistance and propulsion,
and that in reference to the various problems of design, the data
given and methods suggested will be found adequately represen-
tative of present day practice.
STANFORD UNIVERSITY, CALIFORNIA,
January, 1909. iii
188508
PREFACE TO FIRST EDITION.
DURING the last twenty or thirty years the literature
relating to the Resistance and Propulsion of ships has
received many valuable and important additions. Of the few
books in English published in this period, those of the highest
value have been restricted either in scope or in mode of
treatment. For the most part, however, the important addi-
tions to the subject have been published only in the transac-
tions of engineering and scientific societies, or in the technical
press. Such papers and special articles are far from provid-
ing a connected account of the trend of modern thought and
practice, and the present work has been undertaken in the
hope that there might be a field of usefulness for a connected
and fairly comprehensive exposition of the subject from the
modern scientific and engineering standpoint.
Such a work must depend in large measure on the extant
literature of the subject, and the author would here express
his general acknowledgments to those who have preceded
him in this field. In many places special obligations are due,
and it has been the intention to give special references at
such points to the original papers or sources. With the
material drawn from the general literature of the subject
there has been combined a considerable amount of original
VI PREFA CE.
matter which it is hoped will contribute somewhat to what-
ever of interest or value the work as a whole may possess.
A free use has been made of calculus and mechanics in the
development of the subject, the nature of the treatment
requiring the use of these powerful auxiliaries. At the same
time most of the important results and considerations bearing
on them are discussed in general terms and from the descrip-
tive standpoint, and all operations involved in the actual
solution of problems are reduced to simple expression in
terms of elementary mathematical processes. It will thus be
possible, by the omission of the parts involving higher
mathematics, to still obtain a fairly connected idea of most
of the subject from the descriptive standpoint, and to apply
all methods developed for purposes of design.
In the development of such methods the purpose has
been to supply plain paths along which the student may pro-
ceed step by step from the initial conditions to the desired
results, and to arrange the method in such a way as shall con-
duce to the most intelligent application of engineering judg-
ment and experience. In the design of screw-propellers
especially, where the number of controlling conditions is
necessarily large, the purpose has been to present a mode of
solution in which the most important controlling conditions
are represented in the formulae, and in which the determina-
tion of the numerical values of these representatives by
auxiliary computation, estimate, or assumption is forced upon
the attention of the student in such a way as to call at each
step for a definite act of engineering judgment.
In so far as the method may differ from others, it is
intended to present the series of operations in such way as to
favor the recognition of a considerable number of relatively
PREFA CE. vii
simple steps, and the full and free application at each step of
such judgment and precedent as may be at hand. Such
methods are perhaps better adapted to the discipline of the
student than to the uses of the trained designer, and it is
obviously for the former rather than for the latter that such a
work should be prepared. At the same time it seems not
unlikely that even for the trained designer the splitting up of
concrete judgments into their separate factors is often an
operation of value, and one which will the more readily
enable him to adapt his methods to rapidly changing prece-
dents and conditions of design.
The natural limitations of size and the need of homo-
geneity have rendered the work in many respects less com-
plete than the Author might have wished, and many
important developments especially in pure theory have
received but scanty notice. As presented, the work repre-
sents substantially the lectures on Resistance and Propulsion
given by the Author to students of Cornell University in the
School of Marine Construction, and many features both in
subject-matter and mode of treatment have been introduced
as a result of the experience thus obtained in dealing with
these subjects.
CORNELL UNIVERSITY, ITHACA, N. Y.,
February 4, 1898.
CONTENTS.
CHAPTER I.
RESISTANCE.
SECTION PAE
1. General Ideas i
2. Stream-lines 7
3. Geometry of Stream-lines 16
4. Resume of General Considerations 28
5. Resistance of Deeply Immersed Planes Moving Normal to Themselves 32
6. Resistance of -Deeply Immersed Planes Moving Obliquely to their
Normal 37
7. Tangential or Skin Resistance 42
8. Skin-resistance of Ship-formed Bodies 48
9. Actual Values of the Quantities/ and n for Skin-resistance 50
10. Waves 54
11. Wave-formation Due to the Motion of a Ship-formed Body through the
Water 75
12. The Propagation of a Train of Waves 87
13. Formation of the Echoes in the Transverse and Divergent Systems of
Waves 89
14. Wave-making Resistance 93
15. Relation of Resistance in General to the Density of the Liquid 104
16. Influence of Form and Proportion on Resistance 104
17. Modification of Resistance Due to Irregular Movement 107
18. Variation of Resistance Due to Rough Water 108
19. Increase of Resistance Due to Shallow Water or to the Influence of
Banks and Shoals 109
20. Increase of Resistance Due to Slope of Currents .- 123 1
21. Influence on Resistance Due to Changes of Trim 124
22. Influence of Bilge-keels on Resistance 127
23. Air- resistance 129
24. Influence of Foul Bottom on Resistance 131
fa
CONTENTS.
25. Speed at which Resistance Begins to Rapidly Increase 131
26. The Law of Comparison or of Kinematic Similitude 133
27. Applications of the Law of Comparison 147
28. General Remarks on the Theories of Resistance 150
29. Actual Formulae for Resistance 151
30. Experimental Methods of Determining the Resistance of Ship-formed
Bodies 154
31. Oblique Resistance 157
CHAPTER II.
PRO PULSION.
32. General Statement of the Problem 162
33. Action of a Propulsive Element 164
34. Definitions Relating to Screw Propellers 174
35. Propulsive Action of the Element of a Screw Propeller 177
36. Propulsive Action of the Entire Propeller 182
37. Action of a Screw Propeller Viewed from the Standpoint of the Water
Acted on and the Acceleration Imparted 193
38. The Paddle-wheel Treated from the Standpoint of 37., 196
39. Hydraulic Propulsion 201
40. Screw Turbines or Screw Propellers with Guide-blades 203
CHAPTER III.
REACTION BETWEEN SHIP AND PROPELLER.
41. The Constitution of the Wake 206
42. Definition of Different Kinds of Slip, of Mean Slip, and of Mean Pitch 209
43. Influences of Obliquity of Steam and of Shaft on the Action of a Screw
Propeller 218
44. Effect of the Wake and its Variability on the Equations of 36 222
45. Augmentation of Resistance Due to Action of Propeller 227
46. Analysis of Power Necessary for Propulsion 228
47. Indicated Thrust 236
48. Negative Apparent Slip 237
CHAPTER IV.
PROPELLER DESIGN.
49. Connection of Model Experiments with Actual Propellers 240
50. Problems of Propeller Design 252
51. General Suggestions Relating to the Choice of Values in the Equations
for Propeller Design, and to the Question of Number and Location
of Propellers 281
CONTENTS. xi
SECTION PAGE
52. Special Conditions Affecting the Operation of Screw Propellers 288
53. The Direct Application of the Law of Comparison to Propeller Design 296
54. The Strength of Propeller-blades 299
55. Geometry of the Screw Propeller 306
CHAPTER V.
PO\VERLNTG SHIPS.
56. Introductory 322
57. The Computation of W p or the Work Absorbed by the Propeller 323
58. Engine Friction 324
59. Power Required for Auxiliaries 332
60. Illustrative Example . 334
61. Powering with Turbine Machinery 336
62. Powering by the Law of Comparison 337
63. The Admiralty Displacement Coefficient 340
64. The Admiralty Midship-section Coefficient 344
65. Formulae Involving Wetted Surface 344
66. Other Special Constants 346
67. The Law of Comparison when the Ships are not Exactly Similar 347
68. English's Mode of Comparison 352
69. The Various Sections of a Displacement Speed Power Surface 354
70. Application of the Law of Comparison to Show the General Relation
between Size and Carrying Capacity for a Given Speed., , , , 356
CHAPTER VI.
TRIAL TRIPS.
71. Introductory 362
72. Detailed Consideration of the Observations to be Made 364
73. Elimination of Tidal Influence 368
74. Speed Trials for the Purpose of Obtaining a Continuous Relation be-
tween Speed, Revolutions, and Power 372
75. Relation between Speed Power and Revolutions for a Long-distance
Trial 378
76. Long-course Trial with Standardized Screw 380
77. The Influence of Acceleration and Retardation on Trial-trip Data 383
78. The Time and Distance Required to Effect a Change of Speed 384
79. The Geometrical Analysis of Trial Data 394
80. Application of Logarithmic Cross-section Paper 404
81. Steamship and Propeller Data 411
INTRODUCTORY NOTE RELATING TO UNITS OF
MEASUREMENT.
THE units of weight, and of force in general, used in
naval architecture are the pound and the ton, the latter
usually of 2240 Ibs. It will always be so understood in the
present volume.
The units of velocity are the foot per second, the/<?0/ per
minute, and the knot. The latter, while often used in the
sense of a distance, is really a speed or velocity. As
adopted by the U. S. Navy Department it is a speed of
6080.27 ft. per hour. The British Admiralty knot is a speed
of 6080 ft. per hour. The distance 6080.27 ft. is the length
of a minute of arc on a sphere whose area equals that of the
earth. For all purposes with which we are concerned in the
present volume the U. S. and British Admiralty knots may
be considered the same.
On the inland waters of the United States the statute
mile of 5280 feet is frequently employed in the measurement
of speed instead of the knot, and the short or legal ton of
2000 Ibs. for the measurement of displacement and weight in
general.
Revolutions of engines are usually referred to the minute
as unit.
OF THE
[ UNIVERSITY }
RESISTANCE AND PROPULSION OF SHIPS.
CHAPTER I.
RESISTANCE.
1. GENERAL IDEAS.
IN the science of hydrostatics it is shown, for a body
wholly or partially immersed in a liquid and at rest relative
to such liquid, that the horizontal resultant of all forces
between the liquid and the body is zero, and that the vertical
resultant equals the weight of the body. If, however, there
is relative motion between the liquid and the body, the
hydrostatic conditions of equilibrium no longer hold, and we
find in general a force acting between the liquid and the body
in such direction as to oppose the movement, and thus tend
to reduce the relative velocity to zero. This force which we
now consider is therefore one called into existence by the
motion. It will in general have both a horizontal and a
vertical component. The latter, while generally omitted
from consideration, may in special cases reach an amount
requiring recognition. The existence of such a resisting
force, while due ultimately to the relative motion, may be
considered more immediately as arising from a change in the
amount and distribution of the surface forces acting on the
2 RESISTANCE AND PROPULSION OF SHIPS.
body. When there is no relative motion between body and
liquid, the surface forces are wholly normal pressures and
their distribution is such that, as above noted, the horizontal
resultant is zero and the vertical resultant equals the weight.
The instant such motion arises, however, the surface forces
undergo marked changes in both amount and character. The
normal pressures are more or less changed in amount and dis-
tribution, and, in addition, tangential forces which were
entirely absent when the body was at rest are now called into
existence. In consequence the horizontal resultant is no
longer zero, but a certain amount R, the equal of which must
be constantly applied in the direction of motion if uniform
movement under the new conditions is to be maintained.
The entire vertical resultant must, of course, still equal the
weight. This resultant, however, may be considered as made
up of two parts, one due to the motion, and the other to the
statical buoyancy of the portion immersed. The sum of
these two will equal the weight. Hence when in motion the
statical buoyancy will not in general be the same either in
amount or distribution as for the condition of rest.
The amount and distribution of the surface forces must
depend ultimately on the following conditions:
((a) Geometrical form and dimensions
(i) The body -j of immersed portion.
& Character of wetted surface.
(a) Density.
(b) Viscosity.
(3) The relative motion.
Since it is the relative velocity between the body and
liquid with which we are concerned, it is evident that we may
(2) The liquid... j
RESISTANCE. 3
approach the problem from two standpoints according as we
consider the liquid at rest and the body moving through it,
or the body at rest and the liquid flowing past it. The
former is known as Euler's method, and the latter as La-
grange's. Each method has certain advantages, and it will
be found useful to view the phenomena in part, at least,
from both standpoints.
We may also view the constitution of the resisting force
in two ways. The first, as a summation of varying forces
acting between the water and the elements of the surface as
mentioned above. On the other hand considering the water,
we find as the result of this disturbance in the distribution
and amount of the liquid forces a certain series of phenomena
varying somewhat with the location of the body relative to
the surface of the liquid. These we will briefly examine.
As the first and simplest case we will suppose the body
wholly immersed and so far below the surface that the dis-
turbances in hydrostatic pressure become inappreciable near
the surface. It thus results that there are no surface effects
or changes of level.
Let ABK, Fig. I, represent the body in question, con-
sidered at rest with the liquid flowing by. Now considering
the motion of the particles of water, it is found by experience
that two well-marked types of motion may be distinguished.
The particles at a considerable distance from the body
will move past in paths indistinguishable from straight lines.
As we approach the body we shall find that the paths become
gently curved outward around the body as shown at LMN.
Such paths are in general space rather than plane curves.
The nearer we approach to the body the more pronounced
the curvature, but in all these paths the distinguishing feature
4 RESISTANCE AND PROPULSION OF SHIPS.
is the smooth, easy-flowing form and the absence of anything
approaching doubling or looping. Such curves are known as
stream-lines. Passing in still nearer the body, however, we
shall find, if its form is blunt or rounded, a series of large
eddies or vortices seemingly formed at or near the stern.
These float away and involve much of the water extending
from the body to some little distance sternward. The water
between the eddies will also be found to be moving in a more
FIG. i.
or less confused and irregular manner. These eddies with
the water about and between them constitute the so-called
" wake." At high speeds also, and with a blunt or broad
forward end, there may be found in addition just forward of
the body a small mass of water in which the motion is not
well defined and smooth. Again, at the sides and between
the smooth-flowing stream-lines and the body we shall find a
belt of confused eddying water in which the loops and spiral
paths are very small, the whole constituting a relatively thin
RESISTANCE. 5
layer of water thrown into the most violent confusion as
regards the paths of its particles. The extent of the disturb-
ance decreases gradually from the body outward to the water
involved in the smooth stream-line motion. The character-
istic feature of the motion of the water involved in these
vortices, eddies, etc., is therefore its irregularity and com-
plexity as distinguished from the smoothness and simplicity
of that first considered. The volumes KEF and ABC are
sometimes known as the liquid prow and stern. The former,
however, is usually inappreciable in comparison with the
latter. The term "dead-water " is also sometimes applied to
the mass of water thus involved.
We may also consider the path of the particles by Ruler's
method of investigation. This will be simply the motion of
the particle relative to the surrounding body of water consid-
ered as at rest. For the outlying particles, such as those
forming the curves LMN, we shall have now simply a move-
ment out and back as the body moves by. The paths out
and back are not usually the same, and the particle does not
necessarily or usually return to its starting-point. If we now
suppose the body moving toward G, the path of a particle
originally at L will be some such curve as LRS. This is a
single definite path, the particle being moved out from L and
finally brought to 5 without firral velocity, where it therefore
remains. For particles involved in the eddies and whirls,
however, the paths are entirely indefinite, and consist of
confused interlacing spirals and loops. The particle is not
definitely taken hold of at one point and as definitely left at
another, but instead may be carried with the body some
distance and then left with a certain velocity and energy, the
latter gradually and ultimately becoming dissipated as heat.
O RESISTANCE AND PROPULSION OF SHIPS.
Let us now remove the special restriction relating to the
location of the body, and let us suppose the motion to take
place at or near the surface of the liquid. The normal dis-
tribution of hydrostatic pressure near the surface will now be
disturbed, and, as an additional result, changes of elevation
will occur constituting certain series of waves. As we shall
see later, the energy involved in these waves is partly
propagated on and retained within the system, and partly
propagated away and lost.
Now returning to the general consideration of resistance,
it is evident that we may approach its estimation from two
standpoints, (i) We may seek to study the amount and
nature of the disturbance in the distributed liquid forces act-
ing on the immersed surface. (2) We may seek to know the
resistance or force between the body and the liquid, through
its effects on the latter as manifested in the various ways
above described. The latter is the point of view usually
taken. From this standpoint it seems natural to charge a
portion of the resistance to each manifestation, and thus to
look for a part in the production of the curved stream-lines, a
part in the production of the eddies at the bow and stern, a
part in the eddying belt due to tangential or frictional forces,
and a part in the waves.
These various manifestations we shall proceed to take up
in order.
Stream-lines and stream-line motion have played so
prominent a part in the various views which have been held
on resistance, and serve so well to show certain features of
the general problem, that we shall find it profitable to first
examine them in some detail.
RESISTANCE. 7
2. STREAM-LINES.
For the definition and general description of what is
meant by a stream-line we may refer to the preceding section.
We have now to define a stream-tube or tube of flow. Let
PQRS, Fig. 2, be any closed curve in a plane perpendicular
FIG. 2.
to the line of motion LK. Let this curve be located at a
point so far from AB that the stream-lines PUV, etc., are at
P sensibly parallel to KL. The particles comprised in the
contour PQRS at any instant will, in their passage past AB y
trace out a series of paths which by their summation will form
a closed tube or pipe. Such is called a tube of flow. From
these definitions, a particle of water which is within the tube
at PR will always remain within it, and no others will enter.
We shall therefore have a tube of varying section in which
the water will flow as though its walls were of a frictionless
rigid material, instead of the geometrical boundary specified.
Starting at PR with a certain amount of water filling the
cross-section of the tube, we find the motion parallel to KL^
As we approach and pass AB the direction and velocity of
flow will change, but the latter always in such way as to
maintain the tube constantly full. After passing beyond AB
RESISTANCE AND PROPULSION OF SHIPS.
the direction of motion will approach KL, and finally at a
sufficient distance will again become parallel to it. Suppos-
ing the fundamental conditions to remain unchanged, the
entire configuration of stream-lines will remain constant, and
at any one point there will be no variation from one moment
to the next. It follows that the conditions for steady
motion as defined in hydrodynamics are fulfilled, and hence
that the equations of such motion are directly applicable to
the liquid moving within the tube. Assuming for the
present that the liquid is hydrodynamically perfect, that is,
that there are no forces due to viscosity, the equation for
steady motion is
where p = pressure per unit area;
cr = density;
2 = elevation above a fixed datum ;
v = velocity;
h = a constant for each stream-line, called the total
head.
is called the presure-head ;
<j
z is the actual head ;
is called the velocity-head.
2-
The total head, being constant for each line or indefinitely
small tube, but variable from one to another, may be consid-
* We shall not here develop the fundamental equations of hydro-
dynamics, but shall assume the student familiar with them or within reach
of a text-book on the subject.
RESISTANCE. 9
ered as a distinguishing characteristic of the tube or line of
flow. If we take the value of such characteristic far 'from
ABj as at PR, we shall have for p -7- <r simply the statical
pressure-head measured by the distance from PR to the sur-
face of the liquid. For z we have the vertical distance from
PR to the origin O at any fixed depth. The sum of these
two equals the distance from O to the surface. Hence, as is
perfectly permissible, if we take O at the surface, / -f- cr +
= o, and h v* -r- 2g, where V Q is the known velocity with
which the liquid as a whole is moving past AB. Hence with
O at the surface the equation to a line becomes
It is evident that as we move along the tube / and v will
depend on z and on the cross-sectional area of the tube. If
we take a case where the tube is sensibly contained in a hori-
zontal plane, z will be constant and / and v will vary in
opposite directions. The cross-sectional area will be found
to increase somewhat just before AB is reached. Hence in
this neighborhood v will decrease and p will increase. As we
pass on, the area will decrease, becoming a minimum at U.
Here / will be a minimum and v a maximum. The same
changes are then repeated in reverse order as we pass on from
U to V. The reasons for the variation in cross-sectional area
as above stated may be seen as follows:
Consider a tube of flow of large diameter to entirely
inclose AB and the liquid about it. We consider the sides
of the tube so far away that the stream-lines constituting its
boundary are sensibly parallel to KL. All phenomena may
therefore be considered as taking place within the tube. The
IO RESISTANCE AND PROPULSION OF SHIPS.
cross-sectional area available for the flow of the liquid will
evidently be the area taken normal to the stream-lines. At
the entering end and far from AB this will be a right section,
as MN. Just forward of AJB, however, where the stream-
lines curve outward, the orthogonal section will be curved or
dished as indicated by HJ. The area of this will be greater
than that of the right section at MN, and hence the average
area of the tubes of flow must be greater. This effect will
also be more pronounced near the body, where the change in
curvature is greater. As we pass on to /the stream-lines are
again parallel to the sides of the large tube, and hence the
net section available for flow will be the right section of the
tube minus the section of AB. Hence the average cross-
section of the tubes of flow must be less here than at the
entering end. As above, the difference will be more pro-
nounced near the body than far removed from it. Similarly
just behind AB the orthogonal section will be curved and the
average area will be increased, as at HJ.
We have now to show that in any tube of flow in which
the two ends are equal in area, opposite in direction, and in
the same line, the total effect of the internal pressures is o;
that is, that the pressures developed
have no tendency to transport the
tube in any direction whatever.
Let us first consider any closed
tube or pipe, as in Fig. 3, no matter
what the contour or the variation of
sectional area. Suppose it filled
with a perfect or non-viscous liquid
FlG - 3- moving with no tangential force
between itself and the walls of the pipe. The conditions
RESISTANCE.
II
dB
are therefore similar to those for a closed tube of flow
in a perfect liquid. Suppose the liquid within this tube
to have acquired a motion of flow around the tube. There
being no forces to give rise to a dissipation of energy, the
motion will continue indefinitely. We know from mechanics
that the forces in such case
form a balanced system and
hence that there is no resultant
force tending to move the pipe
in any direction. That such is
the case may also be seen by
considering that if there were
a resultant external force we
might obtain work by allowing FlG - 4-
such resultant to overcome a resistance. Such performance
of work could not react on the velocity of the liquid, and
hence we should obtain work without the expenditure of
energy.
Suppose next the pipe to be circular in contour and of
uniform section, as in Fig. 4.
Let a = area of section of pipe;
r = mean radius of contour;
G- = density of liquid =62.5 for fresh and 64 for salt
water ;
f = centrifugal force due to liquid.
Then the volume of any small element is ardO, and the
corresponding centrifugal force is
(TardBv*
df = - - = -- dO.
gr g
12 RESISTANCE AND PROPULSION OF SHIPS.
Let the total angle A OB be 26,. Let 6 be the angle
between OC and any element. Then the component of df
along OC is
crav* cos 6dB
df cos 6 = -- .
<*>
Integrating, we have for the total resultant along OC
Q = ^ sin 0,.
This total force outward from O toward C may be equili-
brated by a pair of tangential tensions at A and B. Let each
of these be denoted by /. Then we have
2<rav*
n < ,
2p sin 0j = Q sin lt
o
or / =
&
That is, the forces due to the movement of the liquid in
any arc give rise to two tangential tensions at the ends of the
arc, and these tensions are independent of its length and
depend simply on a and v. This is also evidently true no
matter what the remainder of the contour, or whether it is
closed or not. Hence in any pipe, closed in contour or not,
containing a portion of uniform area and curved in the arc of
a circle, the forces due to the flow of the liquid through this
portion will be such as would be balanced by two tangential
tensions at the ends of the arc, each equal to crav* -f- g.
Let us now consider a pipe of any irregular contour and
sectional area, as in Fig. 5. Let the ends A and D be of the
same area, but turned in any direction relative to each other.
RESISTANCE. 1 3
Suppose liquid as above to flow through the pipe. Required
to determine the resultant of the internal pressures.
Suppose the circuit completed by means of a pipe AFED
of uniform section and made up of circular arcs AF and DE
and a straight length FE. Then if a circulation is set up in
this closed contour such that the velocities at A and D are
the same as before, the conditions in ABCD will remain
unaffected. We know as above that the entire resultant is o,
and that the effects due to the liquid in AF and DE are
represented by tensions at the extremities of those arcs each
X
B
M A
Vv_
.--'^
, " jf
'. --- ''/ >p,
. - . .
E Jp.
FIG. 5.
of value o-av* -r- g. The effect due to the straight part FE
must be o, and the two tensions at F and E will balance each
other. Hence for the entire resultant of the forces in the
dotted part of the contour we shall have the equal tangential
forces p, = AM and /> 3 = DN at A and D. Now since the
system of forces for the entire contour is balanced, it follows
that the forces due to the liquid in ABCD must be repre-
sented by two forces equal and opposite to AM and DN.
Hence AR and DS must represent in direction, point of
application, and amount the resultant of the forces due to the
water in ABCD.
As a special case let ABCDE, Fig. 6, be a curved pipe of
any varying sectional area, but with its ends A and E equal,
14 RESISTANCE AND PROPULSION OF SHIPS.
opposite in direction, and in the same line. Applying the
above principle, it follows that the resultant will be that of
FIG. 6.
the two forces AR and ES, equal and opposite in direction
and in the same line. Hence in such case the resultant is o
and there is no tendency to move the pipe in any direction,
and in particular no force tending to move it in the direction
AE. This serves to establish the initial proposition relative
to the forces in such a tube of flow. In the most general
case the straight portions at the two ends, while parallel, are
not in the same straight line. The preceding treatment still
holds, however, and we shall have for such case the resultant
represented by two equal and opposite forces not in the same
line and therefore forming a couple.
Let now AB, Fig. 7, be a body about which the liquid
flows without tangential forces in stream-line paths. Let
FIG. 7.
this body be immersed at an indefinite depth so that surface
effects are insensible. We wish to show that the total force
RESISTANCE. 1 5
communicated to the body by the stream-lines is o. This
may be seen by considering PQRS a tube of flow entirely
surrounding ACBD, with its straight ends equal in area,
opposite in direction, and in the same line, and of variable
sectional area, ACBD forming an internal boundary. The
general proposition above then applies, and the resultant of
all forces acting on the internal boundary ACBD will be o.
Otherwise the tube PQRS may be considered as made up of
an indefinite number of small tubes for each of which the
y
resultant may be a couple. It is easily seen, however, that
the forces constituting these couples will, for the whole tube,
constitute a system uniformly distributed over the ends PR
and QS, and hence the entire resultant will be o.
This important conclusion may also be reached by con-
sidering that if a resultant force were communicated from the
liquid to AB, we might obtain work by allowing AB to move
and thus overcome some resistance. By the supposition of
the absence of all tangential force between AB and the
liquid, this would take place without affecting the velocity of
the latter. In fact without tangential force the velocity of
the liquid can in no wise be affected. Hence we should
obtain work without the expenditure of energy, and the sup-
positions leading to this result are therefore inadmissible.
We may at this point note a further result of the relation-
ship expressed by the general equation of hydrodynamics
above.
Suppose z to remain substantially constant and v to
increase continually. The pressure will correspondingly
decrease, and at the limit for a certain velocity we shall have
/ = o. An indefinitely small increase of velocity would lead
to a tendency to set up a negative pressure or tension in the
UNIVERSITY.
OF
1 6 RESISTANCE AND PROPULSION OF SHIPS.
liquid. This could not be actually realized, and the result
would be, instead, a breaking up of the stream into minute
turbulent whirls and eddies. The existence of such turbu-
lence may therefore imply a previous condition in which
there existed a tendency toward an indefinite decrease in the
pressure, and such may be considered as its hydrodynamical
significance. In the actual case the stream becomes unsteady
and breaks up before the pressure becomes actually reduced
to zero, although it is always much reduced as compared with
the pressure for steady flow. A further result of the great
reduction in pressure is usually found in the giving up by the*
water of the air which it holds in solution. The air thus
liberated appears in the form of small bubbles mingled with
the water, causing the foamy or yeasty appearance which
usually accompanies this phenomenon.
. 3. GEOMETRY OF STREAM -LINES.
Among the first to note the relation of stream-lines to the
problem of ship resistance was Prof. Rankine. His investi-
gations covered the examination of their general properties
and the derivation of forms for ships which, under the special
conditions assumed, would give for the relative motion of
ship and water smooth stream-line paths. We will now show
methods of constructing stream-lines for various special cases
as developed by Rankine and other investigators since his
time.
The stream-lines which surround a body are in general
space-curves. The properties of such lines of double curva-
ture are so complex that their investigation is a work of much
difficulty and labor. Many helpful and instructive sugges-
RESISTANCE. I/
tions, however, may be derived from a study of plane water-
lines to which the present notice will be restricted.
In order to imagine a physical condition which would give
such lines, suppose a body partly immersed in a liquid on the
surface of which is a rigid frictionless plane. Let a similar
parallel plane touch the under side of the body. Then let
the liquid between the planes move past the body, or vice
versa let the body move relative to the liquid. The relative
motion of liquid and body may then be considered as taking
place in water-lines contained in horizontal planes. At the
limit we may suppose the planes approached very near, the
body being then simply that portion contained between them.
The liquid will then move in a thin horizontal sheet in plane
water-lines.
It is shown in hydromechanics that for such water-lines
there exists a function fy such that
i .
u = velocity along x =
and v = velocity along y =
dx
That is, that the velocity in any direction is the rate of
change of this function in a direction perpendicular to the
given one. It is also shown that this function is constant for
any one water-line, and that it may therefore be considered
as a characteristic or equation to such line. It is likewise
shown that its value for any one line is proportional to the
total amount of flow between such line and some one line
taken as a standard or line of reference.
18
RESISTANCE AND PROPULSION OF SHIPS.
For the simplest case let us assume two planes as above,
indefinite in extent and very near. At a given point let
liquid be introduced at a uniform rate. Such a point of
introduction is called a source. In such case the liquid will
move in straight lines radiating from the source. Similarly
we might have assumed liquid abstracted at a uniform rate.
Such a point of abstraction is called a sink. In such case
likewise, the liquid will move
in straight lines converging
toward the sink.
In Fig. 8 let O be the point
and OA the line of reference
from which the total flow is
measured. Let a be the amount
introduced per second or other
unit of time, and V the angle
FIG. 8. between OA and any line OP.
Then the flow between OA and OP is that corresponding to
the angle A OP. Hence we have
The quantity a -r- 2n is called the strength of the source or
sink.
It is also shown in hydromechanics that the function ^ for
a complex system of sinks and sources is simply the algebraic
sum of the separate functions, considering i/> for a source as
-|- and for a sink as . Hence for such a system in general
we have
RESISTANCE.
For two sources of equal strength we have
aO l aO,
ib. = - and iL' = .
ri 27T ^ a 27T
Hence
^ +
For a sink and source of equal strength we have
(i ,
In all such cases of simple combination the actual stream-
lines are most readily found by a construction due to Maxwell
and illustrated in Fig. 9.
From O l and <9 2 let radiating lines be drawn at angular
intervals proportional to z~ -=- a. Then the number of lines
will be proportional to a -f- 2ic. Take any point P. Then
it is readily seen that by going across the quadrilateral PQRS
lo R we find another point for which Oi is increased by
2O RESISTANCE AND PROPULSION VF SHIPS.
RESISTANCE.
21
/\~~~~~--- \ """lliJW;
22 RESISTANCE AND PROPULSION OF SHIPS.
the angle RO^S and ^ decreased by the equal angle QO^S.
Hence at R we shall have (6 l + # 3 ) of the same value as at P.
Hence P and R will be points on one of the stream-lines
belonging to the two sources O l and 6> a . It follows that the
series of stream-lines for these sources will be obtained by
drawing continuous curved lines diagonally across the series
of quadrilaterals formed by the intersections of the radiating
lines from O l and O t . Similarly it may be seen that for Q
and 5 the values of (0, 0,) are the same, and hence that
these two points are on one of the stream-lines for a source
0, and a sink a . In like manner, then, the series of stream-
lines for such a doublet, as it is termed, will be found by
drawing continuous curvilinear diagonals in the other direc-
tion across the same set of quadrilaterals. These construc-
tions are illustrated in Figs. 10 and I i, and it is readily seen
in the latter case that the curves are arcs of circles.
Maxwell showed in general that for all cases of combination
of two systems of sinks and sources or their equivalents, the
resulting system of stream-lines could be found geometrically
by laying down those for the two component systems, and
then taking curved lines continuously diagonal to the series
of quadrilaterals thus formed. In this way the result of a
combination of any number of sinks and sources may be
found. In practice the labor involved is very great when
they exceed three or four in number.
In case the sinks and sources are not all of the same
strength, exactly the same method holds.
We have now to show as a special case the result of
the combination of a source and a continuous stream with
straight stream-lines. The latter may be considered as a part
of the system due to a source of infinite strength situated at
KESIS TA NCE. 2$
an infinite distance away. The geometrical method for the
determination of the resulting stream-line system is exactly
similar to the preceding, and comes under the general rule of
procedure as stated above. It is illustrated in Fig. 12. The
straight lines radiating from O are replaced each by a curved
line as shown. Outside of this system of curved lines spring-
ing from O are the deflected lines of the uniformly flowing
stream. It is readily seen that the line ABC separates one
of these systems from the other; also that there is no flow
whatever across this line, and hence that it separates the
liquid introduced by the source O from that originally in the
stream. It is also evident that if, instead of the source O
and its system of stream-lines, we should substitute a friction-
less surface ABC extending away indefinitely, the lines of the
uniform stream would have exactly the same form which they
now have as a result of the combination of the source and
stream.
Taking similarly a uniform stream in combination with a
source and sink, we have the result of Fig. 13. In this case
it is seen that the liquid introduced at (9, and withdrawn at
O^ does not spread away indefinitely, but is confined to the
portion bounded by the line ABC, and that, as before, this
line separates the liquid thus introduced and withdrawn, from
that originally in the stream. Hence also if, instead of the
doublet 6\(? a , we should introduce a thin flat solid with fric-
tionless contour ABC . . . , the resulting system of stream-
lines in the uniform stream would be the same as that here
resulting from the stream and the doublet.
It is thus seen that we may in this way arrive at the
system of stream-lines due to a frictionless solid of certain
form plunged in a uniformly flowing stream.
24 RESISl^ANCE AND PROPULSION OF SHIPS.
RESISTANCE.
26
RESISTANCE AND PROPULSION OF SHIPS.
The form ABC ... is not, however, suitable for the
water-line of a ship, and it is only by an extension of the
method that stream-lines for ship-shaped forms may be
determined. This extension involves the appropriate choice
and location of sources and sinks such that the resulting line
of separation shall as closely as may be desired resemble a
ship's water-line. The most general method of effecting this
extension is that due to Mr. D. W. Taylor,* which we may
briefly summarize as follows:
Instead of a system of separate sources and sinks,
Mr. Taylor imagines a source-and-sink line or narrow slot
through a part of which liquid is introduced, and through the
remainder of which it is withdrawn. The variation of
strength along this line corresponds to a difference in width
of slot. The distribution of strength may be represented
graphically by a line ABC, Fig. 14, where OB represents
the source portion and BP the sink portion, the strength
at any point being proportional to the ordinate of ABC.
FIG. 14.
The areas ABO and BPC must be equal, else the total
amounts introduced and withdrawn would not be the same.
The author then shows by graphical representation and inte-
* Transactions Institute of Naval Architects, vol. xxxv. p. 385.
RESISTANCE.
gration how to find the current function or equation for the
stream-lines corresponding to such a sink-and-source line, and
how by varying the character of ABC, Fig. 14, to vary the
character of the resulting contour ABC . . . , Fig. 13, so as
to bring it to any desired degree of similarity to a given
water-line.
The distribution of stream-lines being known, it is a
simple matter to find, by the help of the equation for steady
motion, 2 (i), the corresponding distribution of pressure.
This is illustrated in Fig. 15. ABC
represents one quarter of the hori-
zontal section of a body producing D
o
a system of stream-lines as shown. A
Then at points along a line OY l f
the excess in pressure over the
normal may be represented by the
ordinate to the curve DE relative
c
to O Y, as axis. Similarly the defect
in speed at the same points may be
represented by a curve such as FG.
In like manner at points along the FlG - I 5-
line BY^ the excess of speed is represented by the curve HI
and the defect of pressure by JK. At a considerable distance
from the body these values all approach indefinitely near to
their normal values and the variations therefrom become
indefinitely small, as indicated by the diagram.
The experimental study of plane stream-line motion has
received special attention at the hands of Hele-Shaw, who has
devised means of showing and of photographing the distribution
for ship-shaped and other forms of various kinds.* As pointed
* Transactions Institute of Naval Architects, 1897-1898.
28 RESISTANCE AND PROPULSION OF SHIPS.
out by him, the study of such systems may be of value in con-
nection with the investigation of the distribution of pressure over
a part of a submerged body such as a rudder.
Since, however, under the special conditions assumed for the
mathematical treatment of stream-lines, the total resultant force
between the body and the liquid is o, the resistance to the motion
of the solid through the liquid will also be o. The investigation
of stream-lines is not therefore sufficient in itself to determine
the actual resistance in any given case and must be considered
rather as a useful auxiliary in the investigation of problems con-
nected with the motion of a body through a liquid contained be-
tween rigid confining boundaries.
In the actual case the liquid is quite free instead of being
thus confined. This introduces into the form of the stream-
lines, especially near the surface, profound changes, so that the
indications received from plane stream-lines must be used with
caution, and only so far as comparison with the results of experi-
mental investigation may seem to justify. See further 14, on
wave-resistance.
4. RESUME OF GENERAL CONSIDERATIONS.
In considering the application of the general principles
developed in the present chapter to the resistance of ships, it
must be borne in mind that for the present we take no
account of any influence due to the presence of a propelling
agent. The resistance which we here consider is the actual
or tow-rope resistance the resistance which the ship would
oppose to being moved as by towing through the water at the
given speed. The modifications due to the presence of a
propeller or paddle-wheel will be considered in 45.
We have before us, therefore, the general problem of the
RESISTANCE. 2 g
resistance of a partially immersed body in an actual liquid, as
water. For convenience we shall examine the subject under
the following heads:
(1) Stream-line Resistance. The liquid being no longer
without viscosity or internal shearing stresses, the stream-line
conditions of a perfect liquid will not be quite realized, and
the maintenance of these lines in the actual liquid will involve
a slight drain of energy from the body. This involves the
transmission of a resultant force between the liquid and the
body opposed to the relative motion of the two, or, in the
present case, to the motion of the latter. This gives the first
item of the total resistance as above.
(2) Eddy Resistance. We come next to the water involved
in the liquid prow and stern. As already noted, this does
not follow open stream-line paths. It is thrown into irregular
involved motion, the energy of which in the actual liquid is
continually degrading into heat. The water involved in this
motion does not, moreover, remain the same, but gradually
changes by the drawing in of new particles and the expulsion
of others.
The maintenance of this liquid prow and stern will give
rise, therefore, to a constant drain of energy from the moving
body, the effect of which is manifested as a resistance to the
movement. This gives the second item as above.
(3) Surface, Skin, or Frictional Resistance. We take
next the belt of eddying water produced by the action of the
tangential forces between the surface and the liquid. These
eddies stand in the same relation to the motion of the body
as those giving rise to the eddy resistance so called. Their
maintenance constantly drains energy from the body, and
this gives rise to a resistance to the movement. This gives
the third item as above.
30 RESISTANCE AND PROPULSION OF SHIPS.
(4) Wave Resistance. Finally we have the wave systems
which always accompany the motion of a body near the sur-
face of the water. As remarked in I and as we shall later
show, the energy involved in these systems is not retained
intact, but a portion is constantly being propagated away and
lost. The maintenance of the wave systems requires, there-
fore, a constant supply of energy from the moving body, and
this, as in the previous cases, giyes rise to a resistance to the
motion. This gives the fourth item as above.
It may be noted that in the most general view (i) and (4)
are not independent. Waves may be considered as the result
of modified stream-lines near the surface, and hence in its
most general significance (i) might be considered as including
(i) and (4). As here classified, however, we mean by (i) the
resistance due to the maintenance of the system of stream-
lines which would be formed were the body moved with the
given speed at an indefinite depth below the surface. The
modification due to wave-formation is then taken care of
by (4). In any case (i) is very small, since, with the veloci-
ties occurring in practice, water behaves nearly as a non-
viscous liquid. Similarly (2) and (3) might with propriety be
termed the eddy-resistance, since both items arise from the
maintenance of systems of eddies. Since their immediate
causes are distinct, however, it is convenient to consider them
separately.
The sum of (i) and (2) is frequently termed the head-
resistance, though it is sometimes considered as of two parts,
called head and tail resistances, the former being due to the
excess of pressure in the liquid prow, and the latter to the
defect in the eddying liquid stern. , Inasmuch as they cannot
be separated, however, we shall use the one term for both.
RESISTANCE. 31
With ships of ordinary form this is so small as to be
relatively negligible. In other cases, as in the motion of the
paddle-wheel, propeller-blade, bilge-keel in oscillation, etc.,
this is one of the chief items of the resistance to the motion
of the given body.
In ships of usual form we are therefore concerned chiefly
with surface and wave resistance.
The sum of (i) and (4) or of (i), (2), and (4) is often
known as the residual resistance. This term has reference to
the usual view of ship-resistance which recognizes but two
chief subdivisions:
(a) Surface or skin resistance.
(b) Residual resistance, including (4), above, and all other
parts not properly classified under (a).
It must not be forgotten that this classification of resist-
ance according to the various effects produced on the water
is somewhat arbitrary. A more logical mode of subdivision
might be made by starting from the body and considering
that each element of immersed surface is subjected to a cer-
tain force. Such force may be resolved into two com-
ponents; one tangential, the other normal. This is entirely
general and includes all effects no matter from what cause
arising. Comparing with the other method of subdivision it
is evident that the longitudinal components of the tangential
forces will give by their summation the surface or frictional
resistance. Hence the longitudinal components of the normal
forces must give by their summation the remaining or resid-
ual resistance, corresponding to (i), (2), and (4), or with ship-
shaped forms almost entirely to (4). This subdivision is
-entirely independent of any supposition with regard to the
effects on the water, and is the most direct or immediate as-
32 RESISTANCE AND PROPULSION OF SHIPS.
pect of resistance itself. The two parts may be termed
tangential and normal resistances.
No matter what view we may take of the subdivision of
resistance, we are obliged to go to experimental investigation
for all satisfactory information as to its amount. We pro-
ceed, therefore, to the description of the various experimental
investigations, and to the discussion of the results which may
be derived therefrom.
5. RESISTANCE OF DEEPLY IMMERSED PLANES MOVING
NORMAL TO THEMSELVES.
If the body in Fig. I is supposed to be reduced to a plane
normal to the line of motion, the result will be a system of
stream-lines and eddies somewhat as in Fig. 16, though the
FIG. 16.
line of demarcation between the two is not fixed nor so defi-
nite as must be suggested by a rough diagrammatic represen-
tation such as the figure gives. The tangential resistance
will be o because all tangential force is at right angles to the
direction of motion. This does not mean that the actual
resistance is independent of the character of the surface, but
simply that the actual-tangential forces have no longitudinal
component. The nature 4 ' of : the surface will presumably
RESISTANCE. 33
slightly affect the stream-line formation and the amount and
nature of the liquid prow and stern, so that ultimately it will
have its effect on the resistance. Viewed immediately,
however, the resistance is due wholly to the eddies and
stream-lines in an imperfect liquid. Experiment shows that
the resistance in such case may be approximately expressed
by an equation of the form
(I)
where R = resistance in pounds;
f = a coefficient;
cr = density of liquid = 62.5 for fresh and 64 for salt
water;
g= acceleration due to gravity = 32.2 ft. ;
A = area in square feet;
if = velocity in feet per second.
For salt water cr -f- 2g is very commonly put equal to I.
In such case the equation becomes
(2)'
The values of the coefficient f which have been deter-
mined experimentally have varied widely. This is doubtless
due to the fact that such experiments have been carried on
by different observers under different conditions, notably as
to amount of surface, depth of immersion, and actual veloci-
ties employed. The subject has not been examined with
sufficient completeness to show satisfactorily the general rela-
tion of the resistance to the three conditions, area, immer-
sion > and velocity. In fact there is considerable doubt as to
whether R should increase strictly as the area, we are not sure
34 RESISTANCE AND PROPULSION OF SHIPS.
that the index of v should for all velocities be 2, and the
relation of /to varying immersion is not satisfactorily known.
The attempt, therefore, to express the results of experiments
under widely varying conditions by the simple formula above,
throwing on / the consequences of all differences between the
true and the assumed law, may with justice lead to widely
varying values for this coefficient.
Among the earliest experiments were those by Col.
Beaufoy made at the Greenland Docks in London between
1793 and 1/98. Taking A in square feet, v in feet per
second, and R in pounds, the mean value of /deduced from
these experiments is about i.i.
Mr. Wm. Froude's experiments on planes about 3 ft.
wide gave with similar units a mean value of about 1.7.
Dubuat at a single speed of about 3.28 ft. per second
found for a plane maintained stationary in a moving stream
a value of /= 1.86. For a stationary liquid and moving
plane, the speed being the same, he found /= 1.43. If in
the first case the liquid moved steadily and without eddies or
internal irregularities, and in the second case the liquid was
strictly at rest, the relative speed being the same in each
case, these two results should be the same. The discrepancy
was doubtless due to the difficulty of making satisfactory
observations on the mean velocity of a flowing stream, and
also to the now well known difference in effect between a
stream moving without and with turbulence.
More recently Joessel's experiments in the river Loire,
made on a plane of sheet iron .98 ft. high by 1.31 ft. long
and immersed so as to have .66 ft. of water over the upper
edge, gave a mean value of /= 1.6. The maximum velocity
was not above about 4.25 ft. per second.
RESISTANCE. 35
Still more recently Mr. R. E. Froude has determined this
coefficient under conditions which insure the highest degree
of experimental accuracy. The resulting value agrees very
closely with the average of Beaufoy's values mentioned
above, and was about i.i.
An extreme divergence from all the values above given is
indicated by the results of experiments on bilge-keels. An
attempt to fit this formula of resistance to these results leads to
a coefficient varying from 5 or 6 to 15 or 1 6. Its excessive varia-
bility, as well as its great divergence from the other values, indi-
cates that we have here involved certain elements or conditions
not represented by the formula, and whose importance has been
hitherto unsuspected.
This problem has received special theoretical investigation by
Bryan,* who shows that in large measure this extra effect may be
accounted for by the following features found in the motion of a
rolling ship fitted with bilge-keels:
(1) The movement of the ship to and fro tends to set up
currents in the water which lag in phase behind the roll of the
ship, and thus run counter in some degree to the motion of the
bilge-keel. These counter currents may be capable of producing
a marked increase in relative velocity between keel and water,
and hence a a marked increase in the resistance opposed to the
motion of the keel.
(2) It appears from a study of the stream-line motion about
a ship rolling with bilge-keels that such motion will produce a
marked change in the distribution of pressure over the surface of
the ship, and that this change of pressure will be of such character
* Transactions Institute of Naval Architects, 190x3.
36 RESISTANCE AND PROPULSION OF SHIPS.
as to resist rolling and thus increase the apparent resistance met
by the bilge-keel.
(3) We may first note generally that if the immersion is suffi-
ciently great to reduce the surface disturbance to a negligible
amount, the resistance will be practically independent of any
further increase of depth. This is readily seen from the fact
that the resistance arises, not from the pressure, which of
course increases with the depth, but rather from a difference
in the distribution of such pressure, and that this is practi-
cally independent of depth so long as surface disturbance is
absent. Still otherwise we may view the resistance as due to
the energy absorbed by the stream-line and eddy systems;
and since below the minimum depth above mentioned the
configuration of these will be constant, the energy absorbed
and the resulting resistance will likewise remain the same.
These conclusions are borne out by Beaufoy's experiments.
It will be observed that, according to definition, head-
resistance is the resistance due to the eddy and stream-line
formation supposing the immersion of the body so great that
surface effects are negligible. It is not the resistance due to
the actual stream-line and eddy formation if near the surface,
for this would include the wave-resistance as well. In the
present case, therefore, tangential resistance being absent, we
must consider the change in the total resistance as the plane
approaches the surface as due to the resistance arising from
the wave-formation. In any experiment, therefore, if the
plane is near the surface, the resistance actually measured will
consist of two parts: head-resistance proper and wave-making
resistance.
As is known furthermore, a slight wave may involve con-
siderable energy, especially if its length is great. Hence appar-
RESISTANCE.
37
ently slight surface effects may count for much in the actual total
resistance. These wave systems will be generated primarily by
the periodic or alternate discontinuous movement of the ship due
to which large bodies of water will be impressed with like dis-
continuous motion, and a portion of the energy of which will
become dissipated in the form of surface disturbance as noted,
and a part in the form of eddy and vortex motion.
The summation of these various influences, combined possibly
with others, may presumably become sufficient to account for
the extreme difference as noted, between the values of the coeffi-
cient resulting from the application of the formula to experiments
on planes in continuous motion and for which it is intended, and
resulting from its application to experiments on ships carrying
bilge-keels in alternate or discontinuous motion, and to the
circumstances of which it is not properly applicable.
6. RESISTANCE OF DEEPLY IMMERSED PLANES MOVING
OBLIQUELY TO THEIR NORMAL.
If the plane, instead of standing at right angles to the
direction of motion, is inclined at an angle 6, we have a result
somewhat as indicated in Fig, 17. In such case the force in
FIG. 17.
38 RESISTANCE AND PROPULSION OF SHIPS.
the line of the normal EB as determined by Joessel's experi
ments is given by the formula
. a .
39+ -61 sm 6 2g
. The longitudinal component, or the resistance in the line
of motion, is therefore
R 9 = 1.622 - p 2 - : TJ Av*. (2)
.39 + -61 sin 6 2g
P 9 sin 9
^ = .39 + .61 sin ....... <3)
sin'
^ = .39 + .61 si^ ........
According to Beaufoy's results we may put approxi-
mately
^ = sin<? ....... (5)
90
D
and j = sin 9 6. ...... (6)
^90
The experiments of Wm. Froude also agreed in giving
= sin 6 ....... (7)
90
Later experiments, especially on air-resistance, have thrown
some further light on the law of variation of normal resistance
with angle of inclination. As pointed out by Johns,f the changes
* Where not otherwise explained, the nomenclature of this section is the same
as that of the preceding.
f Transactions Institute of Naval Architects, 1904, p. 232.
RESISTANCE.
39
In density due to the slight variations in pressure produced by the
movement of a body through air are so small as to be negligible,
and hence we may transform results derived for air into corre-
sponding values for water by an appropriate density ratio factor.
In this manner experiments on air by Dines * on a plate 12 inches
35
Angle of Inclination
FlG. I 8.
square gave a value for water of 1.16 for the head-resistance
coefficient, as discussed in 5, and experiments on a rectangle
3 inches by 48 inches with the long edge inclined gave for the
normal resistance the approximate formula
= i. sn
Likewise from Langley's experiments the corresponding value
of the coefficient reduced from air is 1.24.
* Transactions Institute of Naval Architects, 1904, p. 232.
40 RESISTANCE AND PROPULSION OF SHIPS.
These experiments on air bring out clearly also a curious
anomaly in the law for variation with angle of inclination, and
show that the law of variation with the sine of the angle is only
roughly approximate, holding fairly well, however, for small
angles. The actual law is found to have the general form shown
in Fig. 1 8, with a marked hump at about 35 degrees inclination
and a depression beyond. Up to inclinations of some 20 degrees,
the law of variation with the sine is reasonably close, but beyond
and particularly between 30 and 45 degrees, such law is far from
representing the actual facts of the case.
Joessel's experiments were also directed toward the
determination of the location of the center of the system of
distributed pressures on an oblique plane. The results were
as follows:
Let / be the length of plane in the direction of motion,
and x the distance to the center of pressure from the leading
edge. Then
* = (.195 + -305 sin 0)/. .... (8)
In the narrow range of speeds, o to 4.25 ft. per second,
this law seemed to be independent of the velocity.
Referring again to Fig. 17, it is known that the stream
flowing toward AC will divide and pass around, a part by C
and a part by A. Let BK be the plane which separates one
of these two streams from the other. Then Lord Rayleigh
has shown that under certain special or theoretical conditions B
is determined by the following proportion:
AB 2 +4 cos 6 -2 cos 3 + (71-6} sin d
~AC~
RESISTANCE. 41
The tangential force is evidently less than if both streams
flowed in the same direction, as when the plane^ is moved
parallel to itself Let k be the ratio between the tangential
forces in the two cases. Then Cotterill * has shown that
4 cos (TT 2#) sin
k = - .
4 + K sin #
The graphical representation of these relations is shown
in Fig. 19.
The extreme irregularity in the curve of Fig. 18 is presumably
due to some discontinuity of law occurring at an inclination of
1.00 r
si
O .80
5
rial .60
uj go
o
u.
.40
O
o
UJ on
JLfl
\
\
X*
_\
102030405060708090
VALUES OF Q IN DEGREES
FIG. 19
about 35 degrees, and due to some relatively abrupt change in
the division of the stream between the two parts which flow, one
around C, Fig. 17, and one around A. Further examination of
these points is much needed.
* Transactions of the Institute of Naval Architects, vol. xx. p. 152.
42 RESISTANCE AND PROPULSION OF SHIPS.
The importance of reducing eddy resistance by an appro-
priate choice of form for a ship-shaped body, and the avoid-
ance where possible of all surfaces normal to the direction of
motion and of all sudden changes of direction in the surface,
is readily seen. It thus appears as shown by Taylor * for a
ship 300 feet long at 20 knots, using the results of 7 for
tangential resistance, that the resistance of one square foot
moving normal to its own direction is about 770 times as
great as that of the same area exposed to tangential or skin-
resistance only.
We may also add in this connection that if the lines of a
ship are too full aft for her speed, large, unstable eddies are
liable to appear about the stern, suddenly shifting about from
one quarter to the other and causing sudden and unsym-
metrical changes in the total resistance and in the action of
the rudder, thus rendering such ships very difficult to steer.
7. TANGENTIAL OR SKIN RESISTANCE.
The origin of this form of resistance has been referred to
in I. Experiments for the determination of its amount
have been made by Beaufoy, Tidman, and Wm. Froude, and
later in the U. S. Naval Tank at Washington, and elsewhere.
In the Froude experiments boards A inch thick, 19 inches deep,
and of different lengths from 4 to 50 feet were covered with various
substances and to wed lengthwise in a tank of fresh water, their speed
and resistance being carefully measured by appropriate apparatus.
For the discussion of the results of the experiments the tangen-
tial resistance is supposed to vary according to the formula
R = fAv n ,
where R = resistance in pounds;
/ = a coefficient ;
* " Resistance and Screw Propulsion," p. 27.
RESISTANCE.
43
A = area in square feet ;
v = velocity in feet per second or knots, according to
the value of /used;
n = an exponent.
A summary of the experimental results is shown in the
following table:
TABLE I. SHOWING RESULTS OF EXPERIMENTS ON
SKIN-RESISTANCE.
Length of Surface or Distance from Cutwater.
Nature of
Surface.
2 feet.
8 feet.
20 feet.
50 feet.
A
B
C
A
B
C
A
B
C
A
B
C
Varnish
2.OO
.41
390
1.85
325
.264
1.85
.278
.240
1.83
.250
.226
Paraffin. . . .
I Q*>
^8
a7O
1. 04
.314.
.260
I Q1
271
207
Tinfoil
2.16
30
.295
1. 09
.278
.263
I.qo
.262
.244
1.83
.246
.232?
Calico
1.93
87 .725
I.Q2
.626 1 .504
1.89
531
447
1.8 7
474
.423
Fine sand. ..
2.00
.81
.600
2.OO
.583
450
2.00
.480
384
2.06
.405
337
Medium sand
2.00
.90
730
2.OO
.625
.488
2.OO
534
465
2.00
.488
4S6
Coarse sand .
2 .00
1. 10
.880
2.OO
.714
-520
2.OO
.588
.490
Column A gives the value of n for the particular length
and character of surface.
Column B gives for the whole surface the mean resistance
in pounds per square foot at a speed of 600 feet per minute
or 10 feet per second.
Column C gives for the same speed the resistance per
square foot at a distance from the forward end equal to that
stated in the heading.
It is thus seen that the resistance per unit area decreases
as we go from forward aft. Thus for varnish the resistance
per square foot at distances of 2, 8, 20, and 50 feet from the
bow are respectively .39, .264, .24, .226. It necessarily
results, as is shown by the table, that the mean resistance per
square foot decreases with increased length. The explana-
tion of this as given by Mr. Froude is as follows:
44 RESISTANCE AND PROPULSION OF SHIPS.
The board or surface under test moves surrounded by a
skin of eddying water, the maintenance of the energy of
which gives rise to the resistance under consideration. The
forward part of the board is constantly entering water at rest,
while the after portions come in contact with water already
acted on by the forward end, and possessing, therefore, to a
greater or less extent, this eddying motion. In these eddies
the movement of the water-particles next the board is
forward, more or less of the water being involved, and with a
greater or less velocity according to the length of time they
have been acted on, and to the other circumstances of the sur-
face and motion. It follows that the eddying motion will be
more and more pronounced as we go from the bow aft.
Hence a part of the board near the forward end advancing
into still water will naturally meet a much greater resistance
than the same area located near the after end, and entering
water which already possesses in large measure the eddying
motion.
We will now consider in detail the character of the equa-
tion for tangential resistance as given above, especially as to
the relation between the .quantities involved and the circum-
stances of the experiment.
The quantity A is taken as the area without modification
or change. It is seen as above that the length factor of area
is the one of importance, so that we may consider, so far as
its effect on the quantities of the formula is concerned, A to
be represented by length. Suppose now that we have given
a series of values of R for a range of values of quality, length,
and speed. Each such value will give a single equation
involving the two unknowns /"and n. These may be both
variable from point to point, and not knowing on what they
RESISTANCE. 45
may depend we cannot consider any two equations as simul-
taneous, and thus use them for the determination of f and #.
In other words, without other conditions the relations of/
and n to the circumstances of the experiment are indetermi-
nate. Some arbitrary assumption must therefore be made in
regard to one, and the other determined in accordance there-
with. Taking any set of values of R for fixed quality, and
length at varying speeds, it seems most natural to assume
that /will be constant for all such values of R, and to then
determine n in accordance with this assumption. This is
equivalent to assuming that f is dependent on quality and
length, and independent of speed. We may also consider
this coefficient as providing for effects due to changes of tem-
perature and density of liquid. It may also be well to note
that this coefficient, and in fact the entire equation, is con-
sidered as independent of the depth of immersion.
While thus taking /as independent of speed variation, it
is not independent of the amount taken as the unit of speed.
This appears as follows: In the formula, give v a value I.
Then R=fA, or f= R-i- A. Hence /= the mean value
of R per square foot at the unit speed.
The value of /being thus known, it maybe substituted
in the various equations representing the values of R at vary-
ing speeds, and the values of n thus determined. It is seen
that the values of n may vary with the speed, and they will
likewise depend on the particular value of /used.
It thus appears that /will depend on length, quality of
surface, density, and temperature, while n will depend on
speed and / and hence on speed and the other various con-
ditions above.
Applying these principles to the results of Froude's
46 RESISTANCE AND PROPULSION OF SHIPS.
experiments * it is found in general for a given quality,
that / decreases with increasing length at a decreasing rate:
also that / varies with quality in the way we should naturally
expect.
The relation of n to length, quality, and speed is much more
obscure. Actual values vary from about 1.8 to 2.0, with some
evidence of a slight decrease for increase in length, at first rapid
and then slow. There is also evidence of a generally higher value
for rough surfaces than for smooth, the value for sanded surfaces
being about 2 for all lengths.
Early experiments on tangential resistance were limited in
speed to 6 or 8 knots, while the results obtained have been applied
to the estimation of such resistance for speeds of 30 knots and
more. This gap has been in large measure filled in by experi-
ments at Washington in the U. S. Naval Tank, the results of
which have generally confirmed those drawn from earlier experi-
ments, at least for smooth surfaces, and have indicated for n an
average value of about 1.854 and for smooth surfaces a value of
/=.oo 9 7.
With regard to the use of such values, it may be observed that
for any individual case but little significance can attach to the
third decimal place in the value of n, and the second will like-
wise involve some uncertainty.
On the other hand, we may remark that a difference of i
in the hundredths of n will cause for usual speeds a difference
of only about i per cent in the value of v n ^ so that such an amount
of uncertainty is not greater than that to which we are accus-
tomed in most engineering work.
In this connection the ideas of M. Risbec as given in a
* See complete report in British Association Reports for 1874, of which the
table on page 43 is a partial resume.
RESISTANCE. 47
recent paper * will be of interest. This author considers that
it is more rational to assume that fundamentally, skin-resist-
ance, involving as it does the energy of eddying water, will
be proportional to the square of the speed, and to write the
formula
The coefficient f is then to be determined in accordance
with the experimental data, and is considered as including
a certain function of "perturbation," the value of which
offsets the varying values of n in the more usual mode of
consideration. The possibility of such a treatment of the
original data will be obvious from the preceding discussion of
the relation of the quantities in the equation to the conditions
of the experiment. We have simply to find for the various
values of quality, length, and speed the quotient
M. Risbec then represents the quotient/" as a function of
four parameters for each quality of surface, of the speed ^,
and of the length L. The complexity of the resulting equa-
tion and the uncertainty of the exact nature of the param-
eters, especially as affected by their extension to lengths
and speeds far beyond those contained in the fundamental
data, make it questionable whether any advantage is to be
derived from this mode of consideration. It is, however, of
interest as showing a possible mode of considering the rela-
tion of the formula to the characteristics of the experiment.
* Bulletin de 1' Association Technique Maritime, vol. V. p. 45.
48 RESISTANCE AND PROPULSION OF SHIPS.
8. SKIN-RESISTANCE OF SHIP-FORMED BODIES.
While in the preceding section we have referred in general
to the application of the data therein discussed to the resist-
ance of ship-formed bodies, it must be remembered that
fundamentally the data relate simply to thin boards, sensibly
equivalent to planes.
Now at any point of a ship-formed body let the horizontal
component of the tangential force be denoted by/. Let ds
be an element of area, and the inclination to the longi-
tudinal of the water-lines at this point. Then / cos Ods is the
longitudinal component of the tangential force or the tangen-
tial resistance for this element. For the whole ship we shall
have, therefore, for the skin-resistance
R^ C p cos Bds\
or if/ denote a mean value of/, we have
R = / / ds cos 6.
The value of the integral is known as the reduced wetted
surface,* and it thus appears that strictly speaking it is with
this rather than with the actual surface that we are concerned
in dealing with skin-resistance. The difference between the
two, however, is usually less than I per cent, and the uncer-
tainties surrounding the whole question as already discussed
* As shown in the theory of statical naval architecture the value of the
reduced wetted surface is more simply expressed by
Reduced surface =fds cos 6 = fGdx,
where G is the variable girth or length of section, and dx is the element
of length.
UNIVERSITY
Of
.CALIFI
RESISTANCE. 49
show that no significant error will be made by using either
the one or th other. The experiments of Wm. Froude on
,i
the jjreyhound led him to use for A the actual wetted surface,
and the coefficients derived from his results and given in Table
II, p. 51, are intended to correspond to this value of the area.
In continental Europe the reduced surface is frequently used
in this connection.
The value of the wetted surface may be derived by
approximate integration as shown in the theory of statical
naval architecture, or by the use of one of the following
empirical and approximate formulae:
Let 5 = wetted surface in square feet;
D = displacement in tons-
L = length in feet;
H = draft in feet;
B = beam in feet ;
b = block coefficient of fineness.
Then we may have
S=L[i.>jH
For the reduced surface we have, from the foot-note on p. 48,
S r
* Transactions Society of Naval Arhcitects and Marine Engineers, rol. I.
p. 226.
t Transactions Institute of Naval Architects, vol., xxxvi. p, 72.
50 RESISTANCE AND PROPULSION OF SHIPS.
With the same notation as before we have the following
empirical relation between S and S r :
With a correct value of S,, the error of this formula is usually
less than one-tenth of one per cent.
9. ACTUAL VALUES OF THE QUANTITIES / AND n FOR
SKIN-RESISTANCE.
We will now give a resume of the practical values which
have been proposed by various authorities for the coefficient
/"and the exponent ;/.
The general formula is
R =
where R = resistance in pounds;
f = coefficient;
A = area of wetted surface in square feet;
v speed in knots;
n = index.
* Transactions Society of Naval Architects and Marine Engineers, vol. in.
p. 138-
RESISTANCE.
TABLE I, SHOWING FROUDE'S EXPERIMENTAL VALUES.
Nature of Surface.
Length of Surface.
2 feet.
8 feet.
20 feet.
50 feet.
n
/
/
n
f
n
/
2.00
1-95
2.l6
1-93
2.00
2.00
2.00
,OII7
.Olig
.0064
.0281
.0231
.0257
.0314
I.8 5
1.94
1.99
1. 92
2.00
2.00
2.OO
.OI2I
.0100
.008l
.0206
.0166
.0178
.O2O4
1.8 5
1-93
1.90
1.89
2.OO
2.OO
2.OO
.OIO4
.0088
.0089
.0184
.0137
.0152
.0168
1.8 3
1.83
1.8 7
2.06
2.00
.0097
.0095
.0170
.0104
.0139
Paraffin
TABLE II.
VALUES FOR SHIPS BASED ON FROUDE'S EXPERIMENTS.
(SALT WATER.)
Index for the
Index for the
Coefficient of
Variation of
Coefficient of
Variation of
Length
in Feet.
Skin-resistance.
Resistance
with Speed.
Length
in Feet.
Skin-resistance.
Resistance
with Speed.
f
n
f
n
8
.01197
1.825
80
.00933
1.825
9
.01177
"
90
.00928
IO
.OIl6l
"
100
.00923
12
.01131
"
120
.00916
14
.01106
140
.00911
16
.OIO86
'
160
.00907
18
.01069
"
180
.00904
20
.01055
200
.00902
25
.OIO29
"
250
.00897
30
.01010
"
300
.00892
35
.00993
<
350
.00889
40
.00981
400
.00886
45
.00971
ii
450
.00883
50
.00963
41
500
.00880
60
.00950
"
550
.00877
c
70
.00940
"
600
.00874
<
RESISTANCE AND PROPULSION Of SHIPS.
TABLE III.
VALUES FOR SHIPS BASED ON TIDMAN'S EXPERIMENTS.
(SALT WATER.)
Iron Bottom Clean and
Copper or Zinc Sheathing.
Well Painted.
Length
in Feet.
Smooth.
Rough.
''f
n
f
f
10
.OII24
853
.OIOO
9175
.01400
.870
2O
.01075
.849
.00990
.900
.01350
.861
30
.01018
.844
.00903
.865
.01310
.853
40
.00998
8397
.00978
.840
.01275
.847
50
.00991
.8357
.00976
.830
.01250
.843
100
.00970
.829
.00966
.827
.01200
.843
150
.00957
829
.00953
.827
.01183
.843
2OO
.00944
.829
.00943
.827
.01170
.843
250
.00933
.829
.00936
.827
.OIl6o
.843
300
.00923
.829
.00930
.827
.01152
.843
350
.00916
.829
.00927
.827
.01145
.843
400
.00910
.829
.00926
.827
.01140
.843
450
.00906
,829
.00926
.827
.01137
843
500
.00904
1.829
. 00926
.827
.01136
843
TABLE IV.
VALUES FOR PARAFFIN MODELS BASED ON TIDMAN'S
EXPERIMENTS.
(FRESH WATER.)
Index for the
Index for the
Coefficient of
Variation of
Coefficient of
Variation of
Length
in Feet.
Skin resistance.
Resistance
with Speed.
Length
in Feet.
Skin-resistance.
Resistance
with Speed.
f
n
f
2
.01176
1.94
12
.00908
1.^94
3
.01123
12.5
.00901
4
.01083
13
.00895
5
.01050
13-5
.00889
6
.01022
14
.00883
7
.00997
14.5
.00878
8
.00973
15
.00873
9
.00953
16
.00864
10
.00937
17
.00655
10.5
.00928
18
.00847
n
.00920
19
. 00840
ii. 5
.00914
20
.00834
RESISTANCE. 53
The various values for / given in the above tables are based
on the experiments of Froude and Tidman. As other special
values we may mention the following:
As noted in 7, the values commonly employed at the U. S.
Naval Tank are
/=.oo97, w = 1.854.
Colthurst * gives the value /= .0167 for rough-sawn
wood and /= .01 for smooth-planed wood. The same
authority states that for fine " grass" or marine growth /
may rise to a value from .048 to .062. For clean copper the
value .0073 is given, and for fresh-painted iron the value .01.
This last value is also that used by Prof. Rankine. These
values all relate to ships of usual length, or at least over 50 feet.
In this connection the results given in a recent paper by
Chief Naval Constructor Hichborn f are of interest. From
the data given in this paper the following tabular presentation
is derived : Column a shows the percentage increase of total
resistance with foul bottom over that with clean, reckoned on
the latter as base.
Ship. Speed. a.
Charleston 8.8 70
Yorktown 9 72
Philadelphia II 52
San Francisco 9 33
Bennington 7 200
Baltimore 1 1 20
It thus appears, as we should expect, that the term
" foul " is entirely indefinite, and may mean almost any con-
dition in which the resistance is sensibly increased. In the
* Cited by Pollard and Dudebout, Theorie du Navire, vol. m. p. 376.
t Transactions Society of Naval Architects and Marine Engineers, vol. II. p. 159.
54 RESISTANCE AND PROPULSION OF SHIPS.
extreme case of the table the total resistance of the Benning-
ton when foul is three times that when clean. At the low
speeds at which these runs were made, most of the resistance
will be due to the skin, so that while the above figures refer
to total resistance, the ratios for the skin-resistance will not
be far different, though slightly greater.
These possibilities impress in the strongest manner the
extreme importance of a clean bottom especially for economi-
cal cruising at moderate speeds, and for the possible attain-
ment of the highest speeds.
10. WAVES.
As introductory to the subject of wave-resistance we shall
give in the present section a brief account of the commonly
accepted theories of wave-motion in liquids, with a statement
of the more important results, but without the details of the
mathematical development.*
In order to examine in a satisfactory manner the influence
of waves upon a ship, some working theory as to their
dynamical constitution is required. It must be understood
that such a theory is not to be viewed as final truth in itself,
but rather as a convenient method of correlating as nearly as
possible the known facts, and of deducing by appropriate
* Among others, the following works may be consulted by those in-
terested:
Rankine et al.: "Shipbuilding, Theoretical and Practical." London,
1866.
J. Scott Russell: " System of Modern Naval Architecture," vol. i. Lon-
don, 1865.
Encyclopedia Britannica, article Wave.
Pollard et Dudebout: Theorie du Navire, vol. HI. Paris, 1892.
Gatewood : Journal U. S. Naval Institute 1883, p. 223. Annapolis.
Stokes: Cambridge Transactions 1847.
Encyclopedia Metropolitana, article Tides and Waves.
RESISTANCE.
55
mathematical means the probable nature and extent of the
influences which are to be examined. The most simple of the
theories proposed to this end is that known as the trochoidal.
The theory receives its name from the trochoid, which is the
form of the contour of waves resulting from this mode of
viewing their constitution. The trochoid may be defined as
the locus of a point P, Fig. 20, which revolves uniformly
FIG. 20.
about a center O, which center itself moves uniformly along
a straight line OO 6 . In this case if O moves uniformly
along OO K , and OP revolves uniformly counter-clockwise,
the curve Pf^ . . . P t will result. For comparison there is
shown in dotted lines a sinusoid of equal length and height,
and the difference between the two is indicated by the shaded
area.
A trochoid may also be defined as the locus of a point in
the radius of a circle, which circle rolls uniformly along a
straight line. Thus in Fig. 21 let a circle of radius OQ roll
along on the under side of AB. Let Pbe the tracing-point.
Then the locus of P will be a curve CPDE, and the identity
in character between the two curves in Figs. 20 and 21 is
readily seen. If the tracing-point P is on the circumference
of the circle the locus becomes the common cycloid as a
special case of the trochoid. If the point passes beyond Q,
as to R, the locus is still called a trochoid. The specific
56 RESISTANCE AND PROPULSION OF SHIPS.
names prolate-trochoid and curtate-trochoid are given to
these two classes of curves. It is only with the former as
shown in the figures that we are here concerned.
Waves whose contours are very nearly trochoidal are but
rarely met with, though a moderately heavy swell in calm
weather at a considerable time after the subsidence of the
storm to which it owes its existence, quite closely conforms
FIG. 21.
to this type. The trochoidal wave is, moreover, the type
most readily examined by mathematical means, and is there-
fore naturally taken for purposes of theoretical investigation.
The ideal constitution of a trochoidal wave involves the
following suppositions:
(1) That for any given wave all particles revolve in fixed
circular orbits, in vertical planes parallel to the direction of
propagation and normal to the lines of crests and hollows,
with the same constant angular velocity, and in direction with
the propagation when in a crest.
(2) That for any layer of water originally horizontal the
radii are equal and the centers are on the same horizontal
line.
RESISTANCE.
57
(3) That the radii decrease with increase of depth accord-
ing to a law to be stated later.
(4) That all particles originally in the same vertical are
always in the same phase.
It follows that the surface layer of particles and all suc-
cessive layers originally horizontal will take the form of
trochoidal surfaces of continually decreasing altitudes accord-
ing to the law of decreasing radii. It further results that all
corresponding crests and hollows for successive layers will lie
in the same vertical line, and hence that all wave-lengths for
the successive trochoidal subsurfaces will be equal, and that
the velocity of propagation for all must be the same.
This ideal constitution of a wave may be illustrated in
Fig. 22. The diagram represents a section of a certain por-
FlG. 22.
tion of still water divided by horizontal and vertical planes
into rectangular blocks as shown in dotted lines. The full
lines represent the same blocks of water when thrown into
wave-motion. The horizontal planes o, i, 2, etc., for still
water are, in the wave, thrown into the trochoidal surfaces
#, b, c, etc. The corresponding circular orbits are shown on
58 RESISTANCE AND PROPULSION OF SHIPS.
the left, and their centers are seen to be elevated above the
corresponding still-water planes by successively decreasing
amounts as we go from the surface downward. To avoid
confusion in the diagram the circular orbits are elsewhere
omitted. The motion involved in the figure may be viewed
from several standpoints. Thus we may consider the
trochoidal layers and note, as the wave progresses, the change
in thickness and form at any one point. Or again we may
consider the vertical columns of water between the distorted
verticals af,gh, kl, etc., and note that as the wave progresses
these columns shorten and broaden and then lengthen and
narrow, at the same time waving to and fro with the period
of the wave.
Again, each rectangular block of still water is represented
in the wave by a distorted block, and the successive forms
into which one of these blocks is thrown may be seen by fol-
lowing along the series of distorted quadrilaterals between the
pair of corresponding trochoidal surfaces for the wave.
In order that such a wave may actually exist in a liquid it
must fulfil certain hydrodynamic conditions. These are:
(i) That at the upper surface the pressure must be constant
and normal to the surface. (2) That the liquid cannot
undergo expansion or compression during the process of
wave-propagation. These conditions may be made to furnish
a relation btween the length and the velocity of propagation,
and a law for the decreasing radii of the circular orbits.
These are as follows:
Let L = length in feet;
V = linear velocity in feet per second;
R = radius of rolling circle:
QO =. angular velocity;
RESISTANCE.
59
T = periodic time ;
g = value of gravity = 32.2.
Then V--
or F= 2.265 ^L ft- per sec.,
and F= 1.341 VL knots,
also
(2)
Let r a = radius of orbit at surface;
r = radius of orbit at depth of center z below center
of surface orbits;
z = any given depth of orbit center below center of
surface orbits;
e = base of Naperian logarithms.
Then
or
z _ 2irz
r= r.e~~R r.e~~^
(3)
If these various conditions are fulfilled, therefore, it
appears that the geometrical motion here supposed, if once
inaugurated, is hydrodynamically possible. This does not
prove that in any actual wave the particles move as herein
assumed. Observation shows, however, that these supposi-
tions are near the truth, and the actual motion and character-
istics of waves, so far as admitting measurement, are for the
most part in fairly satisfactory agreement with those resulting
from the type of wave-motion assumed. It seems, therefore,
reasonable to accept the theory as a working basis for the
60 RESISTANCE AND PROPULSION OF SHIPS.
investigation of such characteristics of wave-motion as we are
here concerned with.
We may now state a further series of results following
upon the conditions assumed.
The point of maximum slope of any trochoid corresponds
to a value of 3 given by
cos 6 = . (4)
or
If is this maximum slope, then
r nh , N
and sin = -=--, ...... (5)
where h = height of wave = 2r.
The normal at any point intersects the vertical through
the center of the rolling circle at the constant distance R
above it, and therefore always at the point of contact of the
rolling circle with the line on which it rolls.
The pressure on any trochoidal subsurface is constant and
equal to that due to the depth of the corresponding plane
surface in still water.
Let z = depth of the line of centers;
r = radius of surface orbit ;
r = ' orbit with center at depth z\
R= " " rolling circle.
RESISTANCE.
Then the pressure in the crest of a wave is less than that
for an equal depth in still water in the ratio
z
2R
(6)
Similarly the pressure in the hollow of a wave is greater
than that for an equal depth in still water in the ratio
2R
2 r, -)- r
(7)
At any point the resultant force due to gravity and to
centrifugal force acts along the normal and
hence through the instantaneous center of
motion of the rolling circle as specified
above. In Fig. 23 let OA = R. Then if
the force due to gravity is represented by
OA or R, that due to centrifugal force will
be represented by the radius OP, and the
resultant by the line AP.
Let DI = mass of particle;
g = value of gravity = 32.2 ;
co angular velocitv '
L
FIG. 23.
/= resultant force represented by AP.
Then we have
#. ... (8)
The line of orbit centers for any trochoidal surface or
62
RESISTANCE AND PROPULSION OF SHIPS.
sheet is raised above the original still-water level of the
corresponding plane by the distance
L " 2R
2R
(9)
Let H denote the still-water depth of a layer correspond-
ing to the trochoidal sheet with line of centers at a depth z
below the center line of surface orbits. Then
' (10)
2R
The total energy of a wave is always half kinetic and half
potential.
Let E denote total energy in foot-pounds;
& " density;
L " length in feet;
r " radius of orbit at surface;
h '* height of wave = 2r ;
R " radius of rolling circle.
Then *=
vLk'l lh\\
01 R -- I 1 -- 4.935$;.
If we put & = 64, we have
E =%Uf(i- 4-935
The total power involved in a series of waves per foot of
breadth is
Power = .0329 VLtf (i 4.935 UJ J. . . (12)
RESISTANCE. 63
Combinations of Wave Systems.' Given two trochoidal
systems in general with parallel crests, moving in the same
direction. The result of such a combination geometrically,
as is well known, is to produce a series of groups of waves,
each group reaching a maximum of height where two crests
correspond, and a minimum where crest and hollow combine.
Each group therefore culminates in a wave of maximum alti-
tude with comparatively small disturbance betweeen the suc-
cessive groups. It may be shown that such geometrical
combination does not fulfil the necessary hydrodynamical
conditions, and therefore cannot exist permanently as a com-
bination of actual trochoidal systems. The actual result of the
combination of two wave systems is to produce a more or less
mixed resultant system, with characteristics relative to which
those given by the geometrical combination of trochoidal sys-
tems are intended simply as approximations. We do find,
however, actual combinations of wave systems agreeing in all
their general characteristics with those given by the geometri-
cal combination referred to, and it therefore seems fair to use
the method as a working basis for the examination of such
points as we are interested in.
As a special case let us assume that the two series are of
Components
Resultant
FIG. 24.
equal altitude, h, but of different lengths, Z, and. Z,. Then
the geometrical combination will give a result as indicated in
Fig. 24. At A and C the phases are opposite, and the values
64 RESISTANCE AND PROPULSION OF SHIPS.
of h being equal the resultant disturbance is o, and we have
for a little space practically undisturbed water. At a point
B midway between, the phases are the same and the two
effects are added, giving a wave of height 2//, the maximum
possible in the entire series. The group from A to C is then
indefinitely repeated to the right and left as far as the given
conditions hold.
For such a group the length AC is given by the equation
Let v = velocity of propagation of the resultant group as
a whole;
v l = velocity of propagation of the component of
length Z,;
v % = velocity of propagation of the component of
length Z a .
Then v =
i i i
or - = - + -
*7 *7f ' *7f
(H)
As L l approaches Z 2 in value the length of the group AC
will increase and the velocity will approach more and more
nearly to one half that of the components. At the limit
where Z, = Z, the length would become indefinite and the
velocity would become one half that of the components.
Since any given group is the expression of a certain distribu-
tion of energy, it is considered that the velocity of propaga-
tion of the energy is the same as that of the group.
RESISTANCE.
We will now take the special case of two trochoidal
systems of different altitudes, but of equal lengths and there-
fore of equal velocities.
In Fig. 25 let O denote the orbit center for a given sur-
face particle as affected by either component. Strictly
speaking the two centers are not c
at the same level, but the differ-
ence is very small and is usually
neglected. At any given instant
of time let DO A denote the phase-
angle B 1 of the first component, and
DOB the phase-angle 6^ of the
second component. Then AOB
will represent the constant differ-
ence of phase between the two
components. Also let OA = r } and
OB = r^. Then the combination
of the two components will result in a trochoidal system of
wave-length equal to that of its components, and of surface
orbit radius r = OC, and with phase relationships with its
components as given by the angles CO A and COB. In other
words, if the parallelogram OACB is made to revolve uni-
formly about O while the latter moves uniformly along x,
the two speeds being appropriate to the wave-length taken,
then the point A will trace the contour of the first com-
ponent, B that of the second, and C that of the resultant.
Let a denote the linear difference between similar points
on the two components, <*> the angular velocity, J^the linear
velocity, and R the radius of the rolling circle. Then the
angle denoting the phase difference AOB is given by
FIG. 25.
66 RESISTANCE AND PROPULSION OF SHIPS.
coa a 2ita .
A OB = = -75- = :, in angular measure,
3600 .
or A OB = j in degrees.
_z>
From the triangle OAC we derive for the altitude of the
resultant system
= *; + *. + 2* A cos. . . . (15)
As to the degree of fulfilment which the ideal theory
finds in actual waves it may be said in general, and without
entering into details, that, so far as observations can be made,
most of the actual characteristics are in fair agreement with
the results derived from theory. The agreement is by no
means complete, and in fact differs in certain particulars in
such way as to lead to the belief that a truly trochoidal wave
is but rarely if ever met with. On the other hand the agree-
ment is sufficiently close to seemingly warrant the use of the
theory for^ all purposes of the naval architect, at least in so
far as he is concerned with waves in deep water.
In regard to the limiting sizes of waves the following
results may be given:
From reliable observation it may be fairly concluded that
waves longer than 1200 to 1500 feet are very rarely met with,
though there seems ground for accepting reports of excep-
tional waves of a length approaching 3000 feet. The corre-
sponding periods would be 15 to 17 seconds for the first
named, and about 24 seconds for the last.
With regard to height the evidence is much more conflict-
ing. It appears, however, that from 40 to 45 feet is about
RESISTANCE.
6 7
the maximum value which can be accepted as reliable, and
values approaching these figures are met with but rarely.
From 25 to 30 feet may be taken as the ordinary limit. It
should be noted, however, that these heights relate to a
series of regular waves as contemplated by the theory, and
not to exceptional results arising from the interference and
combination of different systems.
With regard to ratio of height to length, it is found
usually to decrease with increase of length. The highest
values of this ratio appear to be about I : 6, such being found
only with comparatively short waves. More commonly the
ratio is from I : 12 to i 125. The corresponding values of
the maximum inclination are respectively about 32, 15,
and 7.
Shallow-water Waves. When we come to waves in
shallow water we find, as nearly as can be determined, the
paths of the particles to be elliptical with the longer axis
horizontal. In order to meet this departure from the tro-
choidal system, the so-called ellip-
tical trochoidal or shallow-water
wave system is used. This ideal
system may be generated as fol-
lows: Given an ellipse, Fig. 26,
with its major axis horizontal,
moving uniformly along X without
angular change. Draw any line
OB at an angle 6 with the verti- FlG - 26 -
cal. Through B draw a vertical and through A a hori-
zontal. They will intersect on the ellipse at P. Let OB
revolve uniformly to the left. Then the locus of the point P
68 RESISTANCE AND PROPULSION OF SHIPS.
thus determined by the movement of the ellipse along OX
and of OB about O will be an elliptical trochoid. Comparing
this contour with that resulting from the circle as a base, as
in Fig. 27, the former is seen to lie slightly within the latter,
thus making a flatter hollow and steeper crest, these differ-
ences becoming more pronounced as the difference between
the two axes of the ellipse is greater
Full line, Circular Trochoid
Dotted line, Elliptical Trochoid
FIG, 27.
In the present connection the feature of especial interest
is the velocity of propagation. For this the following ex-
pression may be derived :
Let V = velocity in feet per second;
r = semi-major axis at surface;
= " minor " *' "
L = wave length ;
g = value of gravity = 32.2.
Then F=-v/t=\/? (16)
This equation shows by comparison with (i), that the
velocity of a shallow-water wave is less than that of a deep-
water wave of equal length, a conclusion borne out by
experience.
Let d = depth of the water. Then when 2nd -^ L is
small it may be shown that
(17)
RESISTANCE.
6 9
It should be noted that the ideal wave system thus
derived, does not, and seemingly cannot, be made to fulfil the
hydrodynamic conditions necessary to its actual existence as
assumed. This does not prevent, however, such form of
motion from being, possibly, a close approximation to what
actually takes place, nor does it seriously interfere with its
usefulness as an approximate working theory of shallow-water
waves.
Waves of Translation. The waves thus far considered
occur naturally in indefinite series, and represent the result
of a widely distributed disturbance. The typical wave of
translation is individual in character, and represents the result
of a local disturbance. In Fig. 28 let AB be a trough or canal
cc,
DD,
FIG. 28.
of a length AB, large relative to its breadth and depth. Let
CD be a false end or partition movable within the canal
along the direction AB. If now CD be moved rapidly from
CD to C^D^ the water will become at first banked up in front
of it; but as CD is retarded and finally brought to rest at
CJ)^ the banked-up water will leave C^D^ and travel on as a
crest or hump elevated entirely above the general level of the
tank, somewhat as indicated in the diagram. Such a wave is
called positive.
Let us again suppose that the partition was at C l D l and
the water at rest. Let it then be moved from C,D^ to CD.
70 RESISTANCE AND PROPULSION OF SHIPS. '
As a result a hollow or depression will be formed existing
wholly below the general level of the surface. This hollow
will then be propagated on in a manner similar to the crest
previously described. Such a wave is called negative.
A positive wave may also be formed by plunging into the
water at one end of the canal a block which will occupy the
volume CDDfv or by the sudden introduction of a certain
additional amount of liquid into the end of the trough.
Similarly a negative wave may be formed by withdrawing a
block, or by withdrawing, as by a lifting-pump, a certain
amount of water.
Positive waves if properly formed are quite permanent,
only slowly losing their energy and form by external friction
and internal viscous forces. If not properly formed the wave
will soon break up into an irregular series, the members of
which are propagated with different velocities, so that the
energy is soon dissipated and the form disappears. The
negative wave is much less permanent than the positive, and
can be made to preserve its integrity for a short time only.
The horizontal movement of any sheet of particles sit-
uated in a transverse plane seems to be the same. The
vertical motion varies according to its distance from the
bottom of the canal, and seems to be very nearly proportional
to such distance. The path of any particle seems to be very
nearly a semi-ellipse, as shown at AGO in Fig. 29. It there-
fore appears that, due to the passage of the wave, a particle
which was formerly at A has been transported and left at O;
furthermore, that all particles in the transverse plane of A will
undergo the same longitudinal translation, and will be left in
a transverse plane containing O. Care must be here taken
to distinguish between the motion of the particle and the
RESISTANCE.
7 1
motion of propagation. While the particle moves through
its path as A CO, the wave-form will move forward its wave-
length OF or LM, Fig. 28.
FIG. 29.
The approximate contour of a wave of translation may be
determined by the following construction:
Let AGO be the ellipse representing the path of a surface
particle, and let AH be the depth of the canal. Then lay off
AF = 27r/i and construct the sinusoid AEF. Next lay off to
the right from points on this curve, as A, c, g, etc., distances
AO, ab, ef, etc., or the intercepts between the ellipse and
the vertical OP. The points O, d, /, etc., thus determined
will give the desired contour of the ideal wave of translation
corresponding to the conditions assumed.
Let v velocity of propagation ;
h = depth of water when at rest ;
k = height of wave above still-water level ;
L = wave-length = OF, Fig. 29;
a = range of translation = AO. Fig. 295
V volume of water constituting wave;
72 RESISTANCE AND PROPULSION OF SHIPS.
b = breadth of canal;
A area of profile of wave.
Then v = Vjffi+t) (18)
When k is relatively small we have
v = V'gh (19)
It is found that k cannot exceed //. On reaching this
value, or before, the wave breaks at the crest. Hence as a
limiting value of v we have
v = V2gh (20)
For the wave-length
L = 27th a, (21)
or where OL is relatively small
L 2Tth (22)
For the value of A we have
A -ah (23)
As with oscillatory waves, the energy is one-half kinetic
and one-half potential, the entire amount being
Energy = 2<?Vz; (24)
where <r = density;
V = volume;
z = elevation of center of gravity of wave above the
still-water level.
RESISTANCE.
73
TABLE I.
SHOWING RELATION BETWEEN Z, T, AND V FOR
TROCHOIDAL WAVES.
L
T
V in Feet
per Sec.
Kin Knots.
L
T
Fin Feet
per Sec.
V in Knots.
IO
1.40
7.2
4.2
350
8.30
42.4
25-1
20
2.OO
10. 1
6.0
400
8.80
45-3
26.8
30
2-42
12.4
7-3
450
9.40
48.0
28.4
4
2.80
14-3
8.5
500
9.90
50.6
30.0
50
3.10
16.0
9-5
550
10.40
53-1
31-4
60
3-40
17.6
10.4
600
10.80
55-5
3 2.8
70
3-70
19.0
II. 2 650
11.20
57-8
34-2
80
4.OO
20.2
12.0 700
11.70
59-9
35-5
90
4.20
21.5
12.7
750
12. IO
62.0
36.7
IOO
4.40
22.7
13-4
800
I2.5O
64.1
37-9
150
5.40
27.7
I6. 4
850
12.90
66.0
39-i
2OO
6.20
32.O
19.0 ij 9OO
13.20
68.0
40.2
250
7.00
35-8
21.2
950
13.60
69.8
41-3
300
7.60
39-2
23.2
1000
14.00
71.6
42.4
TABLE II.
SHOWING VALUES OF FOR VARYING VALUES OF -y.
7*o *
z
r
z
r
z
r
z
T
L
r o
r
r
L
fo
T
r o
.01
9391
.07
.6442
25
.2079
.60
.0231
.02
.8820
.08
.6049
30
.1518
.70
.0123
03
.8283
.09
.5681
35
.1109
.80
.0066
.04
.7778
.10
5336
.40
.O8l0
.90
.0035
.05
.7304
.15
.3897
45
.0592
I.OO
.0019
.06
.6859
.20
.2846
50
.0432
74
RESISTANCE AND PROPULSION OF SHIPS.
TABLE III.
SHOWING THE VARIATION OF PRESSURE VERTICALLY DOWN-
WARD IN THE CREST AND HOLLOW OF A WAVE.
2
~L
Crest.
Hollow.
Actual Depth
Value of Ratio
LIO] (6).
Actual Depth
Value of Ratio
[10] (7).
L
y^
05
0635
730
.0365
1.269
.10
.1233
.766
.0767
I.23I
.15
.1805
794
.0836
I.2OO
.20
.2358
.818
.1642
I.I74
.25
.2896
.837
.2105
I.I52
.30
.3424
.854
2576
I.I35
35
3945
.868
.3055
1. 1 2O
.40
.4460
.879
3540
I.I08
45
4975
.889
.4025
I.OgS
50
.5478
.898
.4521
1.088
.60
.6488
9!3
5512
1-075
.70
7493
.924
.6507
1.064
.80
.8497
932
.7502
I.O56
.90
.9498
939
.8502
1.049
1. 00
1.0499
.946
.9501
1.045
1.50
1.6000
963
1.4500
1.029
2.OO
2.0500
972
1.9500
1.022
TABLE IV.
SHOWING POWER OF OCEAN WAVES IN HORSE-POWER UNITS.
Length of Wave in Feet.
h
25
50
75
100
150
200
300
400
50
.04
.23
.64
1.31
3.62
7-43
20.46
42
40
.06
.36
1. 00
2.05
5.65
11-59
3L95
66
30
.12
.64
1.77
3.64
10.02
20.57
56.70
116
20
.25
1.44
3.96
8.13
21.79
45.98
126.70
260
15
.42
2.8 3
6.97
14-13
39-43
80.94
223.06
457
10
.98
5-53
15.24
31.29
86.22
177.00
487-75
1001
RESISTANCE.
75
TABLE V.
COMPARISON BETWEEN WAVES IN SHALLOW AND DEEP WATER.
Depth of Water as
a Fraction ot the
Wave-length.
Ratio between Quantities for Shallow Water and Corresponding
Quantities lor Deep Water.
Ratio of Axes of
Surface Dibits.
Length for Given
Velocity.
Velocity for Given
Length.
.01
.063
15.90
.251
.02
.124
8.08
352
03
.186
5.376
431
.04
.246
4-065
.496
05
.304
3.289
552
075
439
2.277
663
.10
557
.796
.746
15
.736
358
.858
.20
.847
.ISO
.920
.25
.917
.091
.958
30
955
.047
977
35
975
.026
.987
.40
.987
013
993
45
993
.007
996
.50
.996
.004
.998
.60
999
.001
999
75
9999
.0001
9999
1. 00
99999
.OOOOI
999999
11. WAVE-FORMATION DUE TO THE MOTION OF A SHIP-
FORMED BODY THROUGH THE WATER.
In 2 it is shown that if the surface of the water were
covered by a rigid plane through which the ship could pass
without resistance, and which closed after it, no waves could
be formed, but a disturbance in the distribution of pressure
would result, such that about the bow and stern there would
be an excess and about the middle portions a defect. If the
hypothetical rigid plane is removed, we shall have instead of
such a modification of pressure, and as the natural manifes-
tation of the tendency to produce it, an elevation of the
surface about the bow and stern and a depression about the
7 6
RESISTANCE AND PROPULSION OF SHIPS.
middle portions. The result of these initial or fundamental
tendencies is, however, much modified by propagation and
the interference or combination of their elementary results.
In any event, however, the distribution of normal pressure
over the surface of the ship will be much changed from, that
corresponding to hydrostatic conditions, such change result-
ing in a force directed from forward aft. This resultant will
RESISTANCE.
77
be of such amount that the increased expenditure of energy
necessary to overcome it will just equal the energy necessary
to the maintenance of the system of waves and stream-lines.
Having thus referred to the modified stream-line system
and its attendant phenomena as a sufficient initial cause for
the formation of waves, we may examine their actual charac-
ter and distribution in more detail.
In Figs. 30, 31, and 32 are shown wave patterns as
determined by Mr. R. E. Froude for the various cases men-
7 8
RESISTANCE AND PROPULSION OF SHIPS.
tioned. We have here indicated two quite distinct varieties
of wave form and distribution:
(a) Waves with crests perpendicular to the line of motion,
constituting the transverse series;
(b) Waves with crests oblique to the line of motion, con-
stituting the oblique or divergent series.
FIQ. 4
PLANS OF WAV-E SYSTEMS MADE BY DIFFERENT VESSELS
AT 18 KNOTS SPEED.
Position of Wave Crests indicated by shading.
83 FT. LAUNCH
FIG. 32.
The series of transverse waves are distributed along the
line of motion of the ship, and may be considered as a group
RESISTANCE. 79
of trochoidal waves as described in 10. The speed of the
individual waves is the same as that ^of the ship, and hence
their length from crest to crest corresponds closely to that
of a deep-water trochoidal wave traveling at the same speed
as the ship. A crest is always found at or near the bow, and
it is this crest which may be considered as the initial result of
the wave-forming tendency in this region. To avoid confu-
sion this crest is omitted from the diagram, and only the more
pronounced local elevations constituting the divergent crests
are shown. As we shall explain later, however, the energy
belonging to this wave is drained away sternward relative to
the ship, and gives rise to a series of secondary waves, or
echoes as they are termed, the primary and its echoes thus
forming the series as a whole.
In a long parallel-sided ship the crests and hollows show
for some distance from the bow, gradually decreasing in
altitude as they spread transversely, and thus involve more
and more water with a gradually decreasing energy. If the
parallel body is so long that the waves have by this process
of dissipation become inappreciable, then it will be found
that a similar principal crest will be generated near the stern,
the energy of which, by the same process of sternward propa-
gation as at the bow, will give rise to a stern series of
echoes. The diminution of pressure amidships will be so
slight and will be distributed over so great a distance that
the primary hollow will usually be scarcely noticeable. If
instead of the long parallel body we have the usual form, we
shall have the primary crest at the bow and stern, and a
more pronounced tendency to form a primary hollow amid-
ships. The energy of the waves thus formed will be propa-
gated sternward as before, thus giving rise to echoes and to
8o RESISTANCE AND PROPULSION OF SHIPS.
three more or less plainly marked initial systems. These,
however, will coalesce, and the whole will form a complete
system with crests and hollows of size and distribution
depending on the character of the various components. As
the variations in velocity due to the stream-line motion are
only slight, the altitude of such waves will be small, and at
ordinary speeds they are scarcely noticeable. As the increase
in velocity amidships is, moreover, quite gradual, the corre-
sponding initial hollow will usually be slight, and a correspond-
ingly unimportant element in the combined series. In many
cases, however, at high speeds and when the elements of
combination enter properly, the combined series may show a
very pronounced hollow amidships.
Turning to the divergent series, we have the configuration
shown in the diagrams involving in each case an initial
divergent crest formed on either side of the bow. These
may be considered as the primary result of certain tendencies
to which we shall directly revert. The same drain of energy
and propagation sternward obtains here as with the trans-
verse series, and, as we shall explain, the result is the series
of echoes as shown, arranged in skew or imbricated order;
i.e., with their crest-lines overlapping like the shingles of a
house.
We have now to explain the formation of the primary
divergent wave at the bow. We have already explained the
initial formation of the transverse crest at the bow. This
crest in itself is of small altitude, and extends over a consid-
erable area according to the speed and to the rate of varia
tion of the pressure in the stream-lines. Referring to Fig. 33,
we have in plan a suggestion of the contour or level lines for
the variation of level which would result from the initial
RESISTANCE.
81
impulse corresponding to the conditions from forward aft.
The line LBHDGFK is supposed to lie on the surface of the
smooth water. AB and EF are humps or crests, and CD is a
depression or hollow. Now the tendencies which lead to the
general elevation of the water throughout AB become very
much accentuated near A, the forward end of the surface of
discontinuity between the ship and the liquid. As a result
there will be raised against the surface of the bow at A a.
FIG. 33.
more or less pronounced wall of water. This may be consid-
ered as a locally exaggerated manifestation of the general
tendency which would give rise to the hump as a whole.
Otherwise we readily see in this wall the simple result of
driving the oblique face A rapidly through the water, or, vice
versa, the natural result of placing an oblique face A in a
rapidly flowing stream.
While therefore this special elevation of water is due
fundamentally to the same general cause as the remainder of
the hump, and while it must be considered as a part of this
general elevation AB, yet it is elevated so definitely above
82 RESISTANCE AND PROPULSION OF SHIPS.
the surface of the remainder of the hump, that immediately
we find this local elevation undergoing propagation somewhat
like a positive wave of translation. More exactly or more
generally, we may view this propagation as a result of the
unequal distribution of pressure in. the water in this neighbor-
hood. The pressure is greatest quite near the ship, and
decreases rapidly as we go slightly outward. The water
thus affected will naturally yield in all directions in which
the pressure is less. Hence it will yield upward and out-
ward, and thus give rise to an initial elevation of water and
to a propagation outward in the direction of the most rapid
decrease of pressure. This at first will be nearly normal to
the water-line or surface of the ship.
Let us now attach ourselves to the ship and consider the
stream as flowing by. Relative to the ship the initial propa-
gating impulse will be in the direction OP, sensibly perpen-
dicular to OQ, Fig. 34. Let v be the velocity. Then if at
FIG. 34.
a given instant the stream could be suddenly arrested, the
water at that instant forming the elevation AB would be
propagated along OP with velocity v. In the actual case,
however, the propagation takes place not into still water, but
into water moving sternward with its natural stream-line
RESISTANCE. 83
velocity, approximately parallel to AQ. The elevation of
water thus propagated will have two velocities one, v, along
OP, and another, w, along OQ. Hence the actual propagation
will be the resultant of these two, and its direction OR will
make an angle ft with OQ such that tan ft = v -f- w. Also
the direction OR will make an angle y with the direction of
motion AL such that
v
tan a -I
IV
tan y = tan (a + ft) = -
v
I tan a
w
The velocity v may be considered as depending on the
rate of decrease of pressure from AQ outward, or on the
pressure very near the surface of the ship in this neighbor-
hood. We might consider a portion of .the velocity of
propagation as due to the elevation AB considered as a
positive wave of translation. Inasmuch, however, as the
distribution of pressure seems to be the important factor in
this propagation, it seems preferable to refer the whole
phenomenon, generation and propagation, to this general
cause. It is therefore evident that v will increase and
decrease with the velocity of the ship according to a law too
complex for general expression.
The velocity w is the velocity along the stream-lines. It
will be at this point slightly less than u t the velocity of the
ship, or the velocity of the flow as a whole. For fine ships
or small values of a its value cannot differ widely from
n cos 01. For large values of a this would give too small a
value.
Turning to the value of tan y above, it is seen that the
only quantity depending on speed is the ratio v -=- w. Both
84 RESISTANCE AND PROPULSION OF SHIPS.
of these, as we have seen, depend on the speed of the ship,
and it seems not unnatural to suppose that -v like w may very
nearly vary directly with u, the speed of the ship. In such
case v -i- w, and with it y, will remain very nearly the same
at varying speeds. This has been found experimentally to
be the case, more especially for fine ships at moderate and
high speeds. For very full ships the value of y seems to
increase slightly with the speed, indicating that for such
forms v increases more rapidly than w. Before leaving this
value of tan y it may be well to show the method of finding
the mean value of tan a for the bow. In Fig. 35 let
the origin be at B and x reckoned plus aft. Then if a
denotes the inclination of the curve AB we have
dy
tan a = -7-,
dx
i /.*i i /*, . r.
and mean of tan a = / tan adx = / ay = .
X^ <J Q X^ e/o X l
Integrating similarly for the mean vertically, we have
y i /*
mean of = - - / y^dz = area of half-section AC ' -~- z.x..
x, z,x,J
Hence
area of half-section AC
mean of tan a for bow = .
RESISTANCE. 85
In usual forms BC may be taken at 10 to 15 per cent of
the length.
At very high velocities of the ship the water in the eleva-
tion AB, Fig. 34, instead of forming a simple sharply-marked
mound will curl and break. The general result will be the
same, however, as the breaking is simply equivalent to pour-
ing a mass of water along AB, and the distribution and rate
of variation of the pressure will be such as to give fhe same
general outward propagation as above described.
Having thus discussed the propagation of a single instan-
taneous elevation of water AB and the considerations on which
its direction depends, we next observe that as the propaga-
tion removes the water forming the elevation at any one
instant, it is immediately succeeded by another like elevation,
so that as a result there is a continual elevation propagated
relative to the boat along the direction OR. As the eleva-
tion recedes from^<2, however, the crest tends to broaden and
thus loses altitude and finally becomes insensible. Toward
the outer end its direction may change somewhat, due to the
superposition of the tendencies at that point on the initial
impulse along OP. As the side of the ship is left, the most
rapid decrease of pressure will be directed more nearly trans-
versely, or even aft, toward the hollow amidships. In con-
sequence of this, together with the changed value of w, the
outer end may curve around somewhat toward the stern.
Due, however, to the decreasing altitude, this change is
usually unnoticeable.
We have thus described the development of the primary
member of the divergent system of waves at the bow. A
similar primary is formed at the stern through the action of
the same general causes. In this case the initial wall or
86
RESISTANCE AND PROPULSION OF SHIPS.
special elevation of water is formed under the stern and is
propagated outward into the outlying water, which, due to
the frictional wake and to the stream-line motion in this
neighborhood, has a somewhat lower velocity than at the
bow. While thus the initial direction of propagation of the
elevation would usually be somewhat aft of the direction for
the similar elevation at the stern, yet the other component
being less than at the bow the resulting direction is not far
different at the two ends. At high speeds instead of the
special elevation being formed against the side of the ship it
is formed immediately astern. In this case the two streams
flowing one on either side of the ship combined with that
flowing under the ship may be considered as meeting and
giving rise to an initial elevation along the line AB, Fig. 36.
FIG 36.
This being propagated transversely outward on each side into
the outlying water gives rise as before to the two divergent
waves ABP and ABQ. At certain speeds and with certain
forms of boat this wave is very pronounced, taking the form
rf a flat-topped hill of water sloping away in a fan-shaped
figure and gradually decreasing in altitude. At still higher
speeds this wave recedes still farther astern and usually
RESISTANCE. 87
becomes somewhat less pronounced. Especially- is this the
case with that form of after-body which approximates to a
wedge with flat side down and edge near the water surface at
the stern, as found on most modern torpedo-boats and many
other high-speed craft.
Having thus discussed the formation of the primary
members of the various series of waves involved in th&
problem of resistance, it is time to take up the question of
the propagation of energy already referred to.
12. THE PROPAGATION OF A TRAIN OF WAVES.
This subject has been briefly considered in 10. We
must now examine in more detail the consequences of the
statements there made. In Fig. 37 let the full line represent
Ci
x_
FIG. 37.
a group of waves of which C marks the center and principal
member, the successive crests on either side gradually
decreasing in altitude until they become negligible. Then
observation in agreement with theory shows that such a con-
figuration or group is propagated with only about one half
the velocity of the individual waves A, B, C, etc. Now the
group as a whole may be considered as the expression of a
certain distribution of energy, and hence the velocity of the
group may be considered as the velocity with which the
energy is propagated, as distinguished from the velocity of
the individual waves. It must be considered, therefore, that
in waves of trochoidal character the energy is propagated at
only one half the velocity of the waves themselves. Viewing
88 RESISTANCE AND PROPULSION OF SHIPS.
the matter from this standpoint, let us examine the propaga-
tion of the group of waves in Fig. 37. The wave C repre-
senting the principal member is propagated forward with a
certain velocity. As it moves forward, however, it leaves
behind a part of its energy, and hence gradually decreases in
altitude, and is no longer the maximum member of the
group. The energy thus left behind is absorbed by B, which
therefore increases and becomes after a certain time the
maximum member B and hence the new center of the
group. The wave C in the meantime has gone from C to C lt
while the principal member of the group, as such, has gone
from C to B^ The latter evidently fixes the velocity of the
group. There is thus a continual propagation of the indi-
vidual waves forward, while relative to the waves the indi-
vidual characteristics are propagated backward. The group
velocity is therefore the difference between these two. It
results that any individual wave as A travels along through
the entire group, taking on successively the characteristics of
the various members, and finally dying out at the forward
edge while a new member rises at the after edge; and thus the
process continues. Or, again, we may consider that the lead-
ing waves are continually dwindling and disappearing for lack
of the energy which they leave behind, while the following
waves similarly grow by the access of energy which they as
continually receive. Thus if v is the velocity of the waves
forward, then the velocity of the group and of the energy is
v/2 forward. Relative to the group the velocity of the
waves is v/2 forward, and of the energy o. Relative to the
waves the velocity of the group and of the energy is v/2
backward.
In thus considering the group velocity of a train of waves
RESISTANCE. gg
we must note that the exact proof supposes trochoidal
motion, and hence circular paths for the motion of the parti-
cles. This cannot be the case where the waves vary in
altitude, and under such circumstances the group cannot be
indefinitely permanent, even neglecting viscous degradation.
This is borne out by observation. Again, in shallow water
the orbits being oval or approximately elliptical, the group-
velocity is greater than one half the wave-velocity, and hence
in such case the energy is transmitted at a velocity greater
than one half that of the individual waves. In the extreme
case we have the wave of translation in which the group
becomes reduced to the individual, the velocity of one being
necessarily that of the other. In such case, then, the energy
is transmitted at a velocity equal to that of the individual
wave.
13. FORMATION OF THE ECHOES IN THE TRANSVERSE AND
DIVERGENT SYSTEMS OF WAVES.
Taking first the primary bow transverse wave, we see that
as an individual it must necessarily keep pace with the ship.
Considering, however, that it is constituted approximately as
a trochoidal wave, its energy will be continually draining to
the rear and giving rise to a successive series of echoes.
These from their gradual spreading sideways soon lose all
significant altitude and become negligible.
Turning now to the divergent system, we find in the
draining backward of the energy relative to the ship com-
bined with the natural internal propagation peculiar to this
wave, the explanation of the overlapping or skew arrange-
ment of the crests. We may consider that these divergent
go RESISTANCE AND PROPULSION OF SHIPS.
waves constitute virtually a special series superimposed, as it
were, on the flatter transverse series. In Fig. 38 let AB be
the primary member. Now this crest, considered simply as
a configuration, is by virtue of its fixed relation to the ship
carried forward unchanged in the direction. OP with a
velocity v equal to that of the ship. A part of the energy,
however, will naturally lag behind, the relative amount
approximating more or less closely to one half, according as
the characteristics of the wave approach more or less closely
FIG. 38.
to those of trochoidal form. Relative to still water, therefore,
there will be a propagation of the energy forward at a
velocity less than v, and approaching v/2 as the wave
approaches trochoidal-form. In addition to this propagation
of energy alon^ OP, there will be likewise, due to the
peculiar constitution of this elevation of water, a propagation
or transmission of energy outward along OQ. The resultant
direction of propagation is therefore along some line OR.
The result is that the energy in AB may be considered as
located at a later instant in A^B^ while the primary crest at
RESISTANCE. Q!
A'B' exists by virtue of energy drawn from the ship between
A and A'. Or otherwise, relative to the ship and the con-
stantly renewed primary crest AB, energy is constantly drain-
ing to the rear and outward along OQ, the result being a
transmission obliquely outward in such manner as to give rise
to the series of overlapping crest-lines as shown.
In a similar way a second echo is formed, and so on until
by lateral extension and viscous degradation the altitude
becomes negligible. In like manner the stern series of
divergent waves may give rise to a similar series of echoes.
We have already referred to Figs. 31 and 32 in illustra-
tion of the wave-pattern whose formation we have attempted
to explain. We may now call attention to certain further
details.
We first note the location of the primary bow and stern
waves. At low speeds the bow divergent waves are formed
close to the bow, while as the speed increases their mean loca-
tion draws somewhat farther aft. At low speeds the stern
divergent \vaves are formed on the quarter and diverge
independently as shown in Fig. 31. As the speed increases
they draw aft and together, and finally coalesce into an elon-
gated mound of water spreading away in V shape, as already
noted.
The bow and stern transverse primary waves have the
same general location longitudinally as the divergent pri-
maries. Indeed, as we have already pointed out, the latter
are merely local exaggerations of the general primary eleva-
tion at these points.
The length between the primary bow and stern waves
increases somewhat with the speed. At moderate or high
speeds it is usually considered as slightly greater than the
Q2 RESISTANCE AND PROPULSION OF SHIPS.
length of the ship. The available data is not sufficient to
determine satisfactorily the amount of excess, but it is usually
taken at from 5 to 10 per cent of the length.
Numerous measurements of the wave-lengths for the
transverse system are found to be in close accord with the
length for the natural trochoidal wave of the same speecl as
the ship, and as given by formula, 10 (i). It may therefore
be considered as shown by experience that the wave-length
of the transverse series is in satisfactory agreement with what
it should be, considering them as a series of trochoidal waves
having the speed of the ship. In consequence it seems
allowable to assume for the waves of this system such further
properties of trochoidal waves as may be convenient for their
investigation.
With the divergent system the case is somewhat different.
These waves, so far as their configuration is concerned, travel
forward with the velocity of the ship. If y is the angle of
their crest with the longitudinal, however, the component of
their velocity normal to the crest-line will be u sin y. Now
measurement indicates that the wave-length in this direction
agrees fairly well with that of a natural trochoidal wave
having a velocity u sin y. It follows that in a sense we may
consider this system as made up of a series of bits of tro-
choidal waves with a velocity u sin y and a length corres-
ponding. For this length we shall have
L = if sin* y.
Hence for the length from crest to crest on a longitudinal
line we should have
LI = U* sin y.
RESISTANCE. 93
For the transverse system we have likewise
Hence the longitudinal length of the transverse series should
be somewhat greater than for the divergent series. This is
borne out by the diagrams, reference to which shows that the
crests of the former fall farther and farther astern relative to
those of the latter. While the difference may not be so
great as is indicated by the formulae above, the tendency is
in this direction, and the errors of the formulae may naturally
be referred to the imperfect manner in which such waves
fulfil the trochoidal characteristics.
14. WAVE-MAKING RESISTANCE.
Having thus considered the formation of the various wave
systems attending the motion of a ship through the water,
we may next examine their relation to resistance.
Taking first the bow transverse system, we have seen that
the energy of the waves is, relative to the ship, constantly
passing sternward. At the same time the energy of the
system as a whole must be maintained constant. Hence this
drain of energy sternward must be made up by energy
derived from the ship, and it is simply the transmission of
this energy which gives rise to this part of the wave resist-
ance. The exact fraction of energy which falls astern rela-
tive to the ship will depend on the geometrical character of
the wave; and it must not be considered that the ratio 1/2
is anything more than an approximation, which, however,
may serve for illustrative purposes. Hence let us assume
UNIVERSITY
94 RESISTANCE AND PROPULSION OF SHIPS.
that one half the energy is naturally propagated with the
form, and that one half must be made up from the ship. It
follows that for every two wave-lengths run the ship must
supply the energy necessary to the formation of one wave.
Let E be the energy of the wave, A the wave-length, and
R the resistance due to its maintenance. Then since resist-
ance equals the work done or energy transmitted divided by
the distance, we have
A similar expression would hold for the element of resist-
ance due to each of the divergent waves at the bow, E being
the energy and A, the longitudinal wave-length for this
system.
It thus appears that the work which is done by the ship
at the bow on the water in maintaining these systems of
waves is equivalent to adding one new wave to each system
for every two wave-lengths traveled. Similar considerations
hold for the resistance due to the stern wave systems.
We have thus far considered the bow and stern wave
systems in their individual aspect. We have now to consider
the possible influence of the former upon the latter.
We first note that, so far as the divergent system is con-
cerned, no direct influence is possible, since these waves
travel off obliquely and come in contact with the ship at the
bow only. Again, if the ship is very long for her speed,
especially if she has a long middle body, the bow transverse
system will become negligible before reaching the stern, so
that each system will produce its individual effect. In the
general case, however, the bow transverse system will not
RESISTANCE. 95
have entirely disappeared by the time it reaches the stern,
and there will be formed at this point a combination system
resulting from the remainder of the bow system and the
natural stern system.
The total energy of the entire wave systems may be con-
sidered as that of the bow transverse and divergent systems
plus that of the stern divergent system, possibly modified by
the bow transverse system, plus that of the stern combina-
tion system minus that part of the latter which is received
from the bow system. For the transverse systems alone the
above value is equal to the sum of the energy of the stern
combination system plus the loss of energy in the bow system
when it reaches the stern.
Let s denote the wave-making length of the ship or the
distance between the natural bow and stern primary crests as
discussed in 13, and let A be the wave-length, which we may
take as that of a trochoidal wave of the velocity of the ship.
Let s = A + a. Then a is the distance from the stern pri-
mary crest forward to the nearest crest of the bow system.
Hence a -f- A is the phase difference ratio and 2na -f- A is the
phase difference angle. Let //, be the altitude of the bow
wave, //^ the remaining altitude of the bow system when it
reaches the stern, and /*, the natural altitude of the stern
wave. Then referring to 10 (15) it is seen that the combi-
nation system will be trochoidal and of altitude
h = V k*h? + //,' + 2 ////, cos ^.
A
Referring now to 10 (i i), it is seen that the energy of a
wave per unit breadth measured along the crest is propor-
tional to the product of the wave-length by the square of the
altitude. Let B^ be the breadth transversely at the bow.
This is of course indefinite, since /;, gradually decreases as we
96 RESISTANCE AND PROPULSION OF SHIPS.
go from the bow outward. We may, however, consider h*
as the mean of the squares of the altitudes for a breadth
B l considered as comprising all of the wave whose elevation
is sensible. The energy of the bow primary wave will then
be proportional to h?Bj^. Let B and h denote similar quan-
tities for the combination system at the stern. Then the
energy of the stern combination wave will be proportional to
h*B^. A portion of the latter, however, proportional to
k*h?Bj^, is derived from the bow system. Hence the energy
necessary for the maintenance of the whole system, and to
be supplied by the ship in running a distance 2A, is propor-
tional to h*B. + tfB'h. tfh?B^.. Now assuming that B,
and B are sensibly the same, substituting for the value of k
above and putting the result proportional to the work done
by the ship in overcoming a resistance R w through a distance
2A, we have
, cos
Whence R w ~ ^ + h; + 2khfa cos ]B. . . (i)
Let s = mL, where L = length of ship, in which, as
already noted, m is usually 1.05 to i.io. Then s mL =
n\-\- a and
2ns -r- h = 2nmL A. = 2nn -|- 27ta -j- A.
27ts 2itmL
Therefore cos - = cos - -- = cos
A A A
But A = 2nv* -T-^-, 10 (i), and hence
2 n mL gmL
cos r - = cos ^ ^
A v
Hence R w - ^ + ^ + 2khji, cos < - B. . . (2)
RESISTANCE.
Now in general we may put
97
and
and hence
. . h ~ V*,
v.
Hence we may represent the resistance due to the natural
bow system by a term of the form H*v', and that due to the
natural stern system by a similar term, /V. Then from the
derivation of (2) it follows that we should likewise represent
the total resistance of the combined series by an expression
of the form
R. =
2HJk cos
(3)
This expression is periodic in character, the mean value
being (H*+J*)Bv\ To illustrate the variation from this
10 12 11 16 18 20 23 24
Values of iHTk cos laid off from OX as axis.
J v*
FIG. 39.
value due to the third term we may refer to Fig. 39. We
have here illustrated the values of the expression
2HJk
using the following numerical values of the terms
2HJk= I,
mL 100.
98 RESISTANCE AND PROPULSION OF SHIPS.
At low and moderate speeds the amount of wave-resist-
ance is so small that these fluctuations are entirely imper-
ceptible. As the speed is increased, however, a point is
reached where the wave-resistance forms an important part
of the total resistance, and where evidences of such a
periodic fluctuation in its amount are plainly indicated. This
is illustrated in Fig. 40,* showing the residual resistance-
80
70
60 co
O
50 *-
z
u
40 o
30 2
s
DC
20
10
L
10 11
13
13
14 15 16 17 18 19 30
SPEED IN KNOTS
CURVES OF RESIDUAL RESISTANCE
22 23
Ship
Length
Beam
Draft t
Disp't
A
400'
38.2
20.7
503v
B
"
19.1
5390
C
16.5
4480
D
15.4
FIG. 40.
curves of a ship at several drafts as determined by Mr. R. E.
Froude.
For the waves of the divergent series we may take like-
wise the energy as proportional to the length, breadth, and
* Transactions Institute of Naval Architects, vol. xxn. p. 220.
f Inclusive of 9 inches of keel.
RESISTANCE. 99
square of the altitude. Hence h, B, and A, denoting in
general the same characteristics as above, we should have for
each train
Energy ~ h*B\,
R ~ tiB.
As further illustrating the above principles, we may refer
to the following experiments of Mr. Froude on the influence
of length of parallel middle body on wave-making resistance.
For the purposes of the investigation, an initial model was
taken as representing a ship 160 feet long. Successive,
lengths of parallel middle body representing 20 feet in length
were then added until the total length represented 500 feet.
These models were tried at various speeds and the resistance
measured. The frictional resistance being then computed
and subtracted, the residual resistance was considered as
sensibly due to the maintenance of the wave systems. In
order to show the results graphically, the information was
disposed as in Fig. 41.*
Any given curve above AA represents the residual resist-
ance at a constant speed, as marked, for varying values of L
as shown on the base A A. Similarly the corresponding line
below AA shows the frictional resistance at the same speed
for the same series of ships, the ordinates in this case being
measured downwards.
We may note the following points indicated by this
diagram:
(i) The curves of residual resistance show plainly a
periodic variation, the length of each period or distance from
* Transactions Institute of Naval Architects, vol. xvni. p. 77.
100
RESISTANCE AND PROPULSION OF SHIPS.
maximum to maximum being approximately constant at any
one speed.
(2) The length of the period or spacing between maxima
RESISTANCE IN TONS
I I I I I I t I I .1 I L& a. I i. li I .1 LU I i.! I ,1 I I 1 I 1 1 \
VI 1 I F
3 U U P
RESISTANCE IN TONS
RESISTANCE IN TONS
FIG. 41.
is greater as the speed increases, the value being found to
vary nearly as the square of the speed.
(3) The amplitude of the variation increases as the speed
increases.
RESISTANCE. TO I
(4) The amplitude of the variation decreases as the length
increases.
The increase of (3) is due to the rapidly increasing im-
portance of wave-making resistance in general, after passing
a certain speed, and also more directly to the value of ,
which usually increases with the speed. The decrease of (4)
is due to the decrease in k as L increases. If L were suffi-
ciently great there would be no interference, and hence no
fluctuation or periodic variation in the resistance.
Turning back now to formula (2) we may further examine
the relation of //,, li v and B to the speed.
In perfect stream-line motion in horizontal planes and
without elevation of the surface we have, denoting the
velocity by u,
P -p, = ?L(u;-u*\ See 2 (i)
O
Hence along any one line for a slight change in velocity
0-
dp = udu.
If now we put du = eu, where e represents a certain frac-
tion, we have
Hence with varying initial velocity, the pressure incre-
ments or decrements at any given point will vary as the
square of the velocity. In actual wave-line motion at
moderate speeds the elevation will correspond nearly to the
increment of pressure, and hence we should find the values of
h, and //, nearly proportional to *. At higher speeds the
stream-line motion is less perfect, and it seems probable that
102 RESISTANCE AND PROPULSION OF SHIPS.
the variation of pressure for a given percentage of speed
fluctuation will increase somewhat more slowly than as the
square of the speed. Assuming, however, in general this
relationship, we may as before represent the resistance due to
the natural bow and stern waves by H'*u and J^u. We
shall then have again as in (3)
= U'(
H' + r + 2kHJ
Next as to B. It has been usually assumed that B is
very nearly constant at varying speeds, and that it may be
taken as approximately proportional to the linear dimensions
of the ship. In such case we should have
R w ~ lu\
Some have thought that it should be considered rather as
varying with h, and hence with u*. In such case we should
have
R, ~ *,
and hence independent directly of the dimensions of the
ship. Risbec * suggests a combination of these two terms in
the form
R w ~ Alu* + Bu\
and believes that as u increases B will become more and
more predominant in the value of R w . It may be noted that
Risbec's suggestion is equivalent to
in which J may depend on the dimensions of the ship and n
may vary from 4 to 6.
* Bulletin de 1'Association Technique Maritime, vol. v. p. 52.
RESISTANCE. 103
Similar considerations hold with regard to the resistance
due to the divergent system, each member of which may be
considered of the form
R = Cu* or C>",
or as a summation of such terms, according as the law is
assumed to be constituted. Hence we may consider that in
reference to speed the total wave-making resistance will
probably vary with some index between 4 and 6, or as a sum
of terms with these or intermediate exponents. The prob-
ably decreasing values of P and Q with increasing speed
would, if they were taken as constant, decrease the exponents
of u from the values they would otherwise have, and thus
perhaps partly compensate for the increase in exponent due
to the variation of the term B.
The actual relation, however, of the total wave-making
resistance to the speed and to the dimensions of the ship is
not known, nor can it be satisfactorily determined from the
data available. There seems to be ground, however, for the
belief that in many cases it shows a higher rate of variation
than that given by the exponent 4. See also 26. For the
present, therefore, we are thrown back on more empirical
formulae, intended simply to express as nearly as may be the
results of experience. Such formulae will be found in 29.
With the accumulation of data from model experiments we
may perhaps hope ultimately for a satisfactory determination
of the constants in (i) and (2) or in other like formulae shown
by the data to be more suitable to the purpose in view.
104
RESISTANCE AND PROPULSION OF SHIPS.
15. RELATION OF RESISTANCE IN GENERAL TO THE
DENSITY OF THE LIQUID.
For the head-resistance and skin-resistance per unit area
we have seen that the density of the liquid enters as a factor.
Similarly here the density must be a factor of the energy of
a wave, and thus all parts of the resistance, and hence resist-
ance as a whole, will vary with the density. The variation
in the resistance of a given ship with change of density of the
liquid is, however, in large measure offset by the oppositely
varying amount of immersed body of ship. As between
fresh and salt water the difference is slight, and is usually
neglected except in the case of model experiments or very
careful estimates. The densities of fresh and salt water are
usually taken in the ratio of i.o: 1.032, or less accurately in
the ratio 35 : 36.
16. INFLUENCE OF FORM AND PROPORTION ON RESISTANCE.
In 1876 Wm. Froude published a paper* showing the
results of experiments on the comparative resistance of
models of four ships of the same displacement but of certain
differences of form. These models were called A, B, C, D;
A being that of the Merkara. The dimensions correspond-
ing to the four models were as follows:
Displace-
ment.
Length of
Fore
Body.
Length of
Middle
Body.
Length of
After
Body.
Total
Length.
Beam.
Draft.
Wetted
Surface.
A
i r
144
72
144
3 6o
37-2
16.25
18660
B
l i
179-5
179-5
359
45-88
18
19130
C
i i
154-5
154-5
309
49.4
19.32
17810
D
j 1
95
95
95
285
45.56
17.89
16950
* Transactions Institute of Naval Architects, vol. xvn. p. 181.
RESISTANCE. 10$
The forms of the fore and after bodies of B, C y and D
were derived from those of A by expansion longitudinally,
transversely, and vertically as required. The fore and after
bodies themselves have therefore the same ratios of fullness
and other characteristics as those of A, and the resulting
differences in resistance may therefore be considered as due
to the effect of the presence or absence of the parallel middle
body.
The results may be summarized by saying that at all
speeds B and C had less resistance than A or D. As
between B and C, the latter having the less wetted surface
had the less resistance. As between A and D, the latter,
though having less wetted surface, had at the speeds tried
(from 9 knots upward) greater resistance, though at the
lowest speeds the directions of the two resistance-curves were
such as to indicate an intersection, and lower values for A for
very low speeds. At high speeds B and C changed places
relatively, the smaller wave-resistance due to B's greater
ratio of L to B being more influential at such speeds than its
greater wetted surface. At all speeds A had less resistance
than D. For the former the resistance began to increase
rapidly at about 17 knots and for the latter at about 14
knots. For C this relatively rapid increase in resistance
began to appear at about 19 knots, while for no speed up to
2O knots was such tendency noticeable with B.
It appears therefore in general that resistance will be
decreased if, instead of a parallel middle body, the fore and
after bodies are expanded so as to obtain a ship of the same
displacement as with parallel middle body, even if the ratio B
to L is thereby somewhat increased.
106 RESISTANCE AND PROPULSION OF SHIPS.
Later experiments, notably those by Rota,* have thrown
additional light on the influence of a slight change in proportions
with the same characteristics of form. In such case secondary
forms are derived from a primary by changing all dimensions in
one direction (as transversely) in a constant ratio. As the result
of a large series of experiments Rota reaches the following con-
clusions :
(a) For the same change in displacement obtained by, such
"expansion" the power required for a given speed is affected
in larger degree by a change in breadth than in depth.
(&) The increase in longitudinal dimensions for the same
displacement results in a notable decrease in the residual resist-
ance and on the other hand in a marked increase in the frictional
resistance the latter effect however, less than the former at
usual speeds.
(c) Every form derived by change in the longitudinal dimen-
sions has a particular speed for which the increase or decrease of
displacement resulting from such change is accompanied by a
small or negligible change in the power required. The length
and speed at which such relation obtains are connected approxi-
mately by the formula, v = VL, where v = speed in knots and
L= length in feet.
(d) Change in the displacement resulting from change in
the transverse dimensions, so long as the percentage of change is
moderate, will result in a change of resistance at the same speed
of approximately the same percentage value; that is, 10 per cent
increase in displacement will require about 10 per cent increase
in power for the same speeds.
Changes in the displacement resulting from change in the
* Bulletin de L' Association Technique Maritime, 1900, p. 49.
RESISTANCE. 107
vertical dimensions, so long as the percentage of change is
moderate, will result in a change of resistance at the same speed
of approximately .6 the same percentage value; that is, 10 per
cent increase in displacement will require about 6 per cent in-
crease in power for the same speed.
With the displacement slightly increased by longitudinal
change and the speed in question greater than \/L, there will
be a decrease in power for the same speed; and with the dis-
placement slightly decreased, there will be an increase in power
for the same speed. With the speed in question less than VZ
the above relations are reversed.
17. MODIFICATION OF RESISTANCE DUE TO IRREGULAR
MOVEMENT.
It is well known experimentally in all cases of bodies
moving in a liquid, that a certain amount of water is
momentarily bound to the body and must be considered as
practically taking part in its motion, especially in any rapid
changes involving accelerations and retardations. This
amount, it also appears, may be from 15 to 20 per cent of
the displacement of the ship. It follows that the resistance
to an acceleration will not be that due simply to the mass of
the ship, but rather to the combined mass of ship and water.
At the same time it must be remembered that during retar-
dation a corresponding effort is given out. It is, however, a
general principle that more energy is required to maintain a
system in irregular movement, the velocity oscillating about
a mean value u, than at the same uniform value of the
velocity. This is because in general R varies with u accord-
108 RESISTANCE AND PROPULSION OF SHIPS.
ing to a higher index than I, and because in such case the
mean of a series of such powers of u is greater than the mean
value of u raised to such power. This is readily verified for
an index 2 or 3.
The mean resistance of a ship in irregular motion but at
a mean speed u will therefore in general be greater than if
the velocity could be made uniform at the same amount.
The actual modes of propulsion employed usually involve
slight irregularities in motion, and these may be more or less
increased by irregularities of wind, current, tide, depth of
water, etc. Under usually favorable conditions it is not
likely that the increase of R due to this cause is important,
although its existence and explanation are of interest.
18. VARIATION OF RESISTANCE DUE TO ROUGH WATER.
The resistance of a ship in a seaway is known actually to
be very considerably greater than that in smooth water for
the same average speed. The causes of this may be given
under three heads:
(1) The direct action of the waves and the rolling and
pitching tend to increase the irregularities of motion, thereby
affecting the resistance as described in the last section.
(2) The wave-disturbance of the water tends to confuse
and disturb the regular stream-line motion, thereby entailing
a greater drain of energy from the ship than in smooth water
at equal speed.
(3) The pitching and rolling, notably the former, place
the ship continually in less favorable positions relative to
RESISTANCE.
propulsion, so that on the whole the mean resistance is in-
creased.
These causes also react unfavorably on the propelling
apparatus, lowering its efficiency and increasing the irregulari-
ties of motion. While this last result is secondary,- the final
effect on the propulsion is the same as though the resistance
of the ship were virtually increased.
No rules can, of course, be given for the estimate of such
irregular modifications. It is a matter of experience, how-
ever, that the following qualities are favorable to uninter-
rupted speed in rough water:
(1) Considerable length, good free-board, and steadiness,
especially absence from pitching.
(2) Large size and consequently great weight.
The reasons for the good effects of the former are ap-
parent. The good effects of the latter are due to the increase
f inertia and the consequent decrease in the irregularities of
movement.
19. INCREASE OF RESISTANCE DUE TO SHALLOW WATER
OR TO THE INFLUENCE OF BANKS AND SHOALS.
It is a well-known fact that the resistance of a ship or
boat at any given speed is very much increased by shallow
water or by the proximity of banks, as in the navigation of
canals and narrow rivers. The primary cause of this excess
of resistance is readily found in the disturbed stream-line
motion. As a result of the decreased cross-section within
which the stream-lines must be contained, there arises an
entire change in their form and in the wave configuration
about the boat. The latter in general is very much exagger-
ated relative to its condition in open deep water, and the net
HO RESISTANCE AND PROPULSION OF SHIPS.
consequence is a very considerable increase in the amount of
energy necessary to the maintenance of the configuration of the
water, and hence a corresponding increase in the resistance.
We may next inquire as to the limits within which these
results become of notable significance.
Taking first the question of shallowness of water, the other
dimensions being unlimited, we have a large number of
instances of trial-trips in which the increase of resistance due
to shallow water is more or less clearly shown.
White* gives the following instance:
The Blenheim in water averaging 9 fathoms made 20
knots with 15750 I.H.P. The wave phenomena were most
striking and unusual. In water from 22 to 36 fathoms with
the same power the speed was 21.5 knots.
In the trial of the U. S. S. New York both ways over a
4O-mile course it was found that at the same point each way,
where the depth of water was charted as 37 fathoms, the revo-
lutions fell off slightly while the steam-pressure became increased.
The water over the remainder of the course was from 15 to 20
fathoms deeper than at this point. The mean speed was 21
knots.
Experimental investigation of these relations reported on by
Marriner and H. Yarrow | has brought out the following facts :
It is a matter of common observation that as a boat runs
into shallow water the wave length of the transverse system
increases in length. The relation between speed and length of
wave for deep water, as noted in 10, is
* Manual of Naval Architecture, 1894, p. 467.
f Transactions Institute of Naval Architects, 1905, p. 344.
RESISTANCE.
Ill
A graphical representation of such law is shown by the heavy
curve of Fig. 42. Now when a ship is run in shallow water the
relation between speed and wave lengths follows this line for low
and moderate speeds, and then as the speed becomes higher the
curve rises more quickly, as shown by the light lines, implying a
more rapid increase in length with increase in speed than for
deep water. Finally, at some limiting speed the curve approaches
the vertical, implying an indefinite increase in length for such
speed. If then the speed is increased beyond this value there is
no corresponding wave or wave system. The speed at which
this condition is approached is called the critical speed for that
depth, and the depth at which the length of wave becomes in-
definite for a given speed is called the critical depth for such
112 RESISTANCE AND PROPULSION OF SHIPS.
speed. The relation between speed and depth for this critical
combination is approximately
where v = speed in knots and d = depth in feet.
In going from deep to shallow water the increase in length
might be expected to follow the law of shallow water waves, as
noted in 10. Actually, however, the increase is less rapid than
the formula would indicate. In explanation of this it has been
suggested by Marriner that if the reaction between the ship and
the wave system results in the latter being dragged along, as it
were, and thus prevented from lengthening as rapidly as it other-
wise would, then such reaction would result in an added drag
on the ship manifested as an increase in resistance. This con-
sideration may aid in explaining the great increase in resistance
in approaching the critical speed for a given depth.
In accordance with these various observations on speed and
wave system, it is found furthermore that the maximum wave
disturbance and the point of maximum resistance both occur at
or near the critical combination of speed and depth.
Certain experiments, moreover, seem to indicate that the
hump in the resistance curve is independent of displacement.
This point, however, needs wider assurance through further
investigation.
Certain other experiments by Rasmussen on the torpedo-boat
Makrelen, and reported by Laubeuf,* bear out the same general
conclusions. In these experiments a sharp rise in the speed
horse-power curve was noted at points corresponding closely to
* Bulletin de L' Association Technique Maritime, 1898, p. 207.
RESISTANCE. 113
the relation 1^ = 11.3^, as noted above. The conclusions drawn
by Laubeuf are as follows:
(1) The various curves present a hump more and more
marked as the depth is reduced. At depths greater than 50 feet
there was no noticeable hump within the range of speed em-
ployed (to 20 knots).
(2) The speed corresponding to the hump was, for a given
depth, nearly independent of the two displacements at which
the boat was tested (105 and 118 tons). The power for such
speed, however, increased in marked degree with the displace-
ment.
The important question, however, relates rather to the con-
ditions under which the resistance will be sensibly that due to
water of indefinite depth, or, in other words, for what depth will
the influence of the bottom be negligible? Regarding this prob-
lem various view-points may be taken. Thus it has been sug-
gested that so long as the depth is sufficient to allow the wave
corresponding to the speed of the ship to assume sensibly its
trochoidal form and constitution no sensible increase in resist-
ance will result. The propriety of considering this as more than
a roughly approximate relationship may be questioned, since we
do not know that the -natural wave formed by the ship is more
than roughly trochoidal in form.
By -reference to 10, Table V, it appears that if the depth
equals one half the wave-length the difference between the two
kinds of waves should be quite negligible. This leads by the
aid of 10 (i) to the following table :
114 RESISTANCE AND PROPULSION OF SHIPS.
Minimum Depth,
Speed, Knots. Feet
10 28
12 40
14 55
16 71
18 '. 90
20 in
22 135
24 160
26 188
28 218
3 2 5
32 284
34 322
36 3 6
38 400
40 444
From considerations drawn from a study of the distribu-
tion of stream-lines under certain ideal circumstances, D. W.
Taylor* concludes that a depth of water equal to ten times
the draft is an outside limit, and that no sensible increase
in resistance is likely to be met with so long as the depth is
not less than six times the draft.
The result in the case of the New York, if reliable, indi-
cates that the influence extends to a considerably greater
depth than six times the draft, or than the amount given by
the table above. On the other hand the experiments reported
by Marriner and Yarrow showed that for torpedo-boats at
* Transactions Institute of Naval Architects, vol. xxxvi. p. 234.
RESISTANCE. 115
speeds of 25 to 30 knots, a depth of water of 100 feet was insuf-
ficient and would not permit of realizing contract conditions,
while an increase to 120 feet proved adequate for the purpose
and permitted substantially the same performance as the
Skelmorlie course with 240 feet depth.
The depth for critical speed is seen to be
"3
while, according to Marriner and Yarrow, the worst combination
of conditions was reached at about
10
It therefore seems most rational to assume that the safe depth
may be stated as some multiple of this depth, modified suitably
by the size and form characteristics of the ship. It is readily
seen that the above table representing half wave-lengths gives
simply values of v 2 + $.6. While perhaps none too deep for large
ships of moderate or full form, the values of this table indicate
a depth somewhat greater presumably than is needed for torpedo-
boats and other like relatively small boats of fine form. Further
investigation is needed on these points, and especially hi relation
to the influence of form and dimension on the limiting depth.
Turning now to the conditions existing in canals, we have
not only the restriction due to the bottom, but also that due
to the sides as well. Proceeding at once to a representative
case, we may note briefly the phenomena attending propul-
sion in a canal of which the cross-sectional area is only some
four or five times that of the greatest section of the ship.
In such cases at moderate speeds the bow wave is quite
marked, rising more or less transversely on each side or a
Il6 RESISTANCE AND PROPULSION OF SHIPS.
little ahead of the bow, and followed amidships or astern by
an equally notable hollow. Following astern is a train of
waves, each member of which moves with the velocity of the
boat, while the train as a whole moves at a lesser velocity, as
explained in 12. The slower the boat moves the more are
the phenomena like those noted in connection with motion
on the open sea. As the speed increases the more definitely
does the boat place herself on the rear edge of a wave as in
Fig. 43, and the more pronounced does the series of follow-
ing waves become. As the speed of the boat approaches
that corresponding to the propagation of a permanent wave
of translation, u = Vgh, the waves begin to fake on more
and more the characteristics of waves of translation. They
transmit a greater and greater proportion of energy and the
train becomes shorter and shorter. The resistance, however,
continually increases due to the rapidly growing size of the
waves and their consequently increased demand for energy.
As the boat reaches the critical velocity or passes slightly
beyond it, the train of waves contracts to a single member,
which takes its place with crest amidships or with the boat
slightly on its forward slope. The resistance then decreases,
and the boat may be towed along at this speed with a much
less expenditure of power than just before. The cause of
this sudden change in the law of resistance is found in the
relation of the boat to the wave. The necessary energy to
RESISTANCE
117
form the permanent solitary wave having been expended,
and the boat having been placed in the most favorable posi-
tion relative to such wave, the maintenance of the wave and
the velocity of the boat may be obtained at a comparatively
small further expenditure. The resistance for the most part
is that due to the maintenance of the wave in an imperfect
liquid, with the increased tendencies toward degradation due
to irregularities in the sides and bottom.
Following is a table showing results of experiments made
by J. Scott Russell, the pioneer of experimental investigation
in this direction.
Weight of Boat
in Pounds.
10239
Speed in
Miles per Hour.
12579
Resistance
in Pounds.
112
201
275
250
268.5
250
500
400
280
Speeds at which these conditions obtain cannot, however,
be practically maintained in canals, due chiefly to the wear
and tear on the banks from the wash of the wave, and the
comparatively high speeds necessary for depths of water
exceeding 10 or 12 feet.
We are therefore rather concerned as to the effect of the
banks and bottom on the resistance at moderate speeds, and
Il8 RESISTANCE AND PROPULSION OF SHIPS.
as to the necessary ratio between the section of the canal and
that of the ship in order that the increase in resistance may-
be negligible.
Satisfactory data for a general discussion of these points
are not available. We have, however, certain partial results
as follows:
Elnathan Sweet * gives the following formula as repre-
senting the results of a series of experiments made by him on
the Erie Canal:
.10303^
1 r- .$97 '
R = resistance in pounds;
s = wetted surface ;
v = speed in feet per second;
r = ratio of section of canal to that of boat.
The speed varied from 2 to 3 feet per second, or from
1.27 to 2.09 miles per hour, s varied from 2870 to 3090;
r from 4.28 to 5; depth of water from 7 to 8 feet; draft of
boat from 6 to 7 feet; displacement from 260 to 310 tons.
The formula is a special case of the general formula pro-
posed by J. Scott Russell and Du Buat for resistance in a
restricted channel, viz.,
ID
where A and B are constants to be determined by experi-
ment. The ranges of values of r, v y and s are manifestly too
* Transactions American Society of Civil Engineers, vol. IX. p. 99.
RESISTANCE.
small to warrant the application of these results to conditions
widely different from those covered by the experiments.
Conder* states that in one instance where r =. 3 a speed
of 7 miles in the open was reduced with the same power to
5 ; while in the discussion of the same paper Gordon states
that in the Rangoon, where r 10, a speed of ten miles
was similarly reduced to 7.
M. Camere gives the following results cited by Pollard
and Dudebout:f
Let resistance be represented by a formula
R = fsv\
where the letters have the same significance as above and /is
a coefficient.
r
Values of /at Speeds per Second of
3'-28
4 '. 9 2
6'. 56
I6. 4
9.0
3-45
.00464
.00836
.0280
.00513
.00891
.00564
.00946
More recently M. de Mas has conducted extensive
experiments in France, the results of which have been
made public by Derome.J Among the many interesting
features of this investigation the following may be men-
tioned :
* Minutes of Proceedings of Institution of Civil Engineers, London, vol.
LXXVI, p. 660.
f Theorie du Navire, vol. in. p. 458.
J Proceedings Sixth International Congress on Inland Navigation, 1894.
Quoted also by Leslie Robinson, Proceedings Institute Mechanical Engineers,
i8 9/ .
120
RESISTANCE AND PROPULSION OF SHIPS.
Three boats were tested by towing on the Seine. The
ratio of the section of the stream to the immersed section of
boat was about 55, for which the results should be, presum-
ably, not far from those for a ratio indefinitely large. The
results are given in Table I, from which it appears that the
TABLE I.
Displace-
Resista
nee in Pou
nds at Spe
eds per M
nute of
Boat.
Length.
ment.
Q 8'. 4
ig6'.8
295'.2
393'-6
492'
124.6
286
liq
JC7
783
1464
2467
Rene
QQ A.
112
qe-i
78-}
1466
24.60
Adrian
67 4
148
112
-ie-7
78-3
1466
2460
resistance was independent of the length within the limits
of speed and length employed. The same three boats
were afterward tested in the Bourgogne Canal at speeds
increasing by increments of 49.2 feet per minute, from
49.2 to 246 feet per minute, with the same result as
regards the substantial independence of resistance on length.
The general truth of this proposition must by no means be
assumed from these experiments, though there seems reason
for believing that, due to the peculiar form of canal-boats,
a change of length has relatively small influence on the
resistance.
A number of boats with particulars as in Table II were
tested in both river and canal, with results as given in
Table III.
RESISTANCE.
121
TABLE II.
PARTICULARS OF BOATS.
Name.
Length.
Average Breadth
at Midship Section.
Block Coefficient
of Fineness.
TO; . *J
16 4
Flute
122.8
16 e,
yy
118.4
i6.5
Q7
III .Q
16 i
Q-J
TABLE III.
rt c
Q.
u o
Speed
Speed
1
*!
984 Feet per Min.
196 8 Feet per Min.
Cross-
s ~
= /.
section
IS 5
TT O.
Draft.
Boat.
of
<- y
~ -
Resistance.
Resistance.
Canal.
w
t?"x
R
R
A
rt'7)
5
^
is
Canal
River
T
Canal
River
r
OS'S
R
r
R
T
( Peniche
3I7 ; 9
86.66
3-67
379
225
1.69
1896
664
2.86
5'.25- Flute
86.44 3.68
247
119
2.07
1060
357
2.97
1 Toue
86.44
3-68
240
97 2. .18
IO2T
278
3-67
( Flute
70.30
4-52
154
97 1-59
626
315
1.99
4'. 27 - Prussian
68 68
4 62
119
49
2-45
474
176
2.69
/ Margotat
69.98
4-54
H7
46
2.52
434
148
2.94
3'.28 Flute
54-04
5-88
106
86
1.23
421
284
1.48
li
1
An interesting test was also made in various waterways of
the boat Jeanne, 99 feet long, 16.4 feet wide, and of the
Flute class. The results are given in Table IV.
122
RESISTANCE AND PROPULSION OF SHIPS.
TABLE IV.
TEST OF JEANNE IN DIFFERENT WATERWAYS.
C-* 3
49'. 2 per Min.
98'. 4 per Min.
i47'.6perMin-
i96'.8perMin.
246' per Min.
Draft.
Boat.
.2 $ a
R
R
R
R
R
R
r
R
r
R
r
R
r
R
!<Ja>
r
y
r
r
^
Seine
116.
26.5
26.5
.00
28.7
28.7
.00
146
146
.00
245
245
.00
384
384
1. 00
A
8.31
26.5
.00
28.7
.00
150
3
271
.11
459
i .20
3'. 28
B
5.89
35-3
"
33
103.6
.42
227
56
4 o8
'
.67
692
'
1.83
C
4.64
37-5
i
.41
108.2
.48
238
.63
452
85
79 6
1
2.07
D
3.82
39-7
'
.5
I2 3-5
.70
284
95
558
.28
1019
'
2.66
Seine
89-3
26.5
'
.00
83.8
83 8
.00
172
172
.00
295
295
.00
474
474
1. 00
A
6 -39
3-9
'
.17
105.8
.26
238
38
44 1
.49
756
i. 60
4'.27 -{
B
4-54
48-5
'
74
154.3
.84
342
.98
622
.1011074
'
2.26
1
C
3-57
5-7
4
.02
178.6
13
410
.38
814
75 T 5 01
3- T 7
I
D
2.94
1
.67
2 55-7
5
657
3.82
1389
"
.70 2646
'
5-58
r
Seine
72.5
2^7
28.7
.00
90.490.4
.00
187
187
i .00
320
320
.00
5 11
5"
I. 00
A
5- J 9
39-7
42
141.1
.56
326
1.74
620
'94
1124
2. 2O
5 . 25 |
B
C
3-68
2.90
70.5
72.8
"
.46
.56
244.7
264.6
'
7 1
93
562
688
3.00
3-67
1047
"
3.28
4-79
1839
2965
4
3-59
5.80
While the information relating to many points of this
problem is still quite meager, it is sufficiently well established
that where the ratio of section of canal to section of ship is
not more than six or eight, a very considerable increase in
the resistance will result, even at moderate speeds. Again,
mere sectional area of canal is not sufficient, for we might
have a very wide and shallow canal in which the ratio of sec-
tions would be very great, but in which we should have the
shallow-water resistance as already discussed. In order to
reduce this increase of resistance to a small or negligible
quantity, it seems likely that the transverse dimensions of
the channel should be everywhere from six to ten times
the corresponding dimensions of the greatest section of the
boat.
In this connection it is interesting to note that the influ-
KESIS TA NCR. 123
ence of fine extremities is less and less useful to reduce resist-
ance as the ratio of section of canal to boat is less. It is
readily seen that a point may be reached where the resistance
may be less for a somewhat narrow and full form than for an
equal displacement and equal length, but with fine ends and
a larger midship section, and hence with a greater constric-
tion of available cross-section for stream-line motion.
20. INCREASE OF RESISTANCE DUE TO SLOPE OF CURRENTS.
In ascending rivers where the current is noticeable there
must be a corresponding up grade and a resulting vertical
component to the movement. The total energy expended
in any given time will therefore include, in addition to that
necessary to overcome the water-resistance, an amount Db y
where D equals displacement and b is the total change in
elevation effected during the interval in question. The corre-
sponding resistance is naturally D sin a Da, where a is the
angle of grade and always small. Naturally the amount of
this part of the total resistance in such a case is usually
small, though it may reach such an amount as to prevent the
ascent of a river where the speed of the current is consider-
ably less than the possible speed of the ship in smooth water.
In special cases ex. may reach a value of from .001 to
.0015. At moderate speeds the smooth-water resistance
may not be more than .oiD or even less, so that it is readily
seen that this part of the resistance may reach an amount
considerable in comparison with that for such speeds under
usual conditions.
124 RESISTANCE AND PROPULSION OF SHIPS.
21. INFLUENCE ON RESISTANCE DUE TO CHANGES OF TRIM.
In treating of resistance we have thus far considered only
the horizontal component. It is quite evident, however, if
the ship were to be maintained in her statical trim and draft
and towed through the water at any given speed, that the
resultant of all the changes of pressure would in general
have a vertical as well as a horizontal component. In many
cases, especially at high speeds, it becomes important to take
cognizance of the existence of this vertical force. Considered
as a single vertical resultant, it usually acts through a point
forward of the statical center of buoyancy. In the actual
case, the ship being free to yield to this force, she will change
trim, the bow rising somewhat more than the stern sinks,
thus decreasing the displacement by a small amount. At
certain high speeds these modifications, especially the change
of trim, become very noticeable. At still higher speeds,
especially with the cut-up form of after body mentioned in
11, it would appear that the trim may begin to go back
toward its normal value, though the displacement will prob-
ably continue to decrease with the increase of speed.
From another point of view we may consider that the
motion of the boat, especially at high speeds, gives rise to a
wave system, and that the boat naturally accommodates itself
in trim to the characteristics of this system. In general at
high speeds such system will involve a hollow under or near
the stern and a crest near the bow, thus placing the boat on
the back slope of the wave and naturally giving rise to the
change of trim. An instance is given by White* of measure-
* Manual of Naval Architecture, p. 470.
RESISTANCE. 125
ments taken by Yarrow on a torpedo-boat about 80 feet long
at a speed of 18.5 knots. The change of trim was about one-
half inch per foot of length or forty inches between bow and
stern. Relative to the water surface forward the bow raised
rather more than one foot, while relative to the surface aft it
settled less than six inches. The remainder of the change in
trim is accounted for by the change in the level of the water
itself due to the wave formation.
Like the other phenomena attending the motion of a solid
in a liquid, this change of trim and of displacement cannot at
present be treated by abstract mathematics. The investiga-
tion of the performance of models and such comparative
methods are alone of use in aiding to form an estimate of the
amount of such change in any given case.
The influence of the change of trim on resistance is prob-
ably slight. Directly it is a result and not a cause of
resistance. Indirectly it may be a cause of a modification of
resistance on account of the difference which it may cause in
the form of the immersed bodv. With usual forms the influ-
ence of such slight changes of this character as usually occur
is not likely to be important. It is probable that the prin-
cipal loss arising from a change of trim is due to decreased
efficiency in the propulsive apparatus rather than to actual
increase in the resistance.
In connection with these considerations it may be pointed
out that there is reason to believe that for extremely high
speeds relative to length, such as are attained by some
torpedo-boats and fast launches, and notably by recent
torpedo-boats and torpedo-boat destroyers, the coefficients in
the formulae of resistance which hold for larger ships and for
more moderate speeds undergo considerable change. A
126 RESISTANCE AND PROPULSION OF SHIPS.
direct extension of the results for lower speeds to such
extreme values is not therefore allowable, and the various
constants and characteristics of empirical formulae for resist-
ance and power must be determined by direct experiment
under such conditions, or under others comparable with
them. See 26.
In the boats mentioned speeds of 30 to 35 knots and upward
have been attained on lengths of from 180 to 200 feet. The
resistance at such speeds is shown to vary with a lower index
than for more moderate speeds, as, e.g., 18 to 24 knots, and
the performance as a whole seems to indicate a general
modification in the relation of the resistance to the attendant
conditions.
As causes of this change in relationship we may look to
the change in displacement, and consequently in wetted sur-
face, and to the changed location of the wave system relative
to the boat. ^
Photographs of boats at very high speeds show that under
such conditions the decrease in wetted surface may be quite
considerable, though data are lacking to furnish a satisfactory
basis for estimate in any given case.
The change in the resistance of a boat in a canal when it
approaches a speed such that it can be forced over on to the
top or forward slope of the primary wave has already been
mentioned. *
It is not to be expected that these conditions can be
paralleled in open water; but we know that as the speed
increases the location of the bow primary wave falls farther
and farther aft, and photographs taken under these conditions
as well as the decreased change of trim seem to show that at
a speed sufficiently high the boat is forced at least partly
KESIS TA NCE. I2J
over the primary bow wave, so that such wave, once formed,
might require a less expenditure for the maintenance of its
energy and the speed of the boat than would be indicated
by an extension of the law for lower speeds to this extreme
point.
It is not likely that this favorable region in the law of
resistance can at present be taken advantage of by other
than boats of the types mentioned. For larger ships, e.g., of
L = 400 feet, such speeds would exceed 40 knots, and the
power required would be so great that with present engineer-
ing resources its development on the weight available and its
economical application would be impracticable.
22. INFLUENCE OF BILGE-KEELS ON RESISTANCE.
Bilge-keels are usually located near the turn of the bilge,
and stand normal to the surface rather than vertical. They
usually run for one half or two thirds the length of the
ship, and are supposed to follow as nearly as may be the
natural stream line path in order to interpose the minimum
resistance. Experiments by Wm. Froude and others have
shown that the additional resistance due to bilge-keels is
slight. Some of the model experiments by Mr. Froude
indicated that the additional resistance due to the bilge-keels
was less than that due to their surface friction as usually
computed ( 8). It has been suggested that this may be due
to the fact that the keels are in a skin of water moving
forward more or less with the ship, and that in consequence
the resistance per unit area is less than that normally due to
the relative velocity of ship and water as a whole. A more
just view is perhaps to consider that with the bilge-keels the
128 RESISTANCE AND PROPULSION OF SHIPS.
average skin-resistance coefficient for the ship as a whole is
somewhat less than for the ship without the keels; so that
while with the keels the total area is greater, the coefficient
is less, and the product is but slightly increased. The reason
for the decrease in the average coefficient may be readily seen
by comparing two bodies of cylindrical form moving axially
through the water. Let one be smooth and the other deeply
corrugated or channelled longitudinally, both of the same
outside diameter. Both will tend to set in motion a cylinder
of water of about the same diameter, and with the channelled
cylinder this tendency will be so effective that at and near
the bottom of the channels the relative velocity of the sur-
face and the water will be very slight, and hence the resist-
ance small. The average coefficient of surface resistance for
the channelled cylinder will therefore be less than for the
plain, and while its area is much greater, the total surface
resistance will not be increased in a proportional degree.
Recent Italian experiments on the model of the Sardegna*
showed for speeds of the ship from 16 to 21 knots an increase
of resistance from I to 5 per cent increasing with speed, or
with given power a decrease in speed from practically nothing
to a little over I per cent. These experiments seemed to
indicate a somewhat rapid increase in the resistance due to
the bilge-keels from speeds of about 18 knots upward. It
may be suggested that this was due in large measure to a
change of trim at these speeds and to a disturbance of the
stream-line flow, in virtue of which the keels became placed
oblique to the line of flow and thus offered a certain amount
of head-resistance.
* Rivista Marittima, Oct. 1895.
RESIS TA NCR. 1 2 9
23. AIR-RESISTANCE.
All above-water portions of a ship in motion are, of
course, exposed to air-resistance. This from the irregularity
of the motion of the air and the great irregularity of the sur-
face exposed is very difficult of computation. Except in
rigged ships with a high velocity of the wind, however, this
effect is not considered of great importance, and usually no
attempt is made to estimate its amount. From data obtained
from planes moving in air" it appears that the resistance may
be expressed by the formula
R=fAv\
In this equation A is the projected area of the surface
exposed, the projection being made on a plane perpendicular
to the direction of motion. For an unrigged ship this is
readily found, being simply the area of an end view of the
part of the ship above water. Where the ship is rigged the
amount to be taken is very indefinite, and no estimate can be
made except by comparative methods, for which, unfortu-
nately, the amount of data available is insufficient. Wm.
Froude was led to believe that in the case of the Greyhound
the influence of the rigging was about equal to that of the
hull. This would depend very much, however, on the nature
and amount of rigging, and doubtless on the speed of the
wind. The velocity v is to be taken as the longitudinal
component of the relative velocity of the wind and ship.
Thus in calm v will be the speed of the ship, in a following-
wind it will be less, and in a head wind more.
The value of the coefficient / rests on experimental
determination. The experiments of Wm. Froude and others
130 RESISTANCE AND PROPULSION OF SHIPS.
indicate that with pounds, feet, and seconds as units its value
may be taken as about .0017, and with pounds, feet, and knots
as about .0048. Later experimenters, among others Dines and
Langley, find values somewhat less and ranging for pounds, feet,
and knots about .004. The experimental values are derived
from planes of small or moderate size, and there is considerable
uncertainty as to the values suitable for use on large areas.
In the case of the Greyhound unrigged, with a relative
speed v = 15 knots the effect produced was measured by 330
pounds. For v = 10 knots this would correspond to about
i^o pounds. This is about 1.5 per cent of the water-resist-
ance at the same speed. With rigging it was assumed that
this might be doubled, so that in a calm the ship would
experience an air-resistance of perhaps 3 per cent of her
water-resistance. For moderate speeds, or so long as the
water-resistance varies sensibly as the square of the speed,
we might expect the relation between the air and water
resistance to remain sensibly constant. For higher speeds
the water-resistance will increase more rapidly than the
square of the speed, so that the ratio of air to water resist-
ance will decrease at these speeds.
If instead of calm air we suppose a speed of 10 knots
through the water against a head wind of velocity say 30
knots, we should have v = 40, and the air-resistance would
be some 16 times as much as before, or unrigged about 25
per cent of the water-resistance and rigged about 50 per cent.
While these figures may not be exact, they are sufficiently so
for illustrative purposes, and show plainly the importance
which wind-resistance may assume in extreme cases.
On one hand the constant decrease of sails and rigging on
all types of steam-vessels tends to decrease the importance
of air-resistance. On the other, the constant effort to
RESISTANCE. 13!
increase speed on ocean-liners, war-ships, torpedo-boats, etc.,
calls increased attention to all causes which in any manner
may affect the resistance. Additional experiments in this
direction are greatly needed. Such experiments might be
carried out by allowing a ship to drift in a fairly smooth sea
under the influence of the wind alone, and measuring the
velocity of the ship relative to the water and that of the
wind relative to the ship. Then from model experiments or
other known data relative to the ship her resistance at the
speed attained may be known, and this must equal that due
to the air moving past the ship with the relative velocity
observed. 'Otherwise a ship might be moored and subjected
to the wind, the strain being measured by suitable dynamo-
metric apparatus.
24. INFLUENCE OF FOUL BOTTOM ON RESISTANCE.
There seems no reason for believing that the condition of
the bottom of a ship will essentially affect anything but the
skin-resistance.
The effect of foul bottom on skin-resistance has been dis-
cussed in 9, and it is again mentioned here simply to make
complete the special causes which may affect the total resist-
ance.
25. SPEED AT WHICH RESISTANCE BEGINS TO RAPIDLY
INCREASE.
The analysis of curves of resistance usually shows that for
very low speeds the total resistance varies nearly as the
square of the speed, such relation remaining nearly constant
until a speed is approached at which the resistance begins
132 RESISTANCE AND PROPULSION OF SHIPS.
rather suddenly to increase more rapidly, the index becoming
3 or even 4 for a time, while at still higher speeds the index
may fall again to 2 or sometimes apparently to even less.
Such characteristics in the curve of residual resistance are
shown in Fig. 40. The speed at which the first rapid in-
crease is located depends chiefly, no doubt, on the wave-
making features of the ship; and while it is not expressible
with any great accuracy by any simple formula, yet it is
sometimes considered that for general purposes it may be
taken as proportional to the speed of a wave of length equal
to that of the ship. Hence we should have
/&.
U ~ A / - - =
V 27T
bVL,
where b is some constant. With knots and feet as units, b is
frequently taken as approximately I, in which case we have
u = VI.
This must be considered as giving simply an indication of
the locality of a region where the resistance will begin to
rapidly increase, rather than the a'ctual location of a definite
speed. It must also be remembered that generally the loca-
tion of this region will be higher as the ratio of length to
beam is greater, and as the prismatic coefficient is less.
Normand has developed formulae for what he terms the
maximum normal speed as follows:
(i.oim-b)L
"* - 1
1.39(1.01^ b)L
or ^
RESISTANCE.
where 5 = beam of ship in feet;
H = draft in feet;
L = length in feet;
b = block coefficient;
m = midship section coefficient.
These formulae are intended to take account of form charac-
teristics, and to indicate the maximum speed which, under normal
or ordinary conditions, such ship may be given without marked
departure from the usual resistance and power relations.
26. THE LAW OF COMPARISON OR OF KINEMATIC
SIMILITUDE.
It is the purpose of the law of comparison to furnish for
two ships of similar geometrical form but of different size,
and moving at different speeds, a relation between the
know, principally the wave-making or modified stream-line
resistance.
This being the purpose, care should be had in noting the
fundamental assumptions on which the statement of the law
depends.
Consider the stream-lines in an indefinite liquid moving
past a body as in I. Let the mo'Jon throughout these
stream-lines be steady. That is, let there be no discon-
tinuity as in the breaking of waves at the surface or the
formation of eddies within the body o'f the liquid. The
usual equation for steady motion will then apply, and de-
noting the total head by k we shall have for any stream-
line,
z + + = .. (i)
z <r
134 RESISTANCE AND PROPULSION OF SHIPS.
Now suppose that we have a second body similar geomet-
rically to the first but A, times larger in all directions. The
second body may then be considered as a magnification of
the first in the linear ratio A. Let it be placed, like the first,
in a liquid of indefinite extent moving past it with some
velocity nv, such that the stream-line distributions in the two
cases are also geometrically similar and in the linear ratio A..
It follows that the entire systems in the two cases are
geometrically similar, and may be considered as derived the
one from the other by linear expansion or diminution in the
ratio A,. For a stream-line in the second system situated
similarly to that taken in the first we shall have
\z for z,
nv (l v,
and we may put p, " p,
and k, " k.
Then we have
29 ,
\z A h = , (2)
2g (T
Now we may most readily conceive of a stream-line AC,
Fig. 44, as due to a head from some indefinitely large reser-
voir E situated at an indefinite distance away. This,
evidently, will in no way change the nature of things at
ABC. Now suppose the stream-line continued back to a
place F where the velocity is inappreciable. If the stream-
line is indefinitely small F will be of small finite area, and we
may so consider it as small as we choose, and hence as very
small compared with the dimensions of the reservoir. Then
if for the present ABC represent a stream-line of the first
RESISTANCE
135
system, equation (i) must hold throughout its length, and we
shall have at F
+->=*.
(3)
In this equation let us take z as measured from a base-
line OX. The pressure/ will be simply the statical pressure
due to the height FG, and p -f- a will in feet be equal to this
FIG. 44.
height. Hence 2 -\- p +- v is here equal to XG, the total
depth of the reservoir to the base-line OX. From (3) this is
the value of k. Hence throughout the entire course of the
stream-line represented by (i), the constant k represents this
depth GX.
Now the second system is a magnification of the first in
the linear ratio A. There will be therefore a similar point f
similarly situated in a similar indefinite reservoir whose depth
to a similar base-line will be A/, and this will be the value of
/,, the constant for the stream-line in the second system,
whose equation is given in (2).
I$6 RESISTANCE AND PROPULSION OF SHIPS.
We have thus far simply considered the velocity-ratio n
such that the stream-lines in the two systems will be
geometrically similar. We must now determine what the
value of this ratio will be in terms of the linear ratio A.
Since the stream-lines are similar throughout, and since
the flow is steady and the condition of continuity is fulfilled,
we must have the velocities along similar lines exactly pro-
portional at similar points. That is, if v ot v lt and F , V l
denote velocities in the first and second systems at pairs of
similar points, we must have
V,_v. F, V,
- '-'-
This shows that n is a constant, and that if we can determine
the velocity-ratio for any pair of similar points the constant
value of n will thus be known. The velocity-ratio at a pair of
corresponding points is most readily found by aid of Fig. 44,
by considering A an open end in each system. We shall
then have / = o, and for the first system
.and for the second
=U-\z= A( - A
Hence, dividing, we have
n 1 = A and n = 4/A.
It follows that in order to fulfil the conditions of simi-
RESISTANCE 137
larity throughout the two systems we must have a constant
velocity-ratio equal to t x A. Putting this for n in (2) we have
(4)
whence, comparing (i) and (4), /, = \p.
The same relation will hold for every other stream-line,
and it follows that throughout the entire second system the
pressure will be A times that at corresponding points in the
first system. In other words, the magnification in pressure
will exactly correspond to that in linear dimension. This
relation will therefore hold between the pressures at corre-
sponding points on the surfaces of the two bodies, because
by supposition the surfaces throughout are in contact with
the stream-lines. In the stream-lines of Fig. I the body is
supposed to be indefinitely immersed, so that no surface
changes take place. All of the preceding equations and con-
clusions hold, however, for any stream-line in a perfect
liquid, and therefore for a stream-line system about a body
partially immersed, so long as the conditions of continuity
of flow are fulfilled. Now it is due to the distribution of
pressure throughout such a stream-line system that the
stream-line resistance arises. Hence this resistance will be
the value of the integral of the longitudinal component of the
pressure acting at the inner surfaces of the stream-line sys-
tem. Denote an element of the surface by ds and the angle
between its normal and the longitudinal by 6. Then we
have for our first system
= Cp cos ds.
138 RESISTANCE AND PROPULSION OF SHIPS.
In the second system the entire distribution of pressure
is multiplied by A, and the linear dimensions are increased in
the same ratio. Hence the ratio between corresponding
elements of surface will be A a , and in the second system we
shall have
R' = A cos e Kds = A 3 cos 6 ds =
That is, with two systems under the conditions we have
assumed, the ratio between the resistances due to stream-line
pressure is A 3 . We may also note that this is likewise the
relation between the volumes of the two immersed bodies,
and hence between their displacements.
The law of comparison thus derived involves three con-
siderations, to which careful attention should be given:
(1) The supposition of geometrical similarity;
(2) The definition of corresponding speeds-
(3) The statement of the law.
Geometrical similarity for the two systems being assumed,
corresponding speeds are described as those which will pro-
duce similar stream-line or similar wave configurations, and
are defined as speeds in the ratio of the square root of the
linear-dimension ratio A. , Based on these conditions, we then
have the following statement of the law:
At corresponding speeds the resistances of similar ships are
in the ratio of the cubes of like linear dimensions ; or as the
cube of the linear -dimension ratio ; or as the volume -ratio.
The truth of the law thus derived, it must be remem-
bered, rests on the application of the formula for steady
motion to stream-line flow. It is therefore only applicable
RESISTANCE. 139
to that part of the total resistance which is due to the
stream-line motion, and hence, so far as the above derivation
is concerned, is not applicable to skin or eddy resistance. It
also presupposes the following as necessary conditions for its
exact truth:
(1) A liquid without viscosity;
(2) The absence of discontinuity of flow such as would be
caused by breaking waves or eddies.
So far as the first condition is concerned, the viscosity of
water relative to the velocities with which we are concerned
is so slight that this departure from the exact conditions will
make no essential difference in the application of the law.
With regard to the second condition we must distinguish
between the effect of the eddies, and that due to breaking
waves. Eddies will be formed by possible irregularities in
the surface, and as an expression of the general tangential
action or skin-resistance. Such eddies may, however, be
considered as virtually a part of the ship so far as the trans-
mission of pressure between it and the stream-line system is
concerned. The actual stream-line system involved is there-
fore that which envelops the eddies. If the two ships are
geometrically similar, we may evidently assume with safety
that the existence of these eddies will not essentially disturb
the geometrical similarity of the enveloping systems of stream-
lines. Hence while we are not here authorized to apply the
law of comparison to eddy and skin resistance, it appears
that the existence of the eddies as the expression of these
two forms of resistance should not interfere essentially with
the application of the law to the systems of stream-lines
actually formed. The breaking of a wave involves a discon-
140 RESISTANCE AND PROPULSION OF SHIPS.
tinuity of stream-line flow, and in such case we are not
strictly entitled to extend the law to the stream line or wave
resistance. Within the limits prescribed by this condition,
however, we may consider the law as satisfactorily estab-
lished. We may also consider it as highly probable that
even with breaking waves' the general configurations in the
two systems will, at corresponding speeds, still be essentially
similar, and that the resistances involved will still essentially
follow the same law of comparison. It must be remembered,
however, that while such may be the case, its verification
will depend on experiment rather than on mathematical
reasoning.
It is possible to view the question of the law of compari-
son from an entirely different standpoint, and thus to extend
the scope of its possible application. It will be noted that
the derivation above given establishes the law under certain
limited conditions. It does not follow, however, that the
same law may not be applicable under wider conditions. To
this question we now turn our attention.
Assume a series of ships all of similar geometrical form,
but diverse in dimension. Assume that the resistances of
such ships as a family at varying speeds may be represented
by a general equation consisting of a series of terms all of the
form
Afi-^v".
We shall then have
R = 2At( 3 ~~ 2 \ n (5)
In this equation / is some typical dimension, necessarily
the same throughout the series, v is the speed, and A is one
RESISTANCE. I 4 I
of a series of constants depending on form characteristics,
and hence constant throughout the series. Such a general
equation might therefore consist of a series of terms, as for
example
R = Al^v + Bl*tf + Cl^v* + Dlv*, . . (6)
in which n was given successively the values I, 2, 3, 4. It
is by no means necessary that n should be integral, so that
terms of the form /'-V- 2 , -F/'-V- 8 , etc., may occur. In fact,
so far as we are at present concerned, n may have any value
whole or fractional, and the number of terms may be indefi-
nite.
In regard to the limits for the values of these exponents
we note that if n = o,
if n = 6,
The first would give a term independent of speed and
dependent only on size, and varying directly as volume or
weight. This seems hardly likely, and we may consider
n = o as the lower limit of values of n. The upper limit
n = 6 gives a term independent of dimension and depending
wholly on speed. As indicated elsewhere, it has been
thought that such a term is probable, or at least possible.
In any event, however, it is evidently the upper limit for n
unless we admit the possibility of a term involving a decrease
of resistance with increase of size, which seems quite unlikely.
!42 RESISTANCE AND PROPULSION OF SHIPS.
Hence n may presumably have any value between o and 6 r
whole or fractional.
Taking therefore (6) simply as an illustration of the kind
of terms to be found in such an equation, we note again that
A, , C, etc., are form constants, and remain unchanged
throughout the particular family of ships. We assume then
that it may be possible to express the resistance of any ship
of this family at any speed by substituting in the appropriate
equation using the form constants A, J3, C, etc., and the
given values of / and v. For any other type of form there
will naturally be another series of form constants with their
series of values of exponents for / and v, not necessarily the
same as in the equation for the first type.
We see also that for the same ship at varying speeds this
equation assumes the possibility of representing the resist-
ance as a summation of terms involving the variable v with
various exponents, whole or fractional; while for different
ships at the same speed we should have the resistance
expressed as a sum of terms invoking the variable / with
another set of exponents, whole or fractional.
A little thought will show that these assumptions are
more elastic than those involved in the derivation of the law
from the equations of hydrodynamics. They admit a wide
range of variation, and an indefinite number of terms in the
expression for R, binding the exponents of / and v only to
the one condition that the exponent of / plus one half that
of v shall equal 3. Taking (5) therefore as the symbolic
equation let us compare by its means the resistances of two
ships similar in form with a linear-dimension ratio A, and at
RESISTANCE. 143
speeds in the ratio |/A. We have then, using subscripts I
and 2,
or
Hence for any resistance which follows the law as ex-
pressed in (5) the law of comparison as above stated will
hold. Furthermore, the elasticity of equation (5) makes it
quite possible or even likely that it may properly represent a
larger portion of the total resistance than that due directly to
the stream-line or wave-formation, and hence that the law of
comparison may be applicable to a somewhat greater part of
the total resistance than that to which the first mode of
derivation properly makes it.
Eddy-resistance, for example, is usually considered as
varying with some area, and as the square of the speed, 5,
or as represented by a term Bl*v*. This is seen to fall into
place as one of the general series of terms in (5).
Next as to skin-resistance. This is represented by a
term of the form fAv n , 7, where f is a coefficient which
may decrease with length, A is area, and n is an exponent
usually less than 2. This term is seen to violate the condi-
tions for members of (5), and the amount by which n is less
than 2 and the amount of decrease of /with increase of /and
,ience of A, furnish together a general index of the amount
by which skin-resistance is not amenable to the law of com-
parison. With smooth bottom and the values derived from
144 , RESISTANCE AND PROPULSION OF SHIPS,
Froude's experiments, the difference between the values of
the resistance as computed and as derived by comparison
may be considerable. To take an illustrative case, suppose
/ 3 -T- /, = 4, and hence v 9 -r- v l 2. Also, let the values of
/be .01 and .0093, and the exponent n = 1.85. Then the
ratio between the resistances by comparison will be 64, while
by computation it will be
A J = i6x .93X2^ = 53.64.
The ratio of these is .84, showing that by formula the
value is 16 per cent less than by comparison. If the skin-
resistance were about one half the total, this would involve a
difference of about 8 per cent of the total resistance.
In many cases, however, especially with rough bottom,
the value of the exponent is nearly or quite 2, and the
decrease in f is very small. In such cases this term would
likewise fall practically under the general law with the others.
The question of the form of the terms expressing wave-
resistance has been discussed in 14. If we take the two-
terms A/v* and Bv*, we find that each comes into place as one
of the terms of (5), and hence both fulfil the law of com-
parison.
We will now give a series of equivalent expressions for
the velocity and resistance ratios for two similar ships at
corresponding speeds. We have
_
R, ~ \ij - \A ~
- A > f^ 1 _ f^ 1 &Y _ (A y f^.) 1 _ (.}
~z,v "vy V w V W' ' ' '
RESISTANCE. 145
Any and all of these, and others which may be written
similarly, are equivalent forms of the ratios involved in the
law of comparison for resistance.
The actual confidence felt in the application of this law,
either to the total resistance or to the residual resistance,
must be considered as resting rather on the result of experi-
ment than on any abstract proof necessarily involving as-
sumptions not perfectly fulfilled in the actual case. This
agreement may also be considered as indicating the degree of
closeness with which we may consider either the residual or
total resistance as capable of expression by the sum of a
series of terms as symbolized in (5).
As direct experimental investigation relating to the law
of comparison, we may mention the experiments referred to
in greater detail in 30, in which in two separate cases the
resistances of a model and of its corresponding ship when
compared according to this law were found to be in very
satisf acton' agreement, the greatest errors being within 3^ per
cent.
In this connection we may also note the experimental
determinations made by Wm. and R. E. Froude on the wave-,
configurations made by similar forms moving at correspond-
ing speeds.
The wave systems due to a madel of the Greyhound and
to the boat herself at corresponding speeds were compared
by Wm. Froude. Their similarity was found striking even
to details of the bow wave system. It was also found for the
model that up to a speed of about 1.6 feet per second the
resistance-curve followed almost exactly the curve of skin-
resistance as computed by formula. It was a matter of
observation that up to this speed the model moved without
146 RESISTANCE AND PROPULSION OF SHIPS.
iaising sensible waves. At higher speeds the total resistance
increased somewhat rapidly over that due to skin-resistance
alone, and seemed to follow the plainly visible augmentation
in the size of the wave system. Mr. R. E. Froude also
mapped with care the wave-configuration about two models,
one four times the size of the other, the larger travelling at
twice the speed of the smaller. The smaller represented a
launch 83 feet long at 9 knots per hour, and the larger a ship
333 feet long at 18 knots per hour. The configurations thus
resulting have been already referred to and are shown in
Figs. 31 and 32, and their great similarity in position and
relation to the ship is strikingly seen. In Fig. 32 is also
shown the system for the smaller model at the same speed as
the larger, so that we have here the comparison of two wave
systems made by different similar ships at the same speed.
The general similarity of distribution relative to the water is
quite apparent.
The first comparison indicates therefore that similar ships
at corresponding speeds will produce similar wave-con-
figurations relative each to the ship; while the second
indicates that similar ships at the same speed will produce
approximately similar wave-configurations relative to the
water.
This latter consideration may lead to the suggestion that
the wave pattern for a given type of ship is in large measure
independent of the size of the ship, and simply dependent on
character of form and on speed. Hence the energy involved
in the waves would be likewise dependent only on the same
functions, and likewise the resistance. It is these considera-
tions pushed to their full limit which gives Bv* as the form of
the term for wave-resistance in 14. In one important
RESISTANCE.
147
feature, however, the wave pattern cannot be independent of
the size of the ship. This feature is the composite stern
system which in general depends on the relative positions of
the components due to the primary bow and stern waves,
and this in turn depends on the relation between the wave-
making length and the speed. Hence in part, at least,
wave-making resistance must depend on the dimensions, and
its proper expression may perhaps involve terms in both v*
and v' t as already suggested in 14.
27. APPLICATION OF THE LAW OF COMPARISON.
CASE I. Denote two similar ships by S l and S t . If the
law of comparison is assumed to apply to the entire resist-
ance, we have to form simple proportions as expressed by
any of the various forms in 26 (7) and (8). The given in-
formation must necessarily involve three terms:
(1) A given ship 5, at a given speed;
(2) The resistance R t at this speed ;
(3) A similar ship 5 2 at a corresponding speed.
The fourth term, R 9 , is then readily found from the pro-
portions in 26 (8).
Let Fig. 45 be a graphical representation of the resist-
ance of a ship at varying speeds. We now wish to show that
the same curve may also be considered as a diagram of resist-
ance for all similar ships, the speed and resistance scales
being suitably modified.
To this end we see that any ordinate as AB gives the
resistance R, for the speed v = OA. Suppose now we have
a ship A times as large in linear dimension. Then at a speed
\/\v the resistance will be KR^. Hence if the scales are so
148
RESISTANCE AND PROPULSION OF SHIPS.
changed that OA to the new scale denotes ^/Az>, and AB
denotes A 3 ^, the one will properly correspond to the other;
and the same relation holding for other points, we shall have
to the new scales a diagram of the resistance of the new ship
o A v
FIG. 45.
at varying speeds. We note that to effect this change the
linear amount representing unit speed must be divided by A*,
and the linear amount representing unit of resistance must
be divided by A 8 . A curve of this character, therefore, by
the proper treatment of the scales will represent graphically
for this type or family, the resistance of any ship at any
speed, within the limits determined by the original data and
by the modification of the scales.
CASE II. Let OP lt Fig. 46, denote the curve of total
resistance for a given ship S 1 at varying speeds. Let the
skin-resistance be computed according to the methods of 8,
and set off from OP 1 downward at the various speeds as CB
at A. This will give a curve OQ as the graphical representa-
tion of the residual resistance, while the intercept between
OQ and OP, will show the value of the skin-resistance. The
RESISTANCE
149
residual resistance as given by OQ is then treated directly by
the law of comparison, either by proportion, or by a change
of scales, as just explained. On the latter supposition OQ
may also be considered as representing to appropriate scales
FIG. 46.
the residual resistance of the second ship S, at varying
speeds. The residual resistance of 5, being thus found, the
frictional resistance is computed as in 8 by an appropriate
selection of the constants. The two being added the entire
resistance is thus determined. For graphically representing
the entire resistance of 5, we may proceed as follows: The
values of the skin-resistance of 5 2 having been computed,
find ordinates for its representation to a scale /I 3 times as
large as that used for S^ Lay these off from OQ, using a
speed scale j/A times as large as that for 5,. The result will
be a curve OP t , representing to these changed scales the total
resistance of 5,. If instead of the ship 5, we have a model,
the principles and operations are exactly similar, and we may
thus consider the above as a general explanation of the
150 RESISTANCE AND PROPULSION OF SHIPS.
method of connecting the results of a model experiment with
the ship which it represents.
28. GENERAL REMARKS ON THE THEORIES OF RESISTANCE.
From the preceding sections treating on the resistance of
ships, it is quite evident that pure theory is quite unable to
furnish the indications necessary to the solution of any given
problem. The true function of theory as at present existing
must be considered as that of providing a means to a sys-
tematic study of the phenomena attending the motion of a
ship-formed body through the water, and of establishing the
basis for an intelligent comparison of the results of one vessel
with those of another, in order that experimental data may
be made available for purposes of prediction and design.
It may be here remarked that the so-called " form of
least resistance " in its abstract sense has for the designer
only an indirect interest. It is of course true for any one
condition of displacement, character of surface, state of the
liquid, and speed, that there must be some form of least
resistance, or at least some form than which none can offer
less resistance. Such form, however, will doubtless change
with every change in the four fundamental characteristics
above, and with the question of resistance must always be
associated those of safety, strength, and carrying capacity,
and often adaptation to special conditions. The purpose is
not therefore in any given case to produce a form of least
resistance as such, but rather a form which shall the most
economically combine the several qualifications in the par-
ticular proportion called for by the circumstances of the
problem.
RESISTANCE. !-!
29. ACTUAL FORMULA FOR RESISTANCE.
Of the many formulae which have been proposed for the
direct computation of resistance we shall mention but few.
In practical application it is usually power rather than resist-
ance which is computed, though with the necessary assump-
tions, of course, the one may be derived from the other. In
Chapter V the question of power will be discussed, and
various additional formulae and methods will be given for its
determination, from which, by inverse processes, the resist-
ance might be determined if desired.
We have first
where D is displacement in tons;
v is speed in knots:
R is resistance in tons;
a and k are two variable coefficients the coefficient
of propulsion and the admiralty coefficient.
These coefficients are explained in 46 and 62, and,
as is readily seen, the formula is simply derived from the
so-called admiralty formula for power, and except where used
with the law of comparison, as explained in 62, its results
are only to be considered as roughly approximate.
Middendorf* has deduced from numerous experiments on
screw-steamers the following formula, of which the first term
expresses the residual and the second term the skin resist-
ance:
R =*
Where the surface is quite smooth the exponent of v in
the second term may be made 1.85 instead of 2.
* Biisley, Die Schiffsmaschine, 1886, vol. II. p. 579.
152
RESISTANCE AND PROPULSION OF SHIPS.
The units are pounds, feet, and knots. B is beam; Z,
length ; M, area of midship section ; S, wetted surface ; while
e is a coefficient to be taken from the following table, using
the prismatic coefficient as argument:
f
.70 and under. .
e
2
P
.81
e
1.50
71
I.QQ
.82
1.42
72
1. 08
.83 .
1.32
71
1. 06
.84.,
1.18
7A
I.Q3
.85.,
i. 06
7S
1. 80
.86
.QO
76
' Z7
i.8<;
.87 .
74
77
1.81
.88 .. . ,
55
78
1.7^
.80..
.31
7Q
I 60
.QO. .
02
.80..
. wy
1.62
Biisley considers that this formula may be expected to
give good results except for very long, fine ships, in which
case the values are somewhat too large.
Taylor* suggests for speeds up to that for which v* in
knots divided by the wave-making length is not more than
about 1.2, the following:
R
(3)
where the first term gives the skin resistance and the second the
residual or wave-making. The units are pounds for R, tons for
Z), and feet and knots for the other quantities. D is displace-
ment; L, length; s, surface; /, the coefficient of skin-resistance as
discussed in 8, 9; and b the block coefficient.
* Transactions Society of Naval Architects and Marine Engineers, vol. n. p. 143.
RESISTANCE. !^
Regarding the application of this formula, Johns states*
that where b varies from .60 to .65, R is generally correct for a
speed such that v 2 + L is about .85. Below this the results are
generally about 7.5 per cent larger than experimental results,
while above this they are somewhat smaller. When b varies
from .50 to .55, R is generally correct where v 2 + L = i. Above
this the calculated values are too small, and below they are gener-
ally some 10 per cent too large.
Reference may also be made at this point to diagrammatic
representation of resistance relations by both Johns f and Taylor {
by means of which the results of experimental investigation may
be more accurately and satisfactorily shown than by formula.
In this connection we may also mention Calvert's formula for
the resistance of sections of propeller-blades or bodies of similar
form moving at a slight obliquity to the face.
Let v = speed in feet per second;
B = breadth in feet;
D = depth in feet;
8 = angle of inclination;
m = an exponent depending on 6, as indicated by the
following table :
6
m
5 20
io 35
20. 50
30 6o
40 69
m
5o u 77
60 84
70 90
80 96
90 i. oo
We then have
R=6V l * s B m Dsme (4)
* Transactions Institute of Naval Architects, 1907, p. 181.
f Ibid. % Marine Engineering, July, 1905.
Transactions Institute of Naval Architects, vol. xxvui. p. 303.
154 RESISTANCE AND PROPULSION OF SHIPS.
30. EXPERIMENTAL METHODS OF DETERMINING THE RESIST-
ANCE OF SHIP- FORMED BODIES.
Experiments on resistance have been made in a few cases
on actual vessels. For the most part, however, such experi-
ments have been limited to models usually from 10 to 20 feet in
length.
Where the actual vessels have been used, they have been
towed either from a boom or off the quarter of the towing
ship, in order to bring them clear of the wave system formed
by the latter. The tow-rope strain was then measured by a
suitable dynamometer and the speed by specially constructed
and carefully rated propeller logs. In this way tests were
made by Wm. Froude * on the Greyhound at a time when
tests of models were generally considered of little value.
The comparison of the data thus derived with that given by
a model of the same ship reduced in the ratio I : 16 showed
a very satisfactory agreement throughout, and did much to
establish confidence in model experiments, the value of which
has come to be accepted by the leading maritime nations of the
world.
A later comparison of the resistance of an actual vessel
with that of the model was made by Yarrow in 1883 f
between a torpedo-boat and its model. The range of speeds
covered by the comparison was up to 15 knots. Here also
there was virtual agreement, the actual values being some
3 per cent greater than those determined by means of the
model.
* Naval Science, vol. in. p. 240.
f Transactions Institute of Naval Architects, vol. 24, p. in.
RESISTANCE. ^5
These and other results, as well as the general agreement
(when special disturbing causes are eliminated) found between
predictions based on model experiments and actual results,
have all led to a high degree of confidence in the value and
reliability of model experiments for the determination of the
resistance to be expected in full-sized ships.
The models may be made either of paraffine or wood. In
the former case they are shaped on a special machine in which
the block cast roughly to shape moves under rapidly-revolving
cutters which are moved by a pantagraphic connection with a
tracing-point, the latter being carried by the operator along the
water-lines of the vessel whose form is to be reproduced. In
this way successive water-lines are traced out at the proper suc-
cessive heights on the surface of the future model, the cross-
sections at the end of the operation resembling series of steps,
the inner corners of which are on the surface desired. The
outer corners are then worked off smoothly by hand-tools until
the inner angle of the path cut by the tool barely remains. The
surface is then smoothed and polished, and is ready for the trials.
At the United States experimental tank at Washington the
models are made of wood. A so-called " former " model is first
made by sawing out sections from the body plan, these being
clamped in place, covered by thin strips, and then smoothed
down. A suitable block of wood, made by gluing up lifts of
pine, is then placed, together with the " former " model, on a
special cutting machine. This machine is provided with saws,
cutters, sanders, etc., so connected by pantagraphic means with
a guide which is moved over the surface of the " former " model
that the form of the latter is reproduced in the block of wood.
After completion as far as possible on the machine, the ends are
156 RESISTANCE AND PROPULSION OF SHIPS.
hand finished and the model, after painting and varnishing, is
ready for use.
The tank at Washington consists of a canal or basin some
470 feet long, 42 feet wide, and 14 feet deep. A truck running
on rails spans this canal and carries the necessary apparatus for
holding the model and measuring the pull at the various speeds
attained. The latter are measured by electrical contacts at
known distances on the track, and also through the revolutions of
the truck-wheels. All the elements of the entire determination,
including seconds of time, dynamometer pulls, changes of trim
and draft, are automatically registered on a drum driven by con-
nection with the truck-wheels. The models are taken hold
of in such way that they are free to assume whatever draft
and trim may suit the conditions from moment to moment.
If desired also, special measurements by photography and
otherwise may be made of the wave systems produced.
For investigating the effect of the propeller on resistance
a special truck is provided carrying the propeller connected
with driving-power in such way that it may be driven at any
desired .number of revolutions per minute. The propeller
thus driven and carried entirely independent of the model is
then brought up into its proper position astern of the model,
and the revolutions adjusted until the total thrust developed
is equal to the modified resistance of the model. The con-
ditions are then similar to those which would exist were the
propeller by means of this thrust driving the model at the
same speed. A comparison of the resistance thus found with
its value without the propeller shows the increase due to the
presence of the latter.
With the same apparatus the performance of model
propellers may be investigated, and thus most valuable infor-
RESISTANCE. 157
rnation bearing on the design of full-sized propellers may be
determined. To these matters we shall refer later in 49.
31. OBLIQUE RESISTANCE.
Euler's Theory. Let the resistance for motion longi-
tudinally ahead be denoted by R l A t v*, and that for motion
transversely by R^ = AJJ*. Then if the actual motion of a
vessel is with a velocity u in a direction making an angle 6
with the longitudinal, the components of the velocity lon-
gitudinally and transversely will be u cos 6 and u sin 6.
Euler then assumed that the corresponding components of
the total resistance will be Arf cos a and Aj? sin 2 6, the
resulting total resistance being
cos 4 + ,4,'sm 4 0, . . . (i)
and the tangent of its direction with the longitudinal
A, sin' A,
These equations can only be considered as giving a rough
approximation to the actual values. They serve, however,
to introduce the questions of principal interest which are the
direction and point of application of this total force R.
These have relation to stability of route when the vessel is
moved obliquely as by a tow-line. In Fig. 47, for a certain
direction OV and speed u, let TS be the direction of R, and
hence ST the direction of the applied towing force. Let M
be the location of the point of application of the resistance.
The tow-line must be attached somewhere in the line ST.
If the point of application is toward T from M, it is readily
158
RESISTANCE AND PROPULSION OF SHIPS.
seen that the couple developed by any slight yaw or deviation
from the given direction OF would give rise to a moment
FIG. 47.
tending to return the ship to this direction, and to the fulfil-
ment of the conditions originally assumed. Under these
RESISTANCE 159
circumstances, then, there will be stability of route. If, on the
other hand, the tow-line were attached beyond M toward 5,
the moment due to a yaw would tend to still further increase
the divergence, and there would hence be instability of route,
with a tendency on the part of the vessel to swing about and
go with the port side forward. For each value of the direc-
tion OV there will evidently be a point M, and the collection
of these for all possible directions will give a locus, as shown
in the figure. There is very little experimental data relating
to the form and characteristics of this locus. Certain general
conclusions, however, may be reached as follows:
It is evident that the line ST must touch the locus at M.
Also, since physically there will be but one such point for
any given position of OV, it follows that ST can touch the
locus but once, and hence will be tangent to it at M. Hence
the locus will be the envelope of the series of lines ST.
From the results of Joessel relative to the location of the
centre of resistance for planes, 6, it seems reasonable to
assume that the center for longitudinal motion will be at
some point A, and that as the angle is varied it will somewhat
rapidly fall aft on either side, thus giving a cusp at this
point. Similarly there would be like cusps at B y the center
for motion astern, and at C and D, the centers for motion
nearly transversely, the exact angle depending on the amount
of difference between the fore and after bodies of the ship.
We then assume the remainder of the locus to be as indi-
cated in the diagram. If therefore any point of attachment
on the ship be taken for the tow-rope, the angle between the
latter and the keel for stability of route will be determined
by drawing a tangent from this point to the envelope.
There will be two or four such tangents in general, according
i6o
RESISTANCE AND PROPULSION OF SHIPS.
as the point is within or without the envelope. Observation
will show, however, which one is applicable to any given
case. This determines a, the angle between the keel and the
tow-rope. In regard to 6, the inclination of the direction of
motion, there seems physical reason to believe that the latter
will always lie between the keel and the direction of the tow-
rope.
This does not agree with the indications of (2) for very
small values of 6, but we may well doubt the exactness of
VALUES OF ANGLE a
FIG. 48.
this equation at the limit where one of the component veloci-
ties is very small. We may expect, therefore, that the rela-
tion between a and 6 will be somewhat as indicated in Fig.
48, where values of (a 6) and a are plotted in rectan-
gular coordinates. The value of (a 6) is (-(-) between O
and A t or for a approximately between o and 90, and ( )
for a between A and B. For a =. o, 180 and some angle
near 90, a and will coincide and (a 0) will be o, as indi-
cated. If, therefore, the information expressed by these two
RESISTANCE. !6i
diagrams is at hand, any problem involving the relation
between the point of attachment of a tow-rope, its direction
for stability of route, and the direction of the motion, is
readily solved. Unfortunately we have no such accurate
knowledge for representative cases, and in lieu the problem
must practically be solved by trial and- error. Guyou has
determined by experiment for a launch that the point A lay
about 1 1 per cent of the length forward of the center.
The character of the two diagrams above will depend on
the ship fundamentally, and in the second place on the fol-
lowing conditions:
(1) Trim. The greater the trim by the stern the greater
the difference between the forward and after bodies, and the
greater the difference between the branches AC and CB of
Fig. 47. The farther aft as a whole, relative to the ship, also,
will the envelope be located.
(2) Speed. It is found by experiment that when v in-
creases, a increases slightly. This indicates that the trans-
verse component of resistance increases more rapidly than the
longitudinal. The value of a is not therefore independent of
speed, as in Ruler's equations.
(3) Helm. The position of the rudder is capable of
changing the whole character of the envelope of Fig. 47 by
introducing an additional oblique force at the stern and by
modifying the stream-line motion. It follows that within
certain limits the relative and absolute values of 6 and a may
be varied at pleasure by the appropriate use of the helm.
The manoeuvring of ships when moored by single cable
or in a bridle, or when towed by single line or bridle, are
familiar illustrations of the practical application of these'
principles.
CHAPTER II.
PROPULSION.
32. GENERAL STATEMENT OF THE PROBLEM.
THE fundamental problem of propulsion is to find a thrust
whereby we may overcome the resistance which the ship
meets when moving through the water. Leaving aside pro-
pulsion by sails, the water about the ship presents itself as
the only medium by means of which this necessary thrust can
be developed.
Now water is a yielding medium, and the conditions are
entirely different from those encountered in pushing a boat in
shallow water by means of a pole, where a firm and unyield-
ing hold is obtained on the bottom. With water or any such
fluid medium the development of a thrust must depend
fundamentally on the utilization of its inertia and viscid
forces. We may view the production of a thrust from two
standpoints:
(a) We know that the production of a change of momen-
tum requires the action of a force and the expenditure of
energy, and that conversely the matter acted on will react on
the agent producing the change of momentum, such reaction
being in fact the resistance opposed to the change of momen-
tum. If therefore we provide an agent attached to the ship
which shall produce a change of momentum in matter of any
iiind, such change of momentum being directed astern, or at
162
PROPULSION, 163
least having a sternward component, there will result on the
agent a reaction having a forward component, such reaction
being then available as a propulsive thrust. The fundamen-
tal conditions necessary are, therefore, that a change of
momentum must be produced in matter by an agent attached
to the ship, and that such change must have a sternward
component. In the usual case water is the matter acted on,
and that in which the change of momentum is produced. A
propulsive thrust would, however, be given by a gun firing
projectiles over the stern or by a boy throwing apples or
stones in the same direction, as well as by the means found
more useful for actual propulsion. fc
We now turn to the second method of viewing a thrust.
() We have seen in Chapter I that no body can be
moved through a liquid without experiencing a resistance.
Let us then consider the pair of bodies made of the ship and
a propeller of some kind, and let us attempt to produce rela-
tive motion by so moving the latter relative to the former
that its motion shall have a sternward component. Such
motion will meet with a resistance which, as we know, is
simply the resultant of the distributed system of pressures
and tangential forces acting on the surface of the moving
body. This resistance will react along the line of relative
motion, and will therefore have a forward component, and
may hence be made to yield a propulsive thrust. It results
that the resistance of the ship and that of the propeller, both
taken along the line of motion of the former, or what is the
same thing, along the line of motion of the system as a
whole, must be equal. We shall next examine geometrically
the conditions attendant on this second mode of view.
164 RESISTANCE AND PROPULSION OF SHIPS.
33. ACTION OF A PROPULSIVE ELEMENT.
Let 55, Fig. 49, represent the ship and C the element,
the latter being simply any body so connected with the ship
that its relative motion is along a line AB with a velocity v.
The line AB is not necessarily horizontal, but may have any
FIG. 49.
direction in space, and is simply defined as making an angle B
with the longitudinal HL. As a result of this motion of C
let the ship move with a velocity u along LH. In the usual
way we determine AF = w as the direction and amount of
motion of C relative to the surrounding body of still water.
This we lay off at CK, remembering that its direction in
space will be determined by LH and AB. While the actual
motion of C is thus along CJ , let its form be such that the
resultant of the system of distributed forces acting on its
surface is along some line in space CR, making an angle y
with CJ. Denote the inclination of CR to AB by yw and to
the longitudinal by A, and denote the value of this resultant
by R. For simplicity, Fig. 49 is drawn as though all the
lines were in the horizontal plane. As noted above, such is
PROPULSION. 165
not necessarily the case, but with the angles A, /*, y, and
defined as above, the statement of the configuration is
entirely general.
The resistance to the relative motion of C along AB will
evidently be R cos #, and for the total work performed by
the propelling power we shall have
W = R cos fj. . v = Rv cos // ..... (i)
For the available thrust we shall have the longitudinal
component of R,
T = R cos A = resistance of ship ; . (2)
and for the work done on the ship,
W, = R cos A . u = Ru cos A ..... (3)
The resistance along the line of motion of C relative to
still water is R cos y, and for the work thus expended we
have
IV 9 = R cos y . w = Rw cos y ..... (4)
We must also have
or v cos // n cos /\ -f- w cos y. ... (5)
This last result may also be derived directly from the triangle
AEF.
Definition of Efficiency. In the remaining chapters of
this work we shall have frequent occasion to use the term
efficiency. In its general sense it is simply the ratio of
returns to total investment; or of the useful result received,
to the total expense necessary to its attainment. The fun-
damental definition must be carefully kept in mind, for in
i66
RESISTANCE AND PROPULSION OF SHIPS.
some cases it is not easy to say what is the useful return, or
what the total expense.
In the present case there is no ambiguity, and we have for
the efficiency
u cos
u cos
W v cos /< u cos A -\- w cos y'
(6)
We may note that the three velocties v cos ju, u cos A,
and w cos y are the projections of the velocities v, u, and w
on the direction of total resistance CR, and that the efficiency
is simply the ratio of the two projections of u and v along
this line.
As a particular case, let the foregoing equations relate to
an element of the surface of C instead of the whole body, or
otherwise we may consider the element as a small plane.
This is represented in Fig. 50. In this diagram, as before, we
FIG. 50.
assume that the lines are not necessarily in one plane. The
total force R may here be represented by its two components
PROPULSION. 167
P and Q normal and tangential to the plane. Denote the
inclination of P to the longitudinal by a and to the line AC
by ft. Then the inclinations of Q to the same lines are
go _j_ a and 90 ft, respectively. We have then for the
thrust
T = P cos a Q sin or, . . . . (7)
and for the component of R along AC
R cos ft = P cos ft + Q sin ft (8)
Hence for the useful work
W^ (P cos a Q sin a)u, .... (9)
and for the total work
W= (Pcos/3 + Q sin ft)v\ . . . . (10)
(P cos a Q sin a)u
whence e = 7-5 a , ^ . -^-. . . (u)
(P cos ft -\- Q sin ft)v
It Q = o, we have
?^ cos a u
v cos ft ~ v cos y3 sec a'
Suppose now the element supported on a smooth unyield-
ing surface as indicated by UV. Then when it has moved
relative to the ship the same distance AN as before, the ship
will have moved to A, and the element to C^ In such case
the only work expended is that on the ship. Hence e must
equal I. Hence in (12) v t a, and ft remain the same, while //
must equal v cos ft sec a. Or in other words, v cos ft sec ft
is the speed which the ship would have if C were supported
on the smooth unyielding surface. This is readily seen
l68 RESISTANCE AND PROPULSION OF S H I PS.
geometrically if the lines of Fig. 50 are considered in one
plane ; otherwise an application of spherical trigonometry is
necessary to a geometrical proof.
The difference between the velocity v cos ft sec a and the
actual velocity u is called the slip. This we denote by 5
and the ratio 5 H- v cos ft sec a. by s. Hence we have
u = v cos ft sec a S,
and =. - = i s (13)
v cos ft sec a
This is therefore the limiting value of the efficiency when the
tangential forces Q are o.
It will be noted that the results thus far developed are
entirely independent of any supposition relating to P or Q or
the laws according to which they vary; and further, that the
limiting value of e in cq (TC) is purely a geometrical
function.
Application to Typical Cases. (a) Suppose the element a
small plane normal to the direction of motion and moved in
the direction of motion, and hence normal to itself. In this
case all tangential forces on the element balance, and R is
simply the difference of the two normal forces, one on the
front and the other on the back. Referring to Fig. 49, we
have, therefore, o as the value of the angles A, //, y, and 6.
Hence T = R,
and W Rv = TV;
W l = Ru= Tu;
w = v u = S'
PROPULSION. 169
(b) Let the element be any solid symmetrical body, or at
least of such form that the total resistance R is in the direc-
tion of motion, and let the latter be longitudinal. Then in
Fig. 49 we have o for the value of A, ju, y, and 0*. Hence
the same equations apply as for (a).
(c) Let the element be a plane revolving about a trans-
verse axis attached to the boat as in a radial paddle-wheel.
If we suppose the boat propelled by a wheel consisting of a
large number of such elements, we know that in order to
obtain the necessary reaction the elements must move back-
ward somewhat relative to the water, and that if u be the
speed of the boat and v the peripheral speed of the paddles,
i' will be greater than u. If v were equal to u, it is readily
seen that the element relative to the water would describe a
cycloid, the motion being in fact exactly as though the boat
were carried on wheels of a radius equal to that of the
element, such wheels rolling without slip on a rigid plane
horizontal foundation. In the actual case the motion will be
as though the wheels were reduced in diameter in the ratio
of u to v, and likewise rolled without slipping on such
reduced diameter. As a result the element will now travel
in a curtate trochoid, as illustrated in Fig. 51. In this figure
the paddle is denoted by the heavy line, and the lower por-
tion of the path of a point P is shown by the curve PQRS.
The line of centers is at AB and the slip being taken at 20
per cent, the effective radius OO is 80 per cent of OR. The
rolling circle will then touch the line CD, which therefore
will contain the instantaneous centers of the path of the
point P. The location of the center of the wheel for succes-
sive angular positions of the element being as shown by the
numbers along AB, the corresponding instantaneous centers
I/O RESISTANCE AND PROPULSION OF SHIPS.
PROPULSION.
are located at points vertically underneath and similarly
numbered on the line CD. The location of these centers
serves to show in any position the direction of motion of the
point P relative to still water, and hence approximately the
like motion of the entire blade. Taking the blade as a
whole, the various elements lying between the inner and
outer edges will have varying values of r and hence of v, and
hence will travel in different trochoidal curves, all deter-
mined, however, by the conditions and in the manner above
described.
For this case in Fig. 50 we should have
=*;
Hence in (7) we have
T=Pcos 6 Q sin ..... (15)
The sign of sin 6 changes as the element passes through
the vertical. Since, however, before reaching the vertical
the element is inclined on one side, and after passing it the
inclination is on the other side, it follows that Q itself also
changes sign, and hence that the horizontal component of Q
is always subtractive. Hence the value of T will always be
correctly given by (15), considering 6 as always plus. We
have also in (8)
R cos /* = P.
Hence W, = (P cos B - Q sin ff)u ;
IV= Pv;
_ (P cos e Q sin 0)u
T 7 2
RESISTANCE AND PROPULSION OF SHIPS.
(d) Let the element be a plane moving as on the end of
an oar, but suppose the plane of the oar to remain always
vertical. This case is readily seen to be similar to the radial
paddle-wheel, and so far as the general equations go, those
given above in (c) apply here as well.
(e) Let the element be a plane always vertical and revolv-
ing about an axis fixed to the boat, as in certain forms of
feathering paddle-wheels ( 38). The path of the element
relative to the water will be the same as before, but we shall
have
a = o;
ft =6.
Hence in equations (7)-(i i)
T=P;
R cos /* = Pcos 0+ Q sin 0;
= (P cos (9+ Q sin 0)v\
~
(Pcos 6>+ Qsin ff)v
(/) Let the element be as in (c) and (e), but so connected
that its plane shall always pass
through the upper point of the
circle which it traverses relative to
the boat, as in certain forms of
feathering paddle-wheel. (See Fig.
52.) In this case we have
= 2'
FIG. 52.
PROPULSIOX. 173
Hence in equations (/)-(i i)
T = P cos - Q sin -;
e e
R cos p = P cos - + Q sm - ;
W,= 7, W-'= 7ez/ cos //;
/) /)
c/ "
cos - Q sin -
2 2
and ^ =
-
( P cos - + Q sin - ) v
\ 2 ix/ 2 /
Let us now compare the limiting efficiencies of these
three styles 'of paddle-wheel, elements, assuming that u and v
are the same in each case. The latter condition is necessary
in order to make a comparison possible, and is perfectly
allowable and natural, since it merely requires an adjustment
of the amount of surface to the size or resistance of the ship.
Remembering that our equations apply simply at a given
instant, we have,
for
for (e), e =
v cos e 9
for (/),
The order of excellence, so far as efficiency goes, is seen to
be (e), (/), (<:). For (/) the efficiency is constant or inde-
pendent of 0, and therefore this is "the value for all parts of
the revolution during which the element is acting, and there-
174 RESISTANCE AND PROPULSION OF SHIPS,
fore the value for this element as a whole. For other like
elements the value will vary inversely as v, and hence
inversely as their radial distances from the axis of revolution.
The resultant limiting efficiency is, of course, a mean of
these elementary values.
For (c) and (e), however, the case is different. The
efficiency for an entire stroke will be the useful work divided
by the gross work for the same period. Hence we should
have,
2Pu cos
for (<:), e = ~
SPu
for (e), e =
cos
Now P being really a function of 6 and the velocities, it
is evident that no simpler general expression can be obtained
except by making special assumptions as to the form of P.
It seems reasonable to believe, however, that under similar
conditions the order of excellence with regard to both limit-
ing and actual efficiency might be expected to remain the
same as above.
34. DEFINITIONS RELATING TO SCREW PROPELLERS.
If one line revolves about another perpendicular to it as
an axis, and at the same time move its point of contact along
this axis, then points in this revolving line will describe
helical curves, and the line as a whole or any part of it will
generate a helical surface.
A screw propeller may for our present purpose be defined
as consisting of one or more blades having on the rear or
driving side an approximately helical surface, such blades
PROPULSION. !75
being joined to a common boss or central portion through
which they receive their motion of rotation in a transverse
plane relative to the ship.
Fig. 53 shows a four-bladed right-hand propeller:
FIG. 53.
A propeller is said to be right-hand or left-hand according
as it turns with or against the hands of a watch when viewed
from aft and driving the ship ahead.
The face or driving-face of a blade is the rear face. It is
that face which acts on the water, and which in return
receives the excess of pressure which gives the driving thrust.
The back of a blade is on the forward side. Care must
be taken to avoid confusion in the use of these terms.
The leading and following edges of a blade are respectively
the forward and after edges.
176 RESISTANCE AND PROPULSION OF SHIPS.
The diameter of a propeller is the diameter of the circle
swept by the tips of the blades.
The pitch is the axial distance between two successive
convolutions of the helical surface. In an actual propeller
the term should be understood as strictly referring to a small
element of the driving-face only. With this understanding
the/z'A:// may be defined as the longitudinal distance which
the ship would be driven were such element to work on a
smooth unyielding surface, as for example the corresponding
helical surface of a fixed nut. Pitch is thus seen to be
purely a geometrical function of the propeller. Its value
may vary from point to point over the entire driving-face, or
it may be constant. In the latter case the propeller is said
to be of uniform pitch. If it increases as we go from the hub
toward the outer circumference, the pitch is said to increase
radially. If it is greater on the following than on the leading
edge, the pitch is said to increase axially. The latter mode
of variation is usually implied by the simple term increasing
or expanding pitch.
The pitch-ratio is the ratio of the pitch to the diameter, or
-pitch-ratio = pitch -r- diameter.
The area, developed area, or helicoidal area of a blade is the
actual surface of the driving-face. For the propeller it is, of
course, the sum of the areas of the blades.
The projected area is, correspondingly, the area of the pro-
jection on a transverse plane of one blade or of all the blades.
The disk area is the area of the circle swept by the tips of the
blades.
The area-ratio is the ratio of the helicoidal or developed area
to the disk area, or area-ratio = developed area -f- disk area.
The boss or hub is the central body to which the blades are all
united, and which in turn is attached to the shaft.
PROPULSION.
177
35. PROPULSIVE ACTION OF THE ELEMENT OF A SCREW
PROPELLER.
It follows from the definition in the preceding section
that a helicoidal path must lie in the surface of a cylinder, and
that the pitch is equal to the distance between two successive
convolutions of the helix measured along the same element
of the cylinder. It is shown in geometry that if the surface
of such a cylinder is developed, the helical path becomes a
straight line, the diagonal of the rectangle representing the
portion of the cylinder containing one convolution. Thus in
Fig. 54, if the cylindrical surface be cut along the line AB,
and the lower part rolled out and flattened, the helix AHB
will become the diagonal AC of a rectangle ABCD, Fig. 55.
In this rectangle AB denotes the pitch and AD the circum-
ference of the circle of diameter A D, Fig. 54. The angle
CAD, Fig. 55, is called the pitch-angle, and the ratio AB -f-
AD, Fig. 54, is called the pitch-ratio.
A radius AL moving so as to always rest on LL and
AHB, Fig. 54, at the same time keeping perpendicular to
FIG. 54- FI G 55-
LL, will then describe a helical surface such as we are now
concerned with. Let us consider a small element of this sur-
1 78 RESISTANCE AND PROPULSION OF SHIPS.
face at the radius LA. If this element acted without slip, as
already defined, then it is evident that for one revolution
the ship would be driven ahead a distance AB and the
element would travel in the helical path AHB. In the
actual case due to the yielding of the water the longitudinal
component of the motion will be AB^ instead of AB, while
the circular component will remain the same. The element
will hence actually traverse a helical path AH 1 B 1 instead of
AHB. Let this be represented in the developed diagram,
Fig. 55> by AE. Then DE-=AB^ the amount of longi-
tudinal motion, and CE BB^ the amount of slip. The
angle CAE is often termed the slip angle.
Let diameter AD, Fig. 54, = d;
radius AL " " = r;
pitch A3 " " =/;
pitch-ratio = c\
revolutions = N\
area of element = dA ;
angle CAE, Fig. 55, = 0.
Then, using the nomenclature of Fig. 50, we have
6 = 90 ;
CAD = a\
PAD = ft.
Hence a + ft = 90 ;
and 'an "=
p = 2 nr tan a ;
P P
c = ~j = = 7f tan a.
d 2r
PROPULSION.'^ jyg
We will now apply the general equations of 33 to the
element under consideration.
We have for the thrust
T= Pcos a Q sin a ........ (i)
Also, R cos }JL = P sin a -\- Q cos at; ....... (2)
W, = Tu = (P cos a Q sin d)u ; . . . . (3)
W = Rv cos jJi = (P sin a -f- Q cos a)v ; . . (4)
(P cos a Q sin a)u
~ (P sin a -f- Q cos a)v' **'
For the limiting value when Q = o, as in previous cases,
v sin a v tan a DC
u cos a u DE .
e = : -- = - = -7=^ ( I s).
We have next to derive expressions for the forces P and
Q considered as determined by the motion of the element.
We may first note that these forces will vary directly with
the density of the liquid. Inasmuch, however, as this factor
enters equally into the resistance of the propeller and that of
the ship, it is evident that so far as the relations of the two
are concerned, the design of the propeller will be independent
of the density. That is, the same propeller should drive the
ship with equal efficiency in either fresh or salt water. In
any event the density factor will be provided for by the con-
stants used to express the values of these forces.
The actual direction of motion of the element relative to
still water is determined by the angles a and 0.
Now let Fig. 55 represent to an appropriate scale the dis-
tances moved per minute instead of per revolution.
l8o RESISTANCE AND PROPULSION OF SHIPS.
Then
AD = 27rrN = v\
DC=pN = 27rrNtan or;
DEpN(i s) 27rrNtan a (i s) = u\
AC= 2nrNVi + tan* a.
Taking 5 (i) and 6 (5) as the general expression of P y
we should have
P = a dA AE sin 0.
EK
But sin = -r- and EK = CE cos a = spN cos or.
Hence P=adAAEEK.
Whence, substituting and reducing, we find
P= 4*VWV sin a Vi + (i s)* tan 2 a a dA.
Likewise from 7, taking the relative gliding, velocity of
water and element as AC, and assuming a variation as the
square of the speed, we have
Q=f~AC' l dA = 4?rVW(i + tan 2 a)/^4.
The coefficient f will depend on the length of the ele-
ment, character of surface, and angle 0.
Now let dT, dU, and dW denote for this element the
thrust, the useful, and the total work, instead of T, W lt
and W.
PROPULSION
Then from (i), (3), (4), we have
dT= 4**r*N*dA(as sin a cos a Vi + (i -- s)* tan 2 a
/sin a(i -j- tan 3 a)};
N\l s}dA(as sin' a Vi + (i - j) 1 tan 2 a
f sin a tan <* (i -f- tan a a) )
sin' a Vi + (i - *)' tan a a
+ / cos a ( i + tan' at) ).
Put sin 8 a Vi + (i sf tan 3 a = B\
sin a tan or (i -f- tan 3 a) = C\
cos (i + tan 3 a) = E.
Whence C = E tan" a.
Then the above equations become
dT = n*fN*dA(asB cot a /C cot a) ; . . ' (6)
:8^VA r3 (i - s)dA(asB - fC)\ ....(/)
^n^N^dA(asB-\-fE) (8)
Whence
./?-/<: J sB ~ c
e = ^ l -^^B+TE = (l - S} ^ (9)
~cff+*
Now put 2r = jv/ where 7 is a variable denoting the loca-
tion of the element in terms of the pitch. It is called the
182 RESISTANCE AND PROPULSION ?^ SHIPS.
diameter ratio, and is evidently the reciprocal of the pitch-
ratio c for the element in question. For the propeller as a
whole the values of c and y will be, of course, those of the
outer element. Where we wish to especially distinguish
these limiting or outer values we shall denote them by c l
and jv
We then have
dT = n*p*N*ffdA\^sB cot a - C cot a) ; . . (10)
. . . . (11)
(12)
36. PROPULSIVE ACTION OF THE ENTIRE PROPELLER.
If the expressions in the preceding section for dT, dU,
and dW could be integrated over the surface of the blade, we
should have values for the entire T, U, and W. Assuming
/ constant,, these may be represented by
T = f?p*N % y(asB cot a fC cot a)dA ; . . (i)
U = n*p*N\i - s)fy*(asB-/C}dA ; .... (2)
(3)
Also, dividing (2) by (3), we have, similar to 35, (9),
.. (4)
PROPULSION. jg^
Of the variables involved in this integration we may
remark that y, s, B, C, E, and dA are determined by the
geometry of the propeller and by the slip. They are there-
fore readily known. It is not the same, however, with a
and f. While these are called constants, there is no reason
for assuming that their values will be the same for different
elements of the surface, or for different aggregations of such
elements. In fact such indications as we have point toward
a decided variation in value with different values of the total
area, pitch-ratio, slip, thickness of blade, condition of sur-
face, and location of element.
In regard to the coefficient f it must be remembered that
it here represents not merely a surface effect, but rather the
resultant force in the direction of the plane of the driving-
face. Due to the increase of thickness from the tips inward,
we should expect a corresponding increase in the value of this
coefficient.
The coefficient for direct resistance is taken in the form a
sin 0, and the supposition of constancy in the value of a
throughout the integration virtually assumes that the normal
resistance varies directly with the surface. This, however,
can only be even approximately true up to a certain point.
We have specified in 32 the two general methods of con-
sidering the propulsive action of an element. Thus far we
have made use of the second one only. In considering the
variation in value of this coefficient a, we shall find it useful
to here briefly consider the propulsive action of the element
of the screw propeller viewed from the first standpoint. In
37 and 38 a more general discussion of the application of
this method will be found.
From this point of view the action of the element is ta
1 84 RESISTANCE AND PROPULSION OF SHIPS.
accelerate in a sternward direction the water acted on. This,
for the element, is a cylindrical shell of water of radius equal
to that of the Ibcation of the element, and of thickness
determined by its height. For the propeller it would be a
cylindrical body of water of diameter sensibly equal to that
of the propeller, and with a hollow core of diameter sensibly
equal to that of the hub. As we shall show later, the maximum
amount of acceleration depends on the slip and on the
geometrical configuration of the element or of the propeller.
If, now, we consider the element very narrow, as for example
a part of a narrow lath-like blade, the amount of acceleration
produced either by the element or by the blade as a whole
will be very much less than this geometrical maximum above
referred to. The thrust obtained will therefore be corre-
spondingly less. As we increase the width of such a blade
the factor a will at first be nearly constant for each element
of increase, so that for a time the thrust will increase directly
with the area. At length, however, as we approach a condi-
tion where the maximum acceleration and maximum thrust
are nearly attained, the increase in thrust for each additional
element of area will be less and less, or in other words, the
thrust will increase at a slower and slower rate relative to the
increase of area. We shall therefore finally reach a width of
blade such that no additional increase will add sensibly to the
thrust. It is evident that as we thus increase the area from
our very narrow initial blade the average value of a or the
thrust received per unit area will steadily decrease, at first
slowly and then more rapidly. It even seems possible that,
under certain conditions, after having passed a certain point
the increase of area might be attended by an actual decrease
in total thrust due to the increasing confusion of the streams
PROPULSION. jg 5
of water acted on. These points will be again referred to in
49. We wish now simply to indicate that this coefficient
will probably vary with the total area in the general manner
above described.
Again, the actual values of due to the influence of the
thickness of the leading edge may perhaps be quite different
from the values determined geometrically, and the law of
variation with the sine of the angle is, again, not quite exact.
It is therefore evident that the coefficient a instead of
being constant will probably vary for different locations and
configurations of the element, for different values of the slip,
and for different amounts of total area of blade. If therefore
we wished to make direct use of these expressions for T, U,
and JFas they stand, we should need a series of experimental
values of these coefficients appropriate to the circumstances
of the case in hand. There is, however, no such set of values
available, so that, even if it were desirable, no direct use can
at present be made of these expressions in the form given
above.
For most purposes, however, we are not so much concerned
with the distribution of the values of a and /as with mean values,
or rather with values of the integrals of (i), (2), (3), (4). These
integrals depend on the values of y\ and s, and on the area and
shape of the blade.
It is obvious that an experimental observation on a pro-
peller of given form, proportion, and dimensions, and operated
under a given set of conditions, will give for such propeller and
conditions the values of work, thrust, etc., and hence the values
of the integral expressions above. If, furthermore, such data
covers a series of propellers all of the same diameter and form
of blade, but of different values of pitch-ratio and area-ratio,
l86 RESISTANCE AND PROPULSION OF SHIPS.
and operated at different values of the slip ratio, then for such
series we may find the values of the integral expressions in (i),
(2), (3), and (4), and by interpolation extend such data to include
all intermediate values of the three variables involved.
The propriety of extending the use of data drawn from ex-
periments on comparatively small or model propellers to those
of full size will be considered in Chapter IV. We shall in the
meantime here refer to the typical form of the relations between
the various quantities involved, as drawn from a series of experi-
ments conducted by the author and elsewhere reported.*
The propellers used in these experiments were of the follow-
ing general proportions and dimensions:
Diameter of ah 1 propellers 12 inches
Diameter of hub 2.4 ' '
Form of blade. elliptical
Number of blades 4
Axis of blade at right angles to axis of propellers.
Pitch-ratios included: .9, i.i, 1.3, 1.5, 1.7, 1.9, 2.1.
The values of the pitch were determined by measurements
on the driving-face.
Maximum widths of blade as fraction of radius: .2, .3, .4, .5,
.6, .7, .8
Resulting area ratios: .18, .27, .36, .45, 54, 63, .72
The general character of hub and blade and the relation of
the elliptical contour to the hub are shown in Fig. 56.
The form of cross-section of blade at the root is that of a
circular segment, as shown in Fig. 57.
* Transactions Society of Naval Architects and Marine Engineers, 1906;
Publications of Carnegie Institute, No. 79.
PROPULSION.
I8 7
The maximum thickness varies with blade width in accord-
ance with the following table :
Blade Width,
Area Ratio.
Maximum Thickness.
,2
.18
. 20 inch
-3
.27
.25 "
4
.36
-30 "
-5
45
-35 "
.6
-54
-40 "
7
.63
.45 "
.8
.72
50 "
As the full number of blade areas were made for each value
of the pitch-ratio, the total number of propellers included in the
experiments was 49.
In the reduction of the observations it was found desirable
to relate all results to standard conditions represented either
FIG. 56.
FIG. 57.
by a speed of advance of 100 feet per minute, or a rotative speed
of 100 revolutions per minute.
In Figs. 58-66 are shown typical forms of various relations
as indicated, and the character of which may with advantage
be carefully studied.
As shown in Figs. 58 and 62 the general form of the curves
showing the relation between thrust and slip, or work and slip,
when reduced to a standard of 100 revolutions per minute and
l88 RESISTANCE AND PROPULSION OF SHIPS.
Constant Pitch, Area and .X
Revolutions *
Slip
FIG. 58.
Constant Pitch, Area and
Speed of Advance
Slip
FIG. 59.
Area Ratio
FIG. 60.
Constant Area and Slip
Pit/>h Ratio
FIG. 61.
Constant Pitch, Area
and Revolutions
Slip
FIG. 62.
Constant Pitch, Area
and Speed of Advance
Slip
FIG. 63.
PROPULSION.
I8 9
with area and pitch-ratio constant, shows a nearly straight line
slightly concave to the axis of slip. A special investigation
with certain selected propellers showed that as the slip is in-
creased the curves bend over toward the horizontal in a more
and more pronounced manner, and at 100 per cent slip reach
values only some 40 or 50 per cent in excess of those for 40 per
cent slip. The slope of the lines, such as those in Figs. 58 and
62, increases with area-ratio and with pitch-ratio, according to
some complex law which will be a function of these two variables.
Constant Pitch and Slip
Area P.atio
FlG. 64.
At values of the slip = o, the thrust is seen to be positive.
This is due to the influence of the rounded back, and simply
implies that for a blade with a rounded back the distribution of
the stream lines is such as to give the same result as the face
of an ideal thin blade inclined at a slightly greater angle. This
is equivalent to saying that the effect of the rounded back is to
increase the effective or virtual pitch of the propeller. See
also 42 and 52.
As shown in Figs. 59 and 63 the general form of the curve
showing the relation between these same variables when re-
duced to a standard of 100 feet speed of advance is distinctly
parabolic, increasing beyond limit as the slip is increased. The
190 RESISTANCE AND PROPULSION OF SHIPS.
reason for this difference in the character of the two sets of
curves will be readily apparent on a little reflection.
As shown in Figs. 60 and 64 the form of the curve showing
the relation between thrust and area or work and area, with
slip and pitch constant, shows a line concave to the axis of area
and in the case of thrust becoming nearly or quite parallel to
such axis for large values of the area. This is in line with the
previous discussion regarding the value of a and shows that
after reaching area-ratios of .50 to .60 but little additional thrust
can be obtained by further increase of area. Increase in the
Constant Area and Slip
Pitch Ratio
FIG. 65.
value of the work is more continuous, as we might expect, and
these two variations taken conjointly indicate a decreasing effi-
ciency for excessive values of the area-ratio.
The curves of Figs. 61 and 65 show the nature of the
relation between thrust and pitch-ratio for the former, and work
and pitch-ratio for the latter, each for constant values of the
area-ratio and slip-ratio. The former curve is concave to the
axis of pitch, showing a value of the thrust continuously increas-
ing within the limits covered by the observations, but with a
continuously decreasing rate of increase. The second curve is
nearly straight, usually with a slight convexity to the axis of
PROPULSION.
pitch, though in some cases this feature is obscure and indica-
tions of a double curvature are sometimes found. The general
slope of the lines increases with slip and with area according to
some complex law which is obviously a function of these two
characteristics.
The curve of Fig. 66 shows the typical form of the effi-
ciency curve plotted on variable slip and for constant values
of the pitch and area. Such curves show invariably a low value
of the efficiency for a low value of the slip rising to a maximum
in the neighborhood of 20 or 25 per cent, and then falling off
.10
.20 .30
Blip
FIG. 66.
.40
more gradually for increasing values of the slip. There are
general indications that a propeller of very small area-ratio
reaches its efficiency at a relatively high slip, usually 25 per cent
or over, while that of large area reaches its best value at about
20 per cent, or slightly below.
The highest values of the efficiency seem, furthermore, to be
only slightly dependent on either area or pitch-ratio, most of
the propellers under test reaching at some value of the slip an
efficiency close to 70 per cent, with in general the best results
for fairly high values of the pitch-ratio and moderate areas.
In general propellers of high pitch-ratio reach their best
192 RESISTANCE AND PROPULSION OF SHIPS.
efficiency at a somewhat higher slip than those of low pitch-
ratio.
For low values of the slip, such as 10 to 15 per cent and low
pitch-ratios, increase of area is accompanied first by increase of
efficiency and then by some falling off, while for higher values
of the pitch-ratio and low slip there is a general improvement
of the efficiency up to some value at which it remains nearly
stationary.
For high values of the slip there is, in general, a continuous,
though slow, decrease in efficiency with area through the range
under investigation.
Intermediate between these extremes we find for moderate
values of the slip but slight change of efficiency with area.
For propellers lying within the more common ranges of pitch-
ratio and area and operating at values of the slip between 20
and 30 per cent, it appears that the efficiency will vary but
slightly over the mid-field range thus indicated, and that any
proportions corresponding to operation within these general
limits may be freely chosen without fear of any important drop
in efficiency below the limits of 68 or 70 per cent.
The highest values of the efficiency realized were about .73.
Other experimenters have found values slightly higher,* but it
will be safe to assume from .70 to .73, or .75, as the highest
general range which values of the efficiency are likely to show.
It should be noted that comparisons of experimental results
made with propellers of different forms of blade cross-section,
or with different proportions and arrangement of thickness, indi-
cate a surprisingly large influence due to thickness, especially
in shifting the relation of thrust, work, and efficiency relative to
* Transactions Society Naval Architects and Marine Engineers, 1905, 1906.
PROPULSION. I9 3
slip-ratio. In the propellers upon which the above general con-
clusions are more directly based, the blade thickness was dis-
tributed as above stated, and it must be understood that the
conclusions as a whole only apply to this particular series of
model propellers. It should be stated, however, that along
general lines these conclusions are in accord with a large amount
of other experimental data derived from model propellers of
different sizes, shapes, and forms of blade, and with different
proportions and arrangements of blade thickness.
37. ACTION OF A SCREW PROPELLER VIEWED FROM THE
.STANDPOINT OF THE WATER ACTED ON AND THE
ACCELERATION IMPARTED.
In the treatment of the screw propeller in 35, 36,
we have preferred the method there followed to that which
involves a consideration of the amount of water acted on and
its acceleration, because the former seems more fruitful prac-
tically, and to furnish a more valuable means of relating
theory to practice than the latter. The latter method, how-
ever, has chiefly attracted the attention of mathematicians
and theoretical investigators, and it may be well to consider
briefly some of the leading features of the theory from this
point of view, especially as they are of general interest as well.
The water acted on must necessarily travel in helical
paths. That is, its velocity will have a longitudinal and a
rotary component. The longitudinal component is that
which directly yields the reaction from which comes the
thrust. The rotary component as well as the longitudinal
absorbs work from the propelling agent. Its existence also
194 RESISTANCE AND PROPULSION OF SHIPS.
gives rise to centrifugal force, due to which the pressure is
less in the interior of the column than at its outer boundaries.
The water in coming aft into the zone of influence of the
propeller gradually has its velocity increased, and a certain
amount of acceleration is therefore received before the water
reaches the propeller itself. Mr. R. E. Froude * has shown
in the limiting case where no rotation is produced, and where
the propeller blades are so narrow that little or no sensible
acceleration can be received in the time taken to pass through
the propeller itself, that one half the acceleration will be
received forward of the propeller and one half in the race aft
of the propeller.
With rotation, however, we shall have in and aft of the
propeller the defect of pressure in the interior of the column
due to centrifugal force. In the actual case, therefore,
somewhat more than one half of the acceleration will be pro-
duced forward of the propeller, a certain amount while pass-
ing through, and the remainder aft of the propeller.
It is also evident that the higher the pitch-ratio the more
the rotation, and vice versa. Hence with propellers of high
pitch-ratio we may expect that most of the acceleration will
be received forward of the propeller, while with low values
of pitch-ratio the proportion there received may be but little
more than one half the total. The column as a whole must,
for continuity of flow, decrease in cross-section as it passes
through the propeller, and to such distance aft as the velocity
increases.
If the amount of water acted on could be known and the
total acceleration produced, the thrust could be immediately
determined, and knowing the speed, the useful work would
follow. The total work would be the sum of the useful, plus
* Transactions Institute Navai Architect*, voi. xxx. p 390.
PROPULSION. IQ5
the work represented by the kinetic energy imparted to the
water. The latter representing the waste is seen to depend
on the amount of water acted on and on the initial and final
velocities. It is also clear that we may divide the energy
spent in the wake into three parts, as follows: (i) that due
to longitudinal change of velocity; (2) that due to circular
change of velocity or rotation; (3) that due to turbulence and
eddies. The portion (i) is of course necessary to the produc-
tion of a useful thrust, while (2) and (3), though not funda-
mentally necessary, cannot be entirely avoided. The energy
involved in the race rotation has been made the subject of
examination by R. E. Froude,* who shows by certain ideal limit-
ing cases that its influence on efficiency will be practically negligi-
ble except with large values of the slip and high pitch-ratios.
If in a simple ideal case M denotes the amount of water
acted on and v the longitudinal change of velocity, 42, (4),
then the thrust will be proportional to Mv and the energy
involved to Mv*. Now for a given value of Mv it is readily
seen that Mv 1 will be a minimum when M is a maximum and
v a minimum. That is, for a given thrust the energy due to
longitudinal acceleration is least when the mass acted on is
large and the acceleration is small. For high efficiency, there-
fore, this consideration points to the use of large propellers
acting at small slip rather than the reverse. This corresponds
* n 35 to the result consequent on the assumption of the
decrease or absence of tangential forces. Actually the exist-
ence of race rotation, eddies and turbulence due to skin-resist-
ance, etc., as well as structural considerations, limit the size
and slip for the best efficiency as already shown in 36.
Compare also 49.
It is thus possible to discuss the action of the screw pro-
* Transactions Institute of Naval Architects, vol. xxxni. p. 265.
196 RESISTANCE AND PROPULSION OF SHIPS.
peller from the standpoint of the present section. In the actual
case, however, none of the terms here involved can be exactly
related to known quantities, so that, as in the other mode of treat-
ment, we are thrown upon experiment for the ultimate control of
whatever formulae may be proposed. We give, however, in 38,
as an illustration, the treatment of the paddle-wheel from this
point of view.*
38. THE PADDLE-WHEEL TREATED FROM THE STANDPOINT
OF 37.
The paddle-wheel may be investigated in the same
general way as the screw propeller in 35, 36. The
lesser relative importance of this instrument of propulsion
does not, however, seem to justify the necessary work, and
furthermore we have no such experimental data for the final
control of our equations as with the screw propeller. We
will therefore consider it briefly from the standpoint men-
tioned in 32 and referred to more especially in 37.
The thrust is measured by the sternward momentum of
the water imparted by means of the paddle.
Let z^ be the initial mean velocity of the water in f. s. ;
v^ be the final mean velocity of the water in f. s. ;
A be the cross-sectional area of the stream affected ;
u be the velocity of the ship in f. s. ;
* For the more important details of the various theories which have
been developed along the lines here referred to, reference may be made to
the following:
" The Mechanical Principles of the Action of Propellers." By Prof.
Rankine. Transactions Institute of Naval Architects, vol. vi. p. 13.
" The Minimum Area of Blade in a Screw Propeller necessary to Form
a Complete Column." By Prof. J. H. Cotterill. Ibid., vol. xx. p. 152.
"The Part played in the Operation of Propulsion by Differences in
Fluid Pressure." By R. E. Froude. Ibid., vol. xxx. p. 390.
"The Theoretical Effect of the Race Rotation on Screw-Propeller Effi-
ciency." By R. E. Froude. Ibid., vol. xxxin. p. 265.
"Marine Propellers." By S. W. Barnaby. London and New York.
PROPULSION. 197
D be the diameter of the circle described by the center of
pressure of the paddles (this point may be taken as
sensibly at the center of the paddle radially) ;
N be the number of revolutions per minute. '
Then nDNA is the volume acted on per minute and for
sea- water: 6^.^nDNA -f- 60 is the mass in pounds acted on
per second.
v t z> o is the change in velocity per second. Hence
64.4nDN'A(v l nDNA
thrust = T = 60X32.3 = -*T (V ' - ^
The thrust equals the resistance, and hence, solving for A,
we have
A
~
In this expression the value of (^ z> ) is uncertain.
Most writers, following Rankine, have considered that
z/, v = slip of propeller or paddle-wheel = s? .
This is known to be quite inexact, but we may use it for the
present illustrative purpose. We have, therefore,
60 X
:
Or we may put more generally and more safely
r>
And likewise, area of one paddle ~ A. Hence if we
denote the area of one paddle by #, we may put
r>
198 RESISTANCE AND PROPULSION OF SHIPS.
For purposes of design we may preferably put the equa-
tion in another form, as follows:
We have R~a(DNJs\ ...... (3)
. Useful work or Ru ~ a(DN)*s(i s). . . (5)
Also Ru ~ total work or I.H.P.
Denoting this by /, we have
I ~ a(jDN?s(i s), ..... (6)
By comparison with paddle-wheels which have performed
well, constants may be found thus relating the area of blade
to known and to previously assumed quantities.
For radial paddles the slip s may be taken from 20 to 30
per cent, and for feathering paddles from 15 to 20 per cent.
We have therefore
a = K
(D'NJs(i - s)
The value of K will depend on the units chosen for )',
N f , and /. For numerical convenience we may take these as
follows:
/= I.H.P. absorbed by one wheel;
D' = (diam. of wheel in feet) -r- 10 = D -f- IO;
N r = (revolutions per minute) -~ 10 = N ~ 10.
We may then take K from 2.0 to 2.5.
This fixes the area of one float. The number of floats
PROPULSION. 199
may be made about .SZ> for radial wheels, and from .^D to
./D for feathering wheels. The proportions of floats are
usually
Length = 3 to 4 times breadth.
The greatest immersion of the upper edge at mean draft
should be from .3 to .8 the breadth, according as the boat is
to navigate smooth or rough water and to vary little or much
in draft.
If u is in knots per hour, instead of (4) we shall have
_ itDN(i -s) _ nDN(\ - s)
~~ 6080 H- 60 : 101.3 ' ' ' ' W
Hence DN = J^L^ ....... ( IO )
To illustrate these formulae take the following data:
/= 1000 on both wheels and 500 on one wheel;
u = 1 8 knots;
s = .20 per cent, assumed.
Th-
and D'N' = 7.255.
Hence, taking K = 2.4,
2.4 X 500
= (7.255)' X.i6 = I9 ' 6 ^ ft " r S ^ 2 "I" ft '
The floats might therefore be 8' X 2'. 5.
We may now divide DN in any proportion suitable to the
circumstances of the case. Thus if we take N ' 40, we have
D = 1 8'. i ; or if we put D = 15', we have N = 48.4.
2OO
RESISTANCE AND PROPULSION OF SHIPS.
Kinematic Arrangement of Feathering Paddles In Figs.
67 and 68 are shown the usual kinematic arrangements for
FIG. 68.
feathering paddles. In Fig. 67 the links similar to AB are
all pivoted at A. This link actuates the lever BC and blade
PROPULSION. 201
DE as shown, and similarly for the others. In Fig. 68 HB
is a drive-link actuated by an eccentric with center at A.
The remainder of the links are pivoted to the eccentric strap
as shown, and are connected to blades similar to DE, as in
Fig. 67. In both diagrams the plane of the blade may be
beyond the pivot C as shown, or it may contain C, or it may
lie between B and C. The arrangement of Fig. 68 is more
commonly used, that of Fig. 67 being occasionally employed
for small wheels.
39. HYDRAULIC PROPULSION.
In this mode of propulsion the water is drawn into pumps
or other similarly acting propelling agents situated within the
boat. It is then acted on by the pump-vanes and delivered
with an increased velocity toward the stern. The necessary
reaction is thus obtained and the boat is moved. This mode
of propulsion from its slight use does not merit any extended
consideration. We may, however, instructively glance briefly
at certain features.
Let M be the water acted on per second in pounds. For
simplicity we may assume the water when first brought under
the influence of the propelling agent to be at rest relative to
the surrounding still water. Let v l be its velocity sternward
relative to the same datum when delivered by the propelling
apparatus. The change of velocity per second is then z/,.
The change of momentum which will equal the thrust will be
therefore
202 RESISTANCE AND PROPULSION OF SHIPS.
The useful work is therefore
M
W. = Tu = v,u.
g
The total work will be Tu plus that involved in the accel-
eration and disturbance of the water, and possibly in raising
it against gravity. If for simplicity we neglect the loss due
to eddies and disturbance and suppose the water discharged
under the same average head as that under which it enters,
we shall have for the total work
Hence e =
This is sometimes called Rankine's ideal efficiency, since
it is the limiting value of the efficiency under the suppositions
made. In the actual case the additional losses due to eddies,
Mv?
etc., will raise the waste work to - or more, and decrease
g
the efficiency to
e = - - or less.
u-\-v^
Now for anything approaching moderate or high speeds
the necessary thrust will require .correspondingly large values
of M and v lt or of both. The first means heavy machinery
and the second low efficiency. In any practical case the
value of v l is so large as to reduce the efficiency below that
of a properly installed screw propeller or paddle-wheel.
Ideal conditions for e would require small v lt and hence large
PROPULSION.
203
M\ but these are impossible to realize in practice, and the
efficiency is therefore necessarily small.
For special purposes where moderate speed is sufficient,
where simplicity is an important element, or where an ordi-
nary propeller might be liable to race violently or become
fouled, or where great manoeuvring power is required as in a
steam life-boat, this mode of propulsion may have advantages
which will justify its use. The manoeuvring power is usually
obtained by multiple-discharge nozzles so situated and so
adjustable that the reaction may be directed in any line, and
the boat propelled in either direction or transversely, or
turned in either direction as on a pivot.
40. SCREW TURBINES OR SCREW PROPELLERS WITH
GUIDE-BLADES.
Thornycroft's screw turbine may be taken as an illustra-
tive example of this type of propelling agent. The propeller
proper consists of a hub ABC, Fig. 69, with blades ADEB
very much longer axially and shorter radially than the
common propeller-blade. In consequence the inclination
of DE to the axis is slight. Aft of this is a projection
FGHKLMN fixed to the ship. This consists of a prolonga-
tion of the boss by the spindle-formed body FGH, a cylin-
drical shell or casing KLMN, and a series of blades connecting
204 RESISTANCE AND PROPULSION OF SHIPS.
this casing to the spindle-formed body as indicated at KG
and HN. These blades have a slight inclination in the
opposite direction to that of the propeller-blades ADEB.
The pitch of the latter blades is variable. The product of
the revolutions by the pitch at is intended to be equal to
the supposed velocity of feed, or the velocity of the stern of
the boat through the wake unaffected by the action of the
propeller. The product of the revolutions by the pitch at D
is likewise intended to be equal to the speed of delivery, and
the increase of size in ABC and consequent reduction in area
of stream are intended to be such as to give for this variable
velocity a steady flow from BE to DA. The water on leav-
ing the blades of the propeller has a considerable rotary
component, due to which it impinges on the fixed blades
KG. By their guidance it is brought gradually to an axial
direction, and thus finally delivered. The long projection
FGH is for the purpose of providing for something approach-
ing steady flow after leaving the guide-blades, and to insure
the gradual union of the various streams with each other and
with the outside water.
From the character of the actual wake as explained in
41, it is quite certain that while in a general way the inten-
tion of the design as to the distribution of pitch and sectional
area of stream to suit the velocities of feed and discharge may
be partly realized, yet such result must be far from exact, and
it is doubtful if such variation of pitch can have much influ-
ence on the actual result.
Remembering the relation of efficiency to the character-
istics of a propeller, it is evident that there is no reason for
expecting any marked difference in this feature between this
form and the common propeller. If, then, there is any in-
PROPULSION. 205
crease in ultimate efficiency it will probably be found in the
relation of the propelling agent to the ship. Either the
augment of resistance due to the propeller must be less or the
wake factor greater ( 44), or both. There seems to be no
reason for assuming any essential difference in the wake
factor, but it may be fairly expected that due to the pressure
on the blades KG, the normal direction of which will have
a component forward, there may be some return for the
increase in hull-resistance due to the action of the propeller.
On the other hand the additional attachment furnishes a con-
siderable increase of resistance, and hence it is doubtful if the
net resistance is much less than with a propeller of the usual
form. We should therefore expect about the same ultimate
efficiency. Such conclusio is are borne out by experience.
The presence of the surrounding casing does, however,
prevent the indraught of air and the radial escape of the
water, and thus increases the resistance of the blades to
revolution, and hence the thrust. It results that the requisite
thrust may be obtained from a propeller of smaller diameter
than when of the usual form, especially if the draft is small
and the tips of the blades of a common propeller would come
near the surface of the water. In such cases there seems to
be a field of usefulness for propellers of this character and
many of the Thornycroft type have been installed with very
satisfactory results.
CHAPTER III.
REACTION BETWEEN SHIP AND PROPELLER.
41. THE CONSTITUTION OF THE WAKE.
IN our treatment of propulsion we have to this point
assumed the propelling agent to act in undisturbed water.
In the actual case this is far from ^correct, and we have now
to examine the effect of irregularly disturbed water on the
preceding results.
We will first note the cause and nature of the disturbance.
Taking the screw propeller as the typical propelling agent,
its place of action is at the stern in what is termed the wake,
The water in this immediate neighborhood is subject to the
following disturbing causes:
(i) Stream-line Motion. In 2 we have seen that the
relative motion between the water and the ship is less at the
stern than between the ship and the outlying undisturbed
water, so that relative to the latter the stream-line action
will give a motion forward. Due to the same general cause,
the direction of the relative motion of the water and ship is
aft, with an upward and inward component as it follows the
contour of the vessel. This brings its direction of flow
oblique to the plane of the propeller Due to the difference
in their location, this is more strongly marked with twin-
screws than with a single screw. The influence of this
obliquity of flow will be discussed at a later point. This part
206
REACTION BETWEEN SHIP AND PROPELLER. 207
of the wake is frequently referred to the wave-motion due to
the system of waves generated by the motion of the ship.
We prefer, however, to refer it to stream-line motion, as a
more fundamental aspect of the cause. This naturally in-
cludes all features of the wake contained in the freely-flowing
water, or in that not involved in frictional and dead-water
eddies.
(2) Skin-resistance or Frictional Wake. The nature of
the action between the skin of the ship and the water has
been discussed in /, from which it appears that the whole
surface is accompanied by a layer of turbulent water partaking
more or less of the forward movement. This motion is
greater as we approach the after end, and at the stern we
shall have a considerable mass of water moving forward rela-
tive to the surrounding body of still water.
As to the distribution of the resulting complex wake, we
have the following considerations:
Near the surface the form of the ship is relatively full and
the convergence of the stream-lines is correspondingly more
marked. The disturbance of the normal horizontal distribu-
tion of the water by the resulting wave-motion is also rela-
tively more marked near the surface. As to the velocity of
the frictional wake, there seems little reason to expect any
marked variation with depth for points near the ship's sur-
face. The velocity will, however, rapidly decrease at points
successively farther and farther from the surface.
As to the actual distribution of wake-velocity, an interest-
ing series of experiments has been carried out bv G. A.
Calvert.* Across the stern of a model 28 feet long and
representing a full-bodied cargo steamer was fitted a frame
* Institution of Naval Architects, vol. xxxiv. p. 61.
208 RESISTANCE AND PROPULSION OF SHIPS.
upon which were stretched several vertical wires extending
from the deck to some distance below the keel. These wires
carried Pitot tubes fitted with vanes and free to swing to the
direction of flow. An outrigger extending into undisturbed
water carried a similar tube. The heights recorded in these
tubes were then translated into velocities by the usual
formula, v = V2gh.
This furnished the means of determining the velocity of
the ship relative to any point in the wake and relative to still
water, and hence the velocity of the water at this point of
the wake relative to still water. The wake-velocities thus
found were expressed as percentages of the velocity relative
to still water. In the section of the wake including the
screw aperture, the boss alone of the screw being fitted, the
percentages in a vertical line near the stern-post varied from
64 to 17 per cent from the surface downward; in a horizontal
line near the surface, from 64 to 7 per cent from the stern-
post outward; and in a horizontal line near the keel, from 17
to 10 per cent from the stern-post outward. The average
amount at this section was about 19 per cent.
An attempt was next made to determine the portion of
this due to the frictional wake. To this end a thin plank of
the same length as the model was towed at the same speeds.
Similar measuring appliances being fitted, the wake speeds at
points near the surface of the plank at distances of I, 7, 14,
21, and 28 feet from the forward end were found respectively
16, 37, 45, 48, and 50 per cent of the speed of the plank.
It was also shown that approximately the wake-velocities
decreased in a geometrical progression as the distances from
the surface increased in arithmetical progression.
From various considerations, for the details of which the
REACTION BETWEEN SHIP AND PROPELLER. 2OQ
original paper may be consulted, the author concludes that of
the 19 percent average wake about 5 per cent was due to
frictional wake, 9 per cent to wave-motion, or stream-line
motion as we prefer to term it, and the remaining 5 per cent
unaccounted for otherwise is charged to the influence of
" dead water" or eddying water about the stern due to the
comparative fullness of form.
It thus appears that the wake-velocity in general is ex-
ceedingly variable throughout the cross-section of the wake
stream. Also from this and other experimental determina-
tions to be referred to at a later point, it appears that includ-
ing simply the water immediately astern of the ship, and
hence that likely to be influenced by propellers, its average
value may vary from 6 or 8 per cent to 20 or 25 per cent of
the speed of the vessel. In general it is found that the wake
value is greater as the ship is longer and fuller, and less as it
is shorter and finer. For empirical equations relating its
value to the characteristics of the ship see 50, (9) and (10).
42. DEFINITIONS OF DIFFERENT KINDS OF SLIP, OF MEAN
SLIP, AND OF MEAN PITCH.
It will be remembered that, unless otherwise stated, all
velocities are referred to the surrounding body of still water
as datum. We will first suppose the wake to be uniform.
Let u = speed of ship;
v = speed of wake if propeller were not acting and
the ship were towed at the speed u\
v l = actual speed of water at or just forward of the
propeller with latter in operation;
V2 = final speed of water beyond propeller with latter in
operation ;
210 RESISTANCE AND PROPULSION OF SHIPS.
v= pitch X revolutions = pN = speed of advance
of propeller through wake if there were no
slip.
Then u v^ = speed of advance of propeller relative to the
wake;
u v l = speed of advance of propeller relative to the
water immediately about it.
v u = apparent slip of propeller = S 1 ; . . . (i)
v (u z/ ) = (v u^-^-v^ S,-\-v Q true slip of propeller S^\
or 5, -j- V Q = true slip of propeller = S y ; (2)
v v^ = total slip of water = S 3 (3)
Usually v 9 is directed forward and v^ aft, and hence 5 3 will
be numerically the sum of v and v^.
It should be es"ecially noted that these three kinds of slip
are separate and distinct, though of course not independent.
In particular it should be noted that S y and 5 3 are not the
same.
In the simple ideal case of Fig. 55 we may readily relate
the value of 5 a to 5 3 or s^ as follows:
It is shown in mechanics that the action of a surface in
deflecting a stream omitting skin-resistance and the forma-
tion of eddies involves simply a change in direction without
change in velocity, and the total force interaction between the
stream and the surface is measured in direction and amount
by the resultant change in momentum. In that diagram the
water is considered as approaching the blade with velocity
represented in amount and direction by EA, and as leaving
with the same velocity parallel to the blade. This is repre-
sented in amount and direction by making KA = EA. Then
by the usual composition of motions the resultant change is
REACTION BETWEEN SHIP AND PROPELLER. 211
represented in direction and amount by EK. The longi-
tudinal component of this is EG, and this is therefore the
acceleration which is directly useful in providing thrust.
Now if the slip angle CAE is small we may put
EG EC cos 2 a s^pN cos 2 a = 5 2 cos 2 a. . (4)
It thus appears that the velocity ,S 8 is variable for a given
value of S 2 , increasing as a decreases, and therefore increas-
ing from the hub outward, at all points, however, being less
than 5 3 .
Effect on the Preceding due to the Irregularity of the Wake.
From (2) we have
S, = $, + .;
^0
whence s, = s, + (5)
Now remembering the actual wake as described in 41 it
appears that ? -r- v may vary from perhaps o to 50 per cent
or more. The value of s l is usually found between 10 and
20 per cent. Hence s^ may vary from say 5 or 10 per cent
to 70 per cent or perhaps even more. It thus appears that
the value of the true slip is variable over the surface of the
blade and from point to point in the revolution between the
very wide limits noted above, and these may not perhaps be
the widest extremes. It is indeed quite possible that for
certain elements the slip might be o or even negative, while
for others its value might rise possibly to nearly 100 per cent.
It is thus seen that the idea of slip for the propeller as a
whole, acting in the wake, has lost entirely the simplicity of
meaning which we were able to give to it when dealing with
a single element in undisturbed water.
212 RESISTANCE AND PROPULSION OF SHIPS.
In spite, however, of the wide range of extreme values
which it seems possible that the slip may have, the conditions
for most of the blade and for most of the revolution are such
as to give rise to a much narrower range of fluctuation, and
most of the work is undoubtedly done between a compara-
tively narrow range of variation. We are thus led to the
definition of mean slip. This admits of definition in various
ways.
(a) It might be taken as the arithmetical mean for an
entire revolution, of the various slips of the elements compos-
ing the driving-face.
(b) Since these elementary values are quite unequal in
relative importance according to their location and the value
of the slip, it might seem more fair to give to each element a
weight corresponding to the gross work absorbed by it. Thus
if w lt w^ w^ etc., denote the amount of work absorbed by
each element, and s lt J a , s 9 , etc., the corresponding values of
the slip, then
Wi*i + *S* +
mean slip = - -
(c) Instead of giving to each elementary slip a weight
proportional to its gross work, we may take its thrust or use-
ful work. Denoting the elementary thrusts by /,, /,, etc.,
we should then have
t t s. + t^s. 2ts
mean slip = = =.
An approximate method of applying the weights thus indi-
cated in (b) and (c) will be given below under the discussion
of mean pitch.
(d) In 35, (12), the total work of an element is expressed
REACTION BETWEEN SHIP AND PROPELLER. 21 3
as a function of the geometrical form of the propeller and the
slip. For the entire propeller with variable slip the total
work will be represented by the summation of such elements.
Suppose now that the slip instead of variable is constant at
a value j, all other conditions remaining as before, and that
the corresponding summation for total work W is the same
as before. Then evidently s may be considered relative to
the variable distribution of slip as a mean or equivalent
value. This is the same as defining mean slip as the slip at
which the same propeller in a uniform stream at the same
number of revolutions would absorb the same total work as
in the actual case.
(e) Instead of total work as in (</), the definition may be
similarly founded on useful work or thrust. This is the same
as defining mean slip as the slip at which the same propeller
in a uniform stream at the same number of revolutions would
give the same useful work or the same thrust as in the actual
case.
In all of our references thus far to an entire propeller we
have assumed it to be of uniform pitch on the driving-face.
On this assumption we have discussed the variability of slip
over the surface and throughout the revolution, and have
given various definitions of mean slip as above. Propellers
are frequently made, however, with pitch variable over the
driving-face, and thus is introduced another element of varia-
tion into the distribution of slip, and also the need for some
definition of the terms mean pitch and mean slip as applied to
such propellers.
For the former we may take a geometrical basis and define
mean pitch as the mean of the distributed values taken over
the driving-face. Where the pitch varies simply from the
214
RESISTANCE AND PROPULSION OF SHIPS.
leading to the following edge, the mean of the two values at
these points is frequently taken as the mean pitch instead of
the more distributed mean as above.
Instead of taking the simple mean of the distributed
values, we might perhaps more properly give to the pitch of
each element a weight proportional to the work which it
absorbs or to the thrust which it develops. Reference to
35> (8) and (6), will show that for usual values of the pitch
ratio this would be approximately realized by giving to the pitch
of each element a weight represented in the first case by the
product of its area by the cube of its radius, and in the second
case by the product of the area by the square of the radius.
The following actual case of a four-bladed model propeller may
be given as an illustration of the determination of the reduced
mean pitch in the second manner. The mean pitch across the
blades was taken at four radial distances nearly equal to .3,
.5, .7, and .9 the radius, and the area was taken as proportional
to the breadth. The work for one blade was then arranged
as follows:
Radius.
r*
Breadth.
Pitch.
r*b
r*bp
1-77"
2.97
4.07
5-26
3-13
8.82
16.56
27.67
3-30"
3-60
3-30
2.16
16.13"
15.90
15.96
15-78
10-33
31-75
54-65
59-76
166.6
504-8
872.2
943-2
156.49
2486.8
15.89
The quotient of the sum of the column r*b into the sum
r~bp gives for this blade the reduced mean pitch equal to
15.89". For the other blades the values similarly found
REACTION BETWEEN SHIP AND PROPELLER. 215
were 15.45", 15.80", and 15. 54". This gives as the final
mean for the entire propeller the value 15.67".
A value of the mean pitch being thus found from either a
simple or weighted mean, a definition of mean slip follows
thus:
Assume a propeller similar to the one given in all respects
except as to pitch, which shall be uniform and of the mean
value as above defined, and let this propeller working in a
uniform stream at the same number of revolutions absorb the
same amount of total work as the given propeller. Then the
slip at which such conditions would be fulfilled would be a
mean or equivalent slip for the given propeller. This is evi-
dently similar to (d) above for the case of uniform pitch.
We may also as in (e) take the useful work or thrust as the
basis for a similar definition of mean slip.
Instead of defining mean pitch on a purely geometrical
basis, we may give it a dynamical definition as follows:
Let the given propeller work in undisturbed water with
given revolutions and speed. Let there be a propeller of
uniform pitch t with the same diameter, area, and shape of
blades, and let it work in undisturbed water at the same
revolutions and speed. Then the pitch at which the latter
propeller would have the same turning moment, or at which
it would absorb the same work, as the first, may be consid-
ered as the equivalent mean pitch of the former. Similarly
the definition may be based on equivalent thrusts instead of
equivalent turning moments.
It may be remarked that the pitch thus determined would
doubtless vary with the revolutions, and with the relation
between revolutions and speed, so that it could not be con-
sidered as a fixed dimension of the propeller.
2l6
RESISTANCE AND PROPULSION OF SHIPS.
A dynamical definition of mean pitch having been thus
taken, the definition of mean slip would most naturally follow
according to (d) or (e) above.
In all such cases where the mean pitch or slip of a pro-
peller is based on its performance in comparison with that of
a propeller of uniform pitch, the influence of the thickness or
rounded back is virtually involved. To this we may briefly
allude.
Given a body of the cross-section of a propeller blade as
in Fig. 70, moved relative to the water in the direction of the
FIG. 70.
face BA. Then the distribution of the stream-lines is such
that relative to the pressure on AB there is an excess from
A to C and a more or less pronounced defect between B
and C. The result is in general represented by a pressure on
the face AB having its center much nearer B than A, and
hence tending to turn the blade about in the clockwise direc-
tion as here viewed. If the direction KL is axial, there will
result in such case a positive or forward thrust. That is, if
the propeller be run in undisturbed water at such revolutions
relative to the speed that the slip of the driving-face is o,
there will in general result a slight positive or forward thrust
REACTION BETWEEN SHIP AND PROPELLER. 21 J
due to this distribution of the stream-line motion, and it is
not until the slip of the driving-face is still further decreased
and made slightly negative that the fore and aft components
of the total surface forces exactly balance and give a zero
thrust. It may also be noted that this zero thrust is actually
the resultant of a component aft due to the tangential forces
or those due to skin-friction and edge-resistance, and of a
component forward due to the normal forces or those due to
the stream-line pressures. If, therefore, the former could be
reduced or eliminated the propeller would, under the condi-
tions just assumed, still show a positive thrust, and it would
require a still further increase of negative slip to reduce the
longitudinal component of the normal pressures to zero.
These results have been frequently noted experimentally, and
have been made the subject of quantitative measurement in
a series of experiments carried on by the author.*
This is what is frequently referred to by the statement
that the addition of thickness results in a virtual increase of
pitch, because if the equivalent pitch be taken as (longi-
tudinal speed for zero thrust) -f- (revolutions), the result will
be greater than that derived by the measurement of the driv-
ing-face, and the excess would be still greater could the
tangential forces be eliminated.
We have thus discussed various possible definitions of
mean pitch and mean slip in order to show the variety of
meaning which may be given to these terms, and the conse-
quent inexactness of significance attending their use without
some agreement as to the basis of definition. Our use of the
equations of 36 will virtually assume the definition of mean
slip as based on (e). That is, we shall assume, as will be
* Carnegie Institution. Publication. Number 79.
2l8 RESISTANCE AA'D PROPULSION OF SHIPS.
explained later, that for the actual variable slip may be sub-
stituted some constant equivalent value for use in the equa-
tions giving the value of the total useful work.
In regard to propellers of variable pitch it may be noted
that such design is usually intended to provide for some
variable distribution of slip over the surface, assuming the
propeller to work in a uniform stream. When, however, we
remember the great variability of the stream or wake, it is
quite evident that any attempt to secure any specified distri-
bution would be entirely futile, and that in any given case the
actual distribution will be quite different from that intended.
Hence any effects resulting from a variable pitch will be quite
accidental, and it is very doubtful if, in the present state of
our knowledge, there is anything to be gained by introducing
such a feature into our designs. These conclusions seem to
be borne out by experience, for propellers of uniform pitch
have shown themselves in practice to be equal in efficiency to
those in which the pitch is variable according to various laws,
We shall therefore pay no further attention to variable pitch
as a feature of screw propellers, but shall in all cases assume
them to be of uniform pitch over the entire driving-face.
43. INFLUENCE OF OBLIQUITY OF STREAM AND OF SHAFT
ON THE ACTION OF A SCREW PROPELLER.
In general the line of the shaft, the direction of the stream,
and the direction of advance are all different. This is illus-
trated in Fig. 71, where AC represents the direction and
speed of advance of the ship relative to still water, BC the
direction and speed of the stream relative to the same, OF
the direction of the shaft, and OA the speed and direction
of the element relative to the ship. Then by the composi-
REACTION BETWEEN SHIP AND PROPELLER. 2 19
tion of relative motion 5 it follows that BA represents the direc-
tion and speed of the stream relative to the ship, and BO that
of the stream relative to the blade or element. Let ^ denote
the velocity of the stream relative to still water as represented
by BC, and denote the angle BCA by 77. Then, as in 33,
we denote the angle between OG and the normal ON to the
element by a. The inclination of z/ to the normal is then
(a 77), and ^ cos (a tj) is the component of z/ in the
direction of the normal, and hence the velocity in this direc-
tion which would be impressed on the element by the stream
of velocity v^ if there were no slip. The result of this in the
direction of advance would be a velocity V Q cos (a rj) sec a.
From 33 the longitudinal velocity without slip in still
water would be v cos fi sec a. Hence denoting tlie total
speed of advance if there were no slip by ', we shall have
u' = v cos j3 sec a -\- v cos (a rf) sec a
= (v cos (3 -\- v cos (a rf)} sec a.
In the present case, with the shaft at an angle e with the
fine of advance, the angle j3 for a given element is constant,
220 RESISTANCE AND PROPULSION OF SHIPS.
while a varies. The exact values of a depend on the solution
of a spherical triangle, as is readily seen, and these values will
vary through a range 2e. In consequence, the values of the
speed of advance without slip, u' ', will vary through a corre-
sponding range. Denote the mean value of u' by / and the.
range of its variation by 2Au' . Also, let 6 denote the angular
location of the element reckoned from the position for which
u' is maximum. Then it is found, if e and v are relatively
small, that we have approximately
u' = u 9 ' + Au' cos 6.
The value of u' thus given is seen to vary from #/ + Au'
through #/, #/ Au' , u to #/ + Au' for the successive
quadrants of an entire revolution. The actual speed of
advance being u, it follows that the difference u' u or the
actual slip 5, and the percentage slip s may likewise be
expressed in the same approximate form. Hence we shall
have
5= S. + AScos 0;
s = s + As cos 0.
Denote the mean value of a by <*. This is seen to be
the angle between the normal and the direction of the shaft.
Then
u 9 ' = (y cos ft -\- v cos (a rf) ) sec <x ;
and S #/ u.
We must now find the value of Au' = AS. We have for
the maximum value of u' approximately
'max. = (^ COS ft + V, COS (flf -f 6 rf) ) SCC (flf + e).
Then z/ max . - it: = Ait' = AS.
REACTION BETWEEN SHIP AND PROPELLER. 221
Considering e small, ^this difference is readily put in the
following approximate form:
Au 1 = AS = u. r
cot a Q e
It thus appears that the values of AS and hence of As
vary over the entire surface of the blade, and hence the entire
distribution of 5 and s will vary from point to point over the
surface and from one position to another in the revolution.
The value of the range of slip 2AS is seen to increase with
e and with av Hence the value of AS will continuously
decrease for locations of the element from the hub outward.
The position from which 6 must be counted is seen from the
following considerations. The origin for 9 is the position for
maximum value of u' ; and since v is small, u' will depend
chiefly on v t and will reach its maximum very near the posi-
tion for which sec a is maximum or a maximum or when
a = a Q -(" e. Hence the o position of 8 is readily seen to be
near that in which the normal is parallel to the plane deter-
mined by the shaft and direction of advance, and in which
the normal and direction of advance lie on opposite sides of
the shaft.
Thus with twin screws, the starboard turning to the right
and the port to the left, let the shafts incline outward from
the engine aft, as usually fitted. Then it follows from the
above that for each blade of each propeller the slip is maxi-
mum near its highest position and minimum near its lowest.
Vice versa with so-called "inboard turning" screws, the star-
board to left and port to the right, the slip will be maximum
near the lowest position and minimum near the highest. Again,
with a right-hand propeller and the shaft inclined downward the
slip of any blade is maximum when it is horizontal and directed
222 RESISTANCE AND PROPULSION OF SHIPS.
to the right, and minimum when horizontal and directed to the
left. With right- and left-hand twin screws, as above noted,
the effect of the obliquity of the shafts is still further increased
by the natural obliquity of the inflowing streams as they follow
the contour of the ship, thus giving a direction of stream relative
to still water somewhat like EC of Fig. 71. Inclination of the
shafts in the opposite direction would tend to correct the varia-
bility of the slip due to these streams, but from structural reasons
this is rarely permissible.
We have discussed in the preceding section the influence
of the variability of the wake on slip, and in the present sec-
tion the influence of obliquity, assuming thus far a uniform
stream or wake. Combining these influences we shall evi-
dently have in any given case variations of slip over the
surface and throughout the revolution ranging through limits,
variable themselves, but probably as wide as from o or a
negative value to 50 or 75 per cent or more. We shall have
therefore the variability noted in 41, 42, with an added
amount due to obliquity. As noted in 41, however, most
of the work will be done between narrower ranges, but due
to obliquity even these may be considerable. It becomes
therefore a matter of importance to inquire to what extent
the formulae of 36 may be invalidated by the existence of
this variability of slip.
44. EFFECT OF THE WAKE AND ITS VARIABILITY ON THE
EQUATIONS OF 36.
We will first take the influence of the variability of the
wake on the efficiency.
Let etc., denote the efficiencies of the various
elements. Then using the previous nomenclature we shall
have for the entire propeller
REACTION BETWEEN SHIP AND PROPELLER. 223
U w,e, _
~~ W w v + w, + . . . JF
If now the value of e varies by a linear law with slip for
the range of variation, we shall have
e = e as ;
e 3 = e + as,.
etc.
Hence we find
U w,s, + w^ + . . .
We know that efficiency does not actually vary by a linear
law with slip, but if most of the work is done between a
relatively narrow range of variation of slip, the variation of
efficiency may be approximately expressed by such a law.
The resulting efficiency in (2) is then seen to be that corre-
sponding to the mean slip as defined in 42 (b). The actual
efficiency will be somewhat less than that given by (2), and
we may consider that this equation simply indicates the
general conditions under which no great loss of efficiency
shall result from variable slip, viz., that most of the work
must be done within a comparatively narrow range of slip
variation.
This general conclusion is borne out experimentally, and we
may consider that the assumptions commonly made regarding
the wake rest on an experimental basis. These are:
That the actual turbulent variable wake may be consid-
ered as sensibly equivalent to a single uniform wake, and that
224 RESISTANCE AND PROPULSION OF SHIPS.
the performance of the propeller may be considered as equiva-
lent to that which would result from the presence of such a
wake.
The true mean slip S^ is then greater than the apparent
slip Sj by the amount of this uniform substituted wake, which
we may denote by z/ . Vice versa it results that the amount
of the uniform substituted wake is equal to the difference
between the true mean slip based on the definition of 42,
(e), and the apparent slip S,.
Stating again the relation somewhat differently, we note
that the true slip is greater by variable amounts than the
apparent slip, and the actual thrust developed is greater than
that which would result in undisturbed water by the screw
working at the apparent slip. At some greater slip in undis-
turbed water, however, the thrust developed would be the
same as that actually obtained. This greater slip may then,
according to 42 (e) be considered as the equivalent or mean
true slip in the actual case, and the difference between the
true and apparent slips thus defined will represent the amount
of uniform wake considered as equivalent to the actual vari-
able wake.
This definition of equivalent wake is based on an equiva-
lence of thrusts, as stated in 42 (e). We might also base
a definition on an equivalence of turning moments or total
works, as stated in definition (d). If now the wake itself
were uniform these two definitions or modes of determination
would evidently lead to the same value of the wake. The
difference in the two values thus determined indicates there-
fore the effect due to the turbulence or irregularity of the
wake. Experiments on this point are not sufficiently ex-
tended to furnish very complete evidence as to the exact
REACTION BETWEEN SHIP AND PROPELLER. 22$
influence which turbulence may play, but its value is believed
to be small, and in any event in default of sufficiently
extended information we are compelled to assume its influ-
ence as negligible. The actual point where the influence of
turbulence or irregularity of wake touches our methods of
design is in its influence on efficiency, to which reference has
been made in the early part of the present section.
We therefore virtually assume in all cases, for the actual
turbulent wake, the substitution without change in efficiency,
of a uniform wake of velocity z/ , the amount of this velocity
being based on an equivalence of thrusts as previously ex-
plained. Let
v n
= w.
This relates v^ to (u -z> ), the speed of advance of the pro-
peller through the water about the stern. Whence
wu
- i+w'
From 42, (5), we have
v w u w
whence
i s.
(3)
(4)
226 RESISTANCE AND PROPULSION OF SHIPS.
Let us next consider the influence of the uniform wake
on the expressions for useful work and efficiency in 36, (2)
and (4).
It is readily seen that the useful work must be obtained
by multiplying the thrust by the speed of the ship relative to
'still water, and hence by (i s^)pN. In all other places,
however, where slip enters into these expressions it will be
the true slip s,. Hence we shall have in such case
- fC}dA ;
W = n*p*N*y\a$B + fE)dA .
The ratio of these two will give an apparent efficiency.
The true efficiency must, of course, be that for the propeller
working at a slip of s 9 . The presence of the wake cannot
change the efficiency of the propeller itself, while it may
increase the amount of useful result L\. The increase in L\
must be credited to the wake rather than to the propeller.
These points will be again referred to in 46. For these
reasons we call the ratio of 7, to Wan apparent rather than
a real propeller efficiency. This we may denote by e lt while
the true efficiency, which we will denote by * a , will have the
value as given in 36, (4). Hence we have
(5)
A6
-7' s
and ^ a = er -' = -^ (6)
3 l T ^ C T I <7ffl V /
REACTION BETWEEN SHIP AND PROPELLER. .22*]
45. AUGMENTATION OF RESISTANCE DUE TO ACTION OF
PROPELLER.
\Ye have thus far considered the influence of the ship,
through the wake which it produces, on the action of the
propeller. We now turn to the influence of the propeller on
the resistance of the ship.
\\ e have already in 37 considered the action of the pro-
peller in producing in front of itself a defect of pressure and
thus imparting a portion of the total acceleration produced,
before the water acted on reaches the propeller itself. This
defect of pressure is for the most part due to the centrifugal
force consequent upon the rotation of the race by the pro-
peller. The rotation will evidently be greater as the blades
stand more nearly fore and aft, and hence as the pitch-ratio is
higher. This defect of pressure will interfere seriously with
the natural stream-line motion. The water instead of being
able to close around the stern and form the natural stern wave
will be more or less disturbed and drawn away to the pro-
peller. The result of this is a diminution of the pressure
which would naturally exist about the stern if the ship were
towed at the same speed. The result of this is, of course,
an increase in the amount of resistance to be overcome by
the propelling agent. This increase when viewed from the
standpoint of the resistance may be termed the augment or
augmentation of resistance.
Where paddle-wheels are used as the propelling agent, a
like augmentation is experienced though it arises from some-
what different causes. The action of the paddle-wheels at
the sides and nearly amidships gives rise to a race of water
moving faster relative to the ship than would be the case if
228 RESISTANCE AND PROPULSION OF SHIPS.
the wheels were not there. The result is an increase in the
skin-resistance. The wheels also undoubtedly exercise a
disturbing influence on the natural stream-line motion, and
they may thus introduce an additional element into the
augmentation of resistance. The amount of augmentation
due to paddle-wheels has not been as extensively determined
experimentally as for screw propellers, but various experi-
ments by Dr. Tidman, R. E. Froude, and Messrs. Denny
indicate that it does not much differ in amount from that for
a screw propeller under like conditions of speed.
46. ANALYSIS OF THE POWER NECESSARY FOR PROPULSION.
Let W= the indicated horse-power;
W f = the power absorbed by the friction of the engine
and shafting, and by any attached pumps.
Then W W f = W p power delivered to propeller.
Let 7" = the necessary thrust, supposing the ship towed
at the given speed u\
T the actual thrust = T -\- the amount of aug-
mentation.
Then T u is called the effective horse-power; and Tu is called
*T*
the thrust horse-power. The ratio ~ is called the propul-
sive coefficient.
The ratio ^-, which is much more definitely related to
W t
propulsive efficiency, has unfortunately received no special
name. We shall here distinguish it as the coefficient //.
As defined in 33, the apparent propeller efficiency is
Tu
REACTION BETWEEN SHIP AND PROPELLER. 229
while as in 44 the true propeller efficiency is
T(u Q _ u v. _ e,
e *~ W p * l u ~ i + w
The efficiency e^ is the ultimate test of the performance of
the propeller considered simply as a propeller. It is given a
certain amount of,work W P . Working in undisturbed water
at the same true slip, 5, = (v u -\- z/ ), it would develop the
thrust Tj and deliver an amount of work T(v 5 a ) =
2\u v 9 ) as in the numerator of e t above.
If, however, we consider the propeller simply as a means
of getting the ship through the water, the useful work will be
T 9 u, since this is the amount which would be required to tow
the ship at the speed u. The ratio T 9 u -r- W p is therefore
the final test of the value of the propeller as a means of
actually propelling the ship. If the ship produced no wake
and the propeller no augmentation of resistance, this ratio
7> -r- W p and the ratio e^ T(u v^ -r- W p would be the
same. The relation between them, therefore, involves the
mutual interaction of ship and propeller. The ratio express-
ing this relation is hence termed the hull efficiency. We
have therefore
coefficient h
Hull efficiency =
true propeller efficiency '
or coefficient h = hull efficiency X true propeller efficiency.
Hence
TT 11 re T * u T ( u ~ *0 ^o u T,
Hull efficiency == ^ - - -^- = y . ^^- - y (i + W ),
and
Coefficient h = $ ^ e, = ~\i + w)e t .
^
230 RESISTANCE AND PROPULSION OF SHIPS.
The hull efficiency is thus seen to be the product of two
factors T -v- T and u -f- (it v ). The first is the ratio of
the true to the augmented resistance, or the reciprocal of
the coefficient of augmentation as above defined. This factor
has been termed by R. E. Froude the " Thrust-deduction
factor." We shall, however, prefer to consider it as the
reciprocal of the coefficient of augmentation, T -i- T . The
effect of this factor, as seen, is to decrease the hull efficiency
in the ratio T -r- T. It therefore implies a loss in efficiency
due to the augmentation of resistance.
The other factor u -f- (u z' ) (i -f- /), as already
defined in 44. The effect of this factor, as seen, is to
increase the hull efficiency in the ratio (i -f- w). It therefore
implies a gain in efficiency due to the location of the propeller
in a forward wake instead of in undisturbed water. This gain
in efficiency is seen to be due simply to the gain in the speed
of advance. That is, if the propeller were working in undis-
turbed water at the same revolutions and true slip S^ and
consequent thrust T, the speed of advance would be (v 5 a ).
In the actual case, due to the fact that the wake water is
borne forward, so to speak, to meet the propeller, the same
true slip 5 a is obtained with a speed of advance #. greater than
(v ,) by the speed of the wake v . Hence the two speeds
of advance are u and u z/ , and their ratio is (i -f- w), as
defined.
Of the two factors thus constituting hull efficiency, one is
seen to be greater and the other less than I. Hence they
tend mutually to offset each other, and the product or the
hull efficiency will usually not vary widely from i. In fact
many determinations by R. E. Froude indicate that the hull
efficiency may usually be taken as I without sensible error,
REACTION BETWEEN SHIP AND PROPELLER. 23!
and that the causes which seem to increase one factor will
correspondingly decrease the other, and vice versa. Hence
the coefficient // may usually be taken as sensibly equal to the
true propeller efficiency *. It should not, however, be for-
gotten that the hull efficiency is by no means necessarily i,
and that circumstances might arise in which such an assump-
tion would involve a sensible error.
This analysis of the total power and the various relation-
ships involved may be illustrated by the diagram of Fig. 72.
A C C, F E
FIG. 72.
The total power, revolutions, and thrust are supposed to
remain constant. The subdivision is then shown both with
and without wake. This is as follows:
AE = I.H.P. = total power;
AB = power absorbed by friction of engine and attached
pumps;
BE = power delivered to propeller = W p \
BC = power absorbed in the eddies, rotation, and sternward
acceleration communicated to the water acted on,
assuming the existence of a wake;
BC l = power absorbed under the preceding head, assuming
that there is no wake and that the propeller works
in undisturbed water;
CE thrust horse-power = 7*, assuming the existence of a
wake ;
C^E = thrust horse-power without wake = T(u z> ) = actual
thrust T multiplied by the reduced speed (u i-j
which would correspond to the assumed power,
revolutions, and slip with no wake;
232 RESISTANCE AND PROPULSION OF SHIPS.
FE power absorbed by the augmentation of resistance.
This for convenience is assumed as the same with or
without wake;
CF = power absorbed by the true or towed resistance in the
case with wake effective horse-power;
C t F = power absorbed by the true or towed resistance in the
case without wake..
Therefore with fixed power, revolutions, thrust, and re-
sistance the wake would increase the speed from (u z/ ) to
u or in the ratio (i -f- w), and hence the useful effect in the
same ratio.
The various ratios are also illustrated as follows:
propulsive coefficient ;
CF
= coefficient h ;
BE
CE
= apparent propeller efficiency ;
Bh
= true
CE
-^p = coefficient of augmentation ;
Cr
CE u u
-=-= = = - - = (i + w) =. wake-return factor;
C^E u l u ?/
CF
= hull efficiency.
It will be noted that the diagram of Fig. 72 is not appli-
cable to the same ship with and without wake, for the power,
revolutions, thrust, and resistance are supposed to remain
constant, while the speed changes from (u v ) to u. The
REACTION BETWEEN SHIP AND PROPELLER. 233
diagram is applicable, however, to the performance of the
propeller with and without wake, under the conditions that
revolutions, thrust, and resistance shall remain constant, and
hence that the propeller at the two speeds (u v ) and u shall
be opposed to the same resistance. This is usually the most
useful mode of analysis, as it serves to connect the propeller
in its actual surroundings with the same propeller in undis-
turbed water, developing at the same revolutions and true slip
the same thrust.
We may, however, as in Fig. 73, illustrate the effect of
the wake on the same ship at constant speed, resistance, and
With
F E
Without
A, B, C, F, E,
FIG. 73.
thrust of the propeller, but varying revolutions and power.
In this diagram ABCFE denote for the ship with wake the
same points as in Fig. 72, while A 1 1 C 1 F l t denote similar
points for the same ship at the same speed and resistance,
but without wake. The power represented by CE is of
course equal to that represented by Cf.^. The propeller
power BE and the total power AE are, however, less than
the corresponding amounts B l E l and A^E^ This saving of
power comes about as follows: As shown in 36, (i), thrust
varies with revolutions and with slip. Now if there were no
wake and the slip were S lt the necessary thrust would be
developed by a certain number of revolutions. With the
wake v a and the same speed u the slip becomes increased to
S l + 7' 5,, and the revolutions necessary to develop the
fixed thrust are correspondingly decreased. Now with con-
stant thrust the turning moment or torque, and hence the
234 RESISTANCE AND PROPULSION OF SHIPS.
mean effective pressure in the cylinders, will all remain very
nearly constant. Hence the propeller horse-power W p and
the indicated horse-power W will vary very nearly as the
revolutions. Hence while the work CE will equal C^E^ the
work BE and the total work AE will be less respectively than
B^E^ and A 1 E 1 very nearly in the same ratio as that in which
the revolutions are reduced. In this way, then, it is seen that
the presence of the wake makes it possible to obtain the
necessary thrust and propulsion of the ship with a lower
number of revolutions and with a correspondingly decreased
engine-power than in undisturbed water.
The wake is due to the motion of the ship, and its kinetic
energy is simply the energy which has been put into it by the
ship in its movement through the water. The energy of the
wake comes therefore from the engine, or more ultimately
from the coal or fuel, as the source of energy. The reduc-
tion of the engine-power and fuel-consumption, as above
explained, may therefore be considered as a return from the
wake, or as a reutilization of a small part of the energy which
has been expended in its formation.
We will now give a numerical illustration of the analysis
illustrated in Fig. 72.
Let s t = .16;
s, = .27.
This would mean that the propeller in undisturbed water
with 27 per cent slip would give the same thrust as obtained
in the actual case at 16 per cent apparent slip, the revolutions
being the same in each case.
Then from 44, I + l - 1 5 I ;
w .151.
REACTION BETWEEN SHIP AND PROPELLER.
235
Let^lE, the I. H. P., be denoted by i
Let AB be 14
Then BE = W P . . . = .86
Let CE = Tu = .67
Then e l = apparent propeller efficiency. . = .779
And e. 2 true propeller efficiency = .677
Also C,E=CE-*r\-\-w = .582
Hence CC t = >o88
Let coefficient of augmentation be = 1 . 165
Then CF = E.H.P = .575
And FE = .095
Then coefficient // . . . = .669
Propulsive coefficient. = -575
Hull efficiency =
The following table shows the results of certain experi-
ments carried out on a Dutch tugboat."* During the trials
the thrusts were measured at the thrust-block by a hydraulic
dynamometer, while the values of the E.H.P. were deter-
mined from model experiments.
I.H.P.
T.H.P.
E.H.P.
Rev.
Speed.
Principal Dimensions.
3I-03
50.56
80.24
132.35
170.83
230.58
260.32
19.76
33.16
53-22
89-43
118.85
161 .40
180.29
15-80
27.42
42-74
70.69
87-75
108.46
120.22
94
in
127.
148
160.
175
180.
6.97
8.07
9.02
10.07
10.47
10.84
II. 01
Length = 72
Beam = 14
Draft forward = 3
Draft aft = 7' 4^
Displacement = 69 tons
9
10"
The ratio of T.H.P. to I.H.P. varies with increase of
speed from .64 to .69, and that of E.H.P. to I.H.P. from
* See Steamship for October, 1897.
236 RESISTANCE AND PROPULSION OF SHIPS.
.54 to .46. The pitch of the propeller was 7.63 feet, and,
'as may be determined, the slip varied from about 1.5 to 19
per- cent. The increase of propulsive coefficient at low
values of the speed and slip is somewhat abnormal, but
whether due to errors in the data or to an increased value
of the ship efficiency at these points does not appear. The
series of experiments is a valuable one, and it may be hoped
that the near future will give us many more of the same
character.
47. INDICATED THRUST.
This term, which frequently occurs in engineering litera-
ture, is defined as follows:
33000 I.H.P.
Indicated thrust in tons = - ^ . . (i)
/7V2240
This is a thrust which would correspond to an absence of
slip and a total efficiency of I. This is a wholly impossible
set of conditions, but as it is considered convenient for the
expression of certain relationships, we will show its connection
with a much better known quantity. Let C be an engine
constant defined by the equation
L.P. cylinder area X 2 X stroke in feet zLA^
C = . (2)
33000 33000 V '
Then the mean effective pressure reduced to the L.P. cylin-
der is the pressure defined by the equation
I.H.P. 33000 I.H.P.
(m.e. P .)= -ar^nxj?-- - (3 >
Now comparing indicated thrust and reduced mean effec-
tive pressure, it is readily seen that there is a constant ratio
REACTION BETWEEN SHIP AND PROPELLER. 237
between them, and hence that the one is in constant pro-
portion to the other. Hence we may remember that the
indicated thrust is simply a quantity proportional to the
reduced mean effective pressure.
We will also show here another expression for the re-
duced mean effective pressure. Let A lt A A, be the areas
of the successive cylinders of a triple-expansion engine, A,
being the value for the low-pressure cylinder. Let p iy /
and /, be the successive actual mean effective pressures in
the same cylinders. Then evidently
, (4)
The values in (3) and (4) are evidently the same. For
multiple-expansion engines of any number of stages the
same general method applies. For triple-expansion engines
with initial pressures from 160 to 180 pounds absolute, or 145
to 165 by gauge, the reduced mean effective pressure is
usually from 30 to 40 pounds per square inch.
48. NEGATIVE APPARENT SLIP.
If there were no wake, the apparent slip s l and the true
slip s t would be the same. With the development of the
wake, however, the true slip remaining the same, the appa-
rent slip decreases until some point is reached where an
equilibrium of conditions is maintained.
From 44 we have
r, = *.(i + w = s t (i s,)w.
Now if, as stated above, s, remains constant and w
238 RESISTANCE AND PROPULSION OF SHIPS.
increases, si will continuously decrease. If w could reach a
sufficient value, si would become o and then negative. The con-
dition that SI=Q is
w
or s 2 = - . . . . (i)
or
The question of the possibility or otherwise or a o or negative
value of si is then a question of the possibility or otherwise of a
wake equal to or greater than this value s 2 +(is 2 ).
When the slip is determined relative to the pitch of the driving-
face of the propeller, it is not likely that such a wake can exist.
With ideal blades consisting of a thin geometrical surface and
acting in an ideal manner, it may be shown dynamically that
such a value of the wake could not be formed. The distribu-
tion of forces on a blade with thickness, however, may produce
a considerable change in the effective pitch, as noted in 42
and 52, and it is entirely conceivable and practically demon-
strable that with such a blade a zero or even negative true slip
may exist relative to the pitch of the driving or after face of the
blade.
It is also clear since the apparent slip is less than the true,
that if the propeller is large for its work, and hence calculated
to realize the needed thrust with a moderate or small true slip,
that the apparent slip may readily approach very small values
and minor errors of observation might easily transform such a
condition into an indication of negative apparent slip.
Such a condition is not desirable or indicative of good pro-
peller efficiency, since it means a low value of the true slip and
REACTION BETWEEX SHIP AXD PROPELLER. 239
consequently a low efficiency, as shown by the type curve of
Fig. 66.
It is probable that most of the cases of negative apparent
slip have arisen from errors of measurement, especially in the
pitch. In particular might this be the case with propellers of
variable pitch, in which the definition of mean pitch is neces-
sarily arbitrary in character, according as we make it purely
geometrical, or give it a dynamical basis by giving to the pitch
of each element a weight proportional to the thrust which it
develops or the work which it absorbs. Due to this necessarily
arbitrary character of the definition of mean pitch with such
propellers, the expressions mean pitch and apparent slip lose
much of the significance which we are able to give to them with
propellers of uniform pitch. See also 42.
CHAPTER IV.
PROPELLER DESIGN.
49. CONNECTION OF MODEL EXPERIMENTS WITH ACTUAL
PROPELLERS.
WE now take up the considerations relating to the appli-
cation of experimental data derived from models to the design
of full-sized propellers.
We must remember that the actual data given by experi-
mental research relate to certain model or small-sized propellers
of known dimensions and geometrical characteristics and oper-
ating at stated revolutions, slip, and speed of advance. If we
may assume that the experiments are properly distributed in
range, as regards variation in the various factors aside from
mere size, then, as noted in Chapter II, we may by suitable
methods of analysis derive results expressed in tabular or geo-
metrical form which will represent the performance of all possible
propellers of the one fixed diameter, and of varying values of the
other characteristics and factors within the experimental range.
Thus for illustration, if the experiments cover a series of model
propellers all of the same diameter, general shape of blade,
general law of- thickness and material of blade, but varying in
pitch-ratio and area-ratio and operated at varying values of the
slip ratio, then we may derive from such experiments a general
240
PROPELLER DESIGN. 241
law, whether expressed graphically or otherwise, which shall
connect these various propellers, as well as all others of interme-
diate values of pitch-ratio or area-ratio or operated at inter-
mediate values of the slip. We have next to consider the
justice of extending 'such results to propellers of other sizes,
and more especially to full-sized propellers for actual ship pro-
pulsion.
The derivation of 36, (n), (12), shows that the whole
question depends on the supposition that the forces P and Q
vary as the area and as the square of the speed, or that the values
of K, L, and K + L = e are independent of actual dimensions
and are functions of geometrical proportion and of slip.
The general factors upon which the performance of any
propeller must depend are obviously the following:
(1) Diameter as the distinctive or characteristic dimension
determining size.
(2) Pitch-ratio.
(3) Developed area of blades or area-ratio taken in conjunc-
tion with (i).
(4) Number of blades.
(5) Shape of blade or distribution of blade-area.
(6) Thickness and the general schedule of its distribution.
(7) Dimensions and proportions of hub.
(8) Material employed.
(9) Slip ratio at which operated.
Turning now to 36, (n), it follows that the value of the
function K would in the most general case depend conjointly on
the various factors (2) -(9) inclusive, and that symbolically it
might be displaced by a function of eight different variables,
each one of which might be supposed to take care of the influence
due to the corresponding characteristic of the propeller. Our
242 RESISTANCE AND PROPULSION OF SHIPS.
present knowledge of the influences due to factors (4), (5), (6),
(7), and (8) is, however, too meager and uncertain to admit of
useful application in this manner, and we are therefore, aside
from (i), reduced to three principal factors, (2), (3), (9), re-
garding which sufficient is known to permit of use in a formula
of this character. We therefore substitute for K the symbolic
function klm and thus have for 36 (n):
U = d 2 (pN)*klm, (i)
where &= factor intended to account for influence due to slip-ratio;
/ = factor intended to account for influence due to pitch-
ratio ;
w=factor intended to account for influence due to area-
ratio.
These various factors are not independent, and in the form
of a simple product, as in (i), they cannot each represent exclu-
sively the influence of a single variable condition. The inter-
relation of these factors will further depend somewhat on the form
of analysis selected for the reduction of experimental results,
as will appear below in connection with the numerical results
given. This does not affect, however, the value of writing the
formula in this manner, since the attention is thus directed to
these factors as representing the principal conditions on which
the performance will depend, and separate or combined judg-
ments are invited regarding the influence which each may have.
Returning now to the question of extending the results of
model experiments to full-sized propellers, it is obvious that if
we may assume efficiency independent of size and the factors klm
likewise, then values of efficiency and of these factors, or of their
joint product 7?, such as may be derived from model experiments,
PROPELLER DESIGN. 243
may be immediately employed in equations such as (i) above
for application to propellers of any dimensions.
As the result of experience in such matters, it has come to
be generally accepted that in extending efficiency values inde-
pendent of absolute dimension, dependance may be placed on the
general trend of change and on the values, especially in a relative
sense, while absolutely some error may be involved. That is,
we may depend on the experimental data to indicate the general
conditions for good efficiency and how the efficiency will vary
for given changes in the factors involved, while some error in
absolute value may possibly or probably result from the exten-
sion to full-sized propellers of efficiency values drawn from model
experiments.
Regarding the relation of the factors k, I, m to size, it must
be admitted that experience is lacking as yet for any satisfactory
determination of the problem. It is more than likely that the
factor m should depend on size, but how much or according to
what law remains for future determination. Such indications as
are available point to some dependance though small in amount,
and in the absence of more precise data it becomes necessary,
in order to make use of model results, to neglect the influence of
size on these factors and in effect, therefore, to assume equation
(i) applicable to propellers of any size and with values of the
factors as derived from an analysis of experiments on small-sized
or model propellers.
Certain further considerations relating to the factor m may
with advantage be noted at this point.
Suppose we start with very narrow, thin, elliptical blades
of maximum width .our, or area-ratio h = .oi. These blades
with given revolutions and slip will develop a certain thrust
T and useful work U corresponding to certain average and
244 RESISTANCE AND PROPULSION OF SHIPS.
distributed values of the coefficients a and /.* Now if the
width and hence the area of these blades be doubled, the area-
ratio being now .02, we shall undoubtedly find that the value
of a will remain sensibly unchanged, and hence the total
thrust will be doubled. If we continue thus to increase the
width, and hence the area, we shall undoubtedly for a time find
the same rate of increase in the thrust, indicating a sensibly
constant value of a. If the path of the blade were rectilinear,
this would undoubtedly hold true up to some increase of area
beyond actual experience. With the screw propeller, however,
the path is helicoidal ; and we readily see that as the width
and area are increased the thrust cannot increase indefinitely,
but must rather tend toward a limit. In other words, as the
area is increased the thrust is increased at a slower and slower
rate. This implies a gradual and continual decrease in the co-
efficient a, while presumably the value of /remains nearly the
same, or at least falls off much less rapidly than a. This de-
crease in a is due to mutual interference in the streams acted on
by the different blades. While therefore a decreases, f re-
mains nearly the same, and the thrust per unit area falls off
accordingly. In this way we shall finally reach a point where
increase of area gives no increase of thrust. The value thus
developed is therefore a maximum, and cannot be exceeded,
no matter what the area. Indeed, certain indications seem
to point toward a possible falling off of thrust with excessive
increase of area.
The ideal maximum thrust obtainable for given diameter,
pitch, revolutions, and slip is that corresponding to the for-
mation of what is termed a complete column ; that is, a col-
* See 35 and 36.
PROPELLER DESIGN. 245
umn of water in which each part has received the full acceler-
ation which, with geometrically perfect action, the propeller
is capable of imparting. The amount of this maximum thrust
may be investigated as follows:
Referring to Fig. 55, it is shown that the value of the
longitudinal acceleration is given by
EG EC cos 2 a = 5 2 cos 2 a = j,/7Vcos 2 a.
Now for an element of the total column consisting of a
shell of water of thickness dr, moving with this acceleration,
the thrust will be
dT = mass of water per second X acceleration EG.
Hence
dT = -7td . dr . velocity of feed X EG.
\ &
We consider the velocity of feed as represented by DG,
or as the speed of advance DE -f- the acceleration EG. The
value of EG above and found in 42 assumes the angle CAE
small. On the same assumption we may also put
CG = EC sin 2 a = 5, sin 2 a = s^Nsin* a.
Hence
DG = pN(\ - J 2 sin 2 a);
and putting in general s for j,, we have
dT = - nd . drsp*N* cos* a(i s sin* a).
But tan a = p -f- n d, and d = 2r = py. Making these
substitutions and reducing, we find
246 RESISTANCE AND PROPULSION OF SHIPS.
Hence for the entire propeller the maximum ideal thrust
will be given by the integration of this expression between
the proper limits forj^. This is seen to be entirely independ-
ent of the breadth of blade or area of its surface. The thrust
thus found would be the maximum possible on the supposi-
tions made, and would correspond to the perfect action of
each element, and to the absence of skin-resistance. It is
therefore greater than could be obtained in any actual case, no
matter what the amount of surface.
On the assumption that the value of a is constant, Cot-
terill* has used the above expression for dT to find the
breadth of blade necessary to form a complete column at
various values of y. This was done by equating the thrust
above to that in a form similar to 35, (6), omitting /and
with an assumed value 1.7 for #, and solving for dA. Since,
however, this value is not accurate for a, and since, more-
over, it cannot remain constant but must fall off in marked
degree with large increase of area, the results thus found can
only be considered as rough approximations to the lower
limits, beyond which we must go in order to approach the
"complete column" condition. So far as these indications
went, however, they showed that propellers of ordinary pro-
portions do not form complete columns, and hence do not
give the maximum thrust corresponding to their diameter,
pitch, revolutions, and slip. This instead of being a fault is
a decided advantage, for it is very sure that a propeller work-
ing near the upper limit of the thrust would be less efficient
than if the surface and thrust were less, all other conditions
remaining the same. This arises from the fact that the last
* Transactions Institute of Naval Architects, vol. xx. p. 152.
PROPELLER DESIGN. 247
increments of thrust must be obtained by the addition of a
disproportionate amount of surface with its accompanying
skin-resistance. This will result in an increase of /relative to
a, and in a corresponding loss in efficiency ( 36, (4)). Hence
while such a propeller may give a large thrust for its diame-
ter, pitch, slip, and revolutions, the proportion of useful to
total work will be comparatively poor.
Let us now return to the expression for the thrust in (i).
By integrating the function of y between various values of the
outer limit we obtain the values of the maximum ideal thrust
for the entire propellers of corresponding pitch-ratios. A
comparison of these with the values actually obtained by ex-
periment under similar conditions of diameter, pitch, revolu-
tions, and slips will be of interest.
The integrations were effected by approximate methods,
and the results are shown by CD, Fig. 74. These are plotted
to represent the maximum ideal thrust in tons for propellers
of 10 feet diameter at 100 revolutions and 20 per cent slip,
and of various values of the pitch-ratio as given on the axis of
abscissae. The curve AB in the same diagram gives similarly
the actual values of the thrust as derived for the same condi-
tions from experimental models, assuming geometrical simi-
larity. This curve gives therefore the actual thrusts for pro-
pellers the same as above, and of area-ratio ^ = .36. With other
values of the slip the results were entirely similar, thus indicating
still more clearly that the propeller of usual proportions is far
from developing the maximum ideal thrust. The ratio of these
two thrusts is shown by the curve EF.
Returning now to the factors k, I, m, of the general equation
(i), we note that the disposition of values among such factors is
to some extent arbitrary, and in the following analysis we shall
248
RESISTANCE AND PROPULSION OF SHIPS.
select some one propeller as an arbitrary standard for which
we assume values of I and m=i.oo. The values of k for
10
I?
1.0 1.1
1.2 1.3
1.4 1.5 1.6
PITCH RATIO
FIG. 74.
1.7
1.8 1.9
1.7
1.6
1.52
cc
various values of the slip will then follow from experimental
results. Then by appropriate methods of analysis, values of /
and m for varying values of pitch-ratio and area may be found
from suitable experimental results.
PROPELLER DESIGN. 249
For the primary source of data for k, I, and m we shall use
the results of the series of tests already referred to in 36. The
shape of all the blades was elliptical, and as noted all had four
blades, so that this series of experiments furnishes data for the
determination of the k, I, and m factors only. We shall take
arbitrarily as follows:
/ for pitch-ratio, i = i ;
m for area-ratio, .4 = 1.
Now in the practical application of this equation it is found
convenient, in order to avoid the use of large numerical values,
to use certain secondary units as follows:
For N a unit of 10 revolutions per min.;
d a unit of 10 feet;
p a unit of 10 feet.
Denote these respectively by NI, d\, p\ t and we have
pi= p + io.
Using these units and measuring U in horse-power, we find
by applying this equation to the actual measured results for a
propeller of pitch-ratio i and area-ratio .4, the series of values
of k related to slip, as shown in Table I, 50.
By suitable methods of analysis we then find values of /
dependent on pitch-ratio and slip, as given in Table II, and of m
dependent on area, pitch-ratio and slip, as given in Table III.
Suitable treatment of the trial results gives also values of the
efficiency for the model propellers, as in Table IV. Intermediate
values may be determined by suitable interpolation.
250 RESISTANCE AND PROPULSION OF SHIPS.
We may turn now to a brief consideration of the influence of
shape and number of blades on the useful work developed by a
propeller.
The typical blade form being elliptical, it will be obvious
that blades with wide tips or of a fan -shaped form with rounded
corners should tend to show a higher average thrust per unit
area or a higher thrust for the same a'rea than one of elliptical
form. This excess is due to the higher velocity at which the
outer elements move and to the lesser obliquity of their normal
to the fore and aft direction. On the other hand, from the view
point of the water laid hold of, it is apparent for an elementary
strip of blade area lying at any given distance from the axis that
there is only such a cylindrical ring of water to be laid hold
of, and if the blade width was previously sufficient to give
to such ring of water its full acceleration, no further advantage
is to be gained by increase of width at this point. The
influence of form can not therefore be properly considered inde-
pendent of area, and while with narrow blades considerable
advantage may be gained by increase at the tip, it must be ex-
pected that such effect will decrease as the average width of
blade is increased. Recent papers by Taylor * and Froude f
give some data regarding this matter, but still further experimental
investigation is needed.
Regarding the difference between three blades and four blades
for the same area, it may be said that with other elements of the
problem equal the narrower blade within certain limits will show
the higher thrust per unit area, and hence a four-bladed propeller
should show higher thrust than a three-bladed of the same area.
The difference, however, has usually been considered very small,
* Transactions Society Naval Architects and Marine Engineers, 1906.
f Transactions Institute of Naval Architects, 1908.
PROPELLER DESIGN. 251
and in most cases of design, a given area has been considered of
equal effect whether divided among three or four blades. Fur-
thermore, from the view point of the water laid hold of, it will be
evident that if the area is such that the three blades will give all
the acceleration to be obtained under the geometrical conditions
of pitch and slip, then four blades will give no more. Certain
data relating to this point are given by Froude in the paper referred
to above under the head of blade form.
Influence due to the Thickness of Blades. Certain light has
been thrown on this subject by experiments at the U. S. Naval
Tank and reported by Taylor.* Various proportions of thick-
ness and various forms of cross-section were tried with general
results, as follows:
For the usual form of section with flat face and rounded back,
both thrust and work absorbed are increased up to a point beyond
usual proportions, as the thickness is increased.
Of these two rates of increase the latter is the greater, and
thus the efficiency decreases with increase of thickness.
The location of maximum efficiency relative to slip is also
subject to change due to variation in thickness, but in manner
too uncertain to admit of formulation into any definite law.
Regarding influence of form of section, it was shown that the
form of the following or rear half of the blade section seems to
exercise a dominant influence over the results, and with equal
thickness and similar rear halves, the performances of propellers
with leading halves of various forms fell very close together.
As regard efficiency, the modification shown in Fig. 81,
55 j g ave results superior to the more common form, especially
at moderate or low values of the slip.
* Transactions Society Naval Architects and Marine Engineers, 1906.
252 RESISTANCE AND PROPULSION OF SHIPS.
In general, the influence due to thickness of section is sur-
prisingly great. Thus to give one instance, doubling the thick-
ness at 20 per cent slip produced an increase of 30 per cent in
the thrust developed and of some 42 per cent in the power ab-
sorbed.
GENERAL REMARKS.
We therefore consider equation (i), together with the tables
relating to k, I, m, and e^ as our fundamental data for propeller
design.
As will appear more fully by illustrative examples, the method
of design proposed does not take the place of judgment and
experience. So far as the method itself is concerned, there
may result an indefinite number of propellers, each more or
less completely fulfilling the conditions imposed, and in the
discrimination between these full scope may be found for the
use of judgment and experience. It is intended rather as an
aid to the intelligent use of experience, and to the systematic
analysis of the results of trial data.
50. PROBLEMS OF PROPELLER DESIGN.
Formula (i), 49, relates simply to the useful work of a
propeller operating in undisturbed water. In the actual case
the propeller operates in the forward wake. In Chapter III
we have discussed this wake, and shown that we may substi-
tute for the actual turbulent wake a uniform stream of
velocity v giving a true slip S y = S t -f- v o , and a speed of
advance of the propeller relative to the water in which it
works, of PN S^ = PN 5, v = u v n . Now in order
to connect the propeller in undisturbed water with the actual
case, we assume that for a given propeller at given revolu-
PROPELLER DESIGN. 253
tions working in the wake of velocity z/ and developing a
thrust of T and a speed relative to the wake of (u z/ e ), the
useful work and efficiency will be the same as if it were work-
ing in undisturbed water at the same revolutions and true
slip, and hence with an equal speed of advance (u z/ ).
That is, referring to Fig. 75, we must consider the useful
work of our formula as based on the true propeller efficiency
and not the apparent, or on the speed of advance of the pro-
peller through the water in which it works, and not relative
to still water. Hence the useful work thus defined will be
represented by C^E T(u v ), and not by CE = Tu.
It is therefore the work Cf, which must be substituted
in the formula, 49, (i).
The following will therefore be the most natural order of
procedure:
The I.H.P. being given or AE, we assume AB. This
gives us BE. We may then select or assume the true efficiency
i 1 i 1 i
A B c C, F E
FIG. 75.
at which we propose to work. Then BE multiplied by this
efficiency will give C\E, the useful work to be substituted in our
formula. Again, it may arise that instead of the I.H.P. we have
the E.H.P., or assume the propulsive coefficient, and thus pass
from AE to CF direct. Then CF multiplied by hull efficiency
= CiE, the useful work, as before. As already noted, unless
special information is available, the hull efficiency is usually
taken as unity.
254 RESISTANCE AND PROPULSION OF SHIPS.
We must next determine the value of pN.
Let u = speed in knots. Then
101.3^ = speed in feet per minute.
lOI.lU
And P*f=T
But from 44, (4),
Hence pN=j-
This determination calls for an estimate of the wake factor
w. Unfortunately the information available as a basis for such
estimate is somewhat meager. Fronde in his classic paper *
on propellers gives a series of values for ships of various charac-
teristics, but beyond this little information is available in the
general literature of the subject. Adding to these a few other
known values, Prof. McDermott f has connected them with the
characteristics of the ships by empirical formulae, as follows:
Let w denote wake factor;
p
prismatic or cylindrical coefficient;
m
midship-section coefficient;
L
" length in feet.
Then
w
IP \
- .161 L* .6) for single-screw ships;
I+W
w
(P \
= .13^ L i.i J for twin-screw ships.
l+w
* Transaction Institute of Naval Architects, 1886.
f Transactions Society of Naval Architects and Marine Engineers, vol.
p. 164.
PROPELLER DESIGN. 255
These equations represent simply certain available data, and
as this is relatively meager, care must be exercised in their use.
Where no special information is otherwise obtainable, however,
they will probably serve to give a fair approximation to the value
desired. Further suggestions on this point, as well as on several
others relating to the choice of values for the various quantities
involved, will be given in the following section. We will now
proceed to illustrate the application of our formulae and tables to
actual problems of design.
We first repeat in collected form, for convenience, the prin-
cipal equations which we may have occasion to use:
(i)
(5)
w Ip , \
^-'^i- --6 j for single screw-ships; (7)
w (p \
'=.131 L i . i ) for twin-screw ships ; . . (8)
I+W
The values of k, I, m, 62, and of 101.3^ may be taken from
Tables I, II, III, IV, V, for the values of slip, pitch-ratio, area-
ratio, and speed given, and by interpolation for intermediate
values.
256 RESISTANCE AND PROPULSION OF SHIPS.
TABLE I. VALUES OF FACTOR k.
Slip Ratio.
k
Slip Ratio.
k
Slip Ratio.
k
.10
.110
.22
.168
-32
-195
.12
.121
.24
-175
-34
.199
.14
.132
.26
.181
-36
.202
.16
.142
.28
.186
-38
.204
.18
.151
-3
.191
.40
.205
.20
.l6o
TABLE II. VALUES OF FACTOR /.
PITCH-RATIO.
Slip Ratio.
.8
9
I .0
i
. i
i .
2
1-3
1.4
.10
i-35
-15
.00
.
5 7 i
-7<
>3
-675
-594
.12
-3 2
.14
.00
*75
-7'
Ji
.682
.610
.14
3
- J 3
.00
.
*79
-7'
j&
.691
.621
.16
.29
-13
.00
*8 3
7
k
.700
.630
.18
.28
.12
.00
386
7<
)o
.707
-637
.20
.28
.12
.00
.
38 9
7<
>5
.714
.644
.22
.28
.12
.00
5 9 i
7<
)9
.719
.649
.24
.28
.12
.00
*93
.8c
)2
.724
-655
.26
.28
.12
.00
395
.8c
)6
.727
.660
.28
.27
.12
.00
3 97
.8c
>9
-73 1
.664
-3
.27
.12
.00
*99 S
.81
[2
-734
.668
3 2
.27
.12
.00
m
900
.8
rOr
-737
.670
34
.27
.12
.00
m
901
.8
:8
-739
-673
-36
.27
.12
.00
m
902
.8
[ 9
.741
-675
-38
.27
.12
.00
93
.8
20
.742
.676
.40
.26
.12
.00
903
.8
2O
.742
.676
i Slip Ratio.
1.5
1.6
I -7
i .
8
1.9
2 .0
.10
.524
.465
.41
2
-3
58
.
3 2 9
- 2 93
.12
-539
.481
-43<
3
-3
*4
.
345
3i;
.14
-55
-493
.44
[
-3
W
.
359
-3 2 5
.16
.560
-503
-45
2
-4
28
.
369
-336
.18
-5 6 9
.512
.46
[
-4
i.7
B
379
-345
.20
-577
-521
. 4 6<
)
-4
26
.
386
352
.22
.584
.528
-47<
5
-4
34
.
394
.360
.24
-590
.536
.48
\
-4
*l
.
401
.366
.26
.596
-542
.49
i
-4
*8
,
406
372
.28
.600
-547
.49
5
-4
54
412
-377
3
.605
-551
-5
i
.4
58
.
417
.381
3 2
.609
555
-50
5
-4
52
.
421
-385
34
.612
.558
-50
?
4
56
424
-389
-36
.614
.560
-5i
2
4
69
428
392
38
-615
.562
-5i
*.
4
7i
43
394
.40
.616
.563
5*
5
4
72
432
.396
PROPELLER DESIGN.
TABLE III. VALUES OF FACTOR m.
PITCH RATIO .8.
257
Area-ratio.
Slip
Ratio.
.20
.30
.40
.50
.60
.10
-73
.900
I. 00
1.05
1.07
. 20
.783
.927
I. 00
1.04
I- 06
3
.819
.942
I .00
1.03
1-04
.40
.840
-95
I. 00
1-02
I 02
PITCH-RATIO .9.
Area-ratio.
&.
.20
30
.40
50
.60
.10
.660
.883
I.OO
1. 06
1.09
.20
.750
.914
I.OO
1.04
1.07
3
793
93
I.OO
1.03
1.05
.40
.820
944
I.OO
1. 02
1,03
PITCH-RATIO i.o.
Slip
Ratio.
Area-ratio.
Ratio.
. 20
30
.40
.50
.60
.10
.620
.865
I.OO
1. 06
I.IO
.20
.725
90S
I.OO
1.05
1.07
3
.773
.927
I.OO
1.04
1. 06
.40
.802
.940
I.OO
1.03
1.05
PITCH-RATIO i.i.
Sin
Area-ratio.
Ratio.
.20
.30
.40
.50
.60
.10
595
.852
I.OO
1.07
1*12
.20
7i5
.895
I.OO
1.05
1. 08
'30
755
.919
I.OO
1.04
1,07
.40
.784
93
I.OO
1.04
1.06
258 RESISTANCE AND PROPULSION OF SHIPS.
TABLE III. VALUES OF FACTOR m Continued.
PITCH-RATIO i 2.
Area-ratio.
Slip
Ratio.
.20
30
.40
SO
.60
.10
-575
.843
I.OO
1. 08
1-13
.20
.690
.890
I. 00
1. 06
1.09
30
743
.912
I.OO
1.05
1.07
.40
.770
.9 2 5
I.OO
1.04
1.07
PITCH-RATIO 1.3.
Area-ratio.
Ratio.
.20
.3
.40
50
.60
.10
.560
.831
I.OO
1.09
1. 14
.20
.680
.880
I.OO
1. 06
I.IO
30
.732
905
I.OO
1.05
1. 08
.40
755
.919
I.OO
1.05
1. 08
PITCH-RATIO 1.4.
Area-ratio.
Ratio.
.20
30
.40
50
.60
.10
552
.823
I.OO
1.09
1.14
.20
.674
.880
I.OO
1. 06
I.IO
3
.726
.903
I.OO
1. 06
1.09
.40
744
-9 J 3
I.OO
1.05
1. 08
PITCH-RATIO 1.5.
Area-ratio.
Ratio.
.20
.30
.40
SO
.60
.10
-549
.818
I.OO
I.IO
**$
.20
.670
.875
I.OO
1.07
I.IO
3
.720
.899
I.OO
1. 06
1.09
,40
>736
.908
I.OO
1.05
1.09
PROPELLER DESIGN.
2 59
TABLE III. VALUES OF FACTOR m Continued
PITCH-RATIO 1.6.
Area-ratio.
Ratio.
.20
30
.40
.50
.60
.10
541
.813
I.OO
I. id
i. iS
.20
.669
.873
I.OO
1.07
i. ii
3
.715
.898
I.OO
i. 06
1.09
.40
73
.902
I.OO
i. 06
1.09
PITCH-RATIO 1.7.
Area-ratio.
Slip
Ratio.
.20
30
.40
So
.60
.10
540
.812
I.OO
I.IO
.*s
.20
.668
.873
I.OO
1.07
I. II
3
,712
.895
I.OO
1. 06
1.09
.40
.724
.901
I.OO
1. 06
1.09
PITCH-RATIO 1.8.
Area-ratio.
Ratio.
.20
.30
.40
.50
.60
.10
.20
540
.669
.813
.872
I.OO
I.OO
I.IO
1.07
1.16
i. ii
3
.710
.892
I.OO
1. 06
I.IO
.40
.719
.899
I.OO
1. 06
1.09
PITCH-RATIO 1.9.
Area-ratio.
Clip
Ratio.
.20
30
.40
50
.60 v
.10
549
.815
I.OO
I. II
1.16
.20
.670
-873
I.OO
1.07
i. ii
3
.710
.891
I.OO
1. 06
I.IO
.40
.719
.899
I.OO
1. 06
1.09
PITCH-RATIO 2.0.
Area-ratio.
Slip
Ratio.
.20
30
.40
.50
.60
.10
.556
.819
I.OO
I. II
1.16
.20
.674
.873
I.OO
1.07
I. 12
3
.711
.891
I.OO
1. 06
I.IO
.40
.718
.899
I.OO
1. 06
1.09
360 RESISTANCE AND PROPULSION OF SHIPS.
TABLE IV. VALUES OF EFFICIENCY.
PITCH-RATIO .8.
Area-ratio.
Slip
Ratio.
.20
3
.4
.50
.60
.IO
.58
.61
.63
63
.63
.20
.64
.65
.66
.66
.66
3
.62
.62
-63
.63
63
.40
.58
.58
.58
.58
.58
PITCH-RATIO .9.
Slip
Ratio.
Area-ratio.
.20
.30
.40
so
.60
.10
.20
3
.40
^64
.60
.61
- .67
.64
59
.64
.67
.64
-59
.64
-67
.64
-58
-63
.66
.64
.58
PITCH-RATIO i.o.
Area-ratio.
Slip
Ratio.
.20
30
.40
So
.60
.10
-57
.61
.64
-64
-64
.20
.67
.68
.68
.68
-67
.30'
.66
.66
-65
.64
-63
.40
.62
.61
.60
.58
.58
PITCH-RATIO i.i.
Area-ratio.
Ratio.
.20
3
.40
.50
.60
.10
.56
.61
.64
.65
.65
.20
.69
.69
.69
.68
.68
3
.68
-67
.66
-65
.64
.40
.64
.62
.60
.59
58
PROPELLER DESIGN.
26l
TABLE IV. VALUES OF EFFICIENCY Continued.
PITCH-RATIO 1.2.
Area-ratio.
Slip
Ratio.
.20
-30
.40
50
.60
.10
0-6
.6l
.64
65
.64
.20
.70
.70
.69
.69
.69
3
-7
.68
.68
.66
.65
.40
.65
-63
.62
.60
-59
PITCH-RATIO 1.3.
Area-ratio.
Ratio.
.20
30
.40
50
.60
.10
55
.61
-64
.65
65
.20
-71
-7i
.70
.70
.69
3
-71
.70
.69
.67
65
.40
.66
.65
.64
.61
.60
I
PITCH-RATIO 1.4.
Area-ratio.
Slip
Ratio.
.20
30
.40
so
6
.10
-5 2
.60
-63
.64
.64
.20
-7 1
-73
-73
.72
.72
3
.72
-73
-72
-7i
.70
.40
.68
.68
.67
-65
.65
PITCH-RATIO 1.5.
Area-ratio.
Slip
Ratio.
.20
.30
.40
50
.60
.10
-55
.60
.64
.65
.66
.20
-71
.72
71
.70
.70
3
.72
-71
.69
.68
.66
.40
.67
.65
.64
.62
.61
262
RESISTANCE AND PROPULSION OF SHIPS.
TABLE IV. VALUES OF EFFICIENCY Continued.
PITCH-RATIO 1.6.
Area-ratio.
Ratio.
.20
30
.40
so
.60
.10
-54
.60
.64
-65
.66
.20
.72
.72
.72
7 1
.70
3
73
.72
.70
.68
.68
.40
.68
.66
.64
-63
.62
PITCH-RATIO 1.7.
Slin
Area-ratio.
Ratio.
.20
.3
.40
5
.60
.10
-54
.60
.64
-65
.66
.20
.72
.72
.72
.71
.70
-3
.72
.72
69
.68
.40
.68
.67
-65
-6 4
-63
PITCH-RATIO 1.8.
Qlj n
Area-ratio.
Ratio.
.20
30
.40
so
.60
.10
-53
.60
-64
-65
-65
.20
.72
-73
-73
.72
-71
3
-73
-73
-7
.69
.40
.69
.67
.66
.64
-63
PITCH-RATIO 1.9.
Qij-n
Area-ratio.
Ratio.
. .20
.30
.40
so
.60
.10
-53
.60
-63
-65
-65
.20
.72
-73
-73
.72
.71
-3
.72
-73
.72
.70
7
.40
.69
.66
-65
.64
PITCH-RATIO 2.0.
Qljn
Area-ratio.
Ratio.
.20
30
40
so
.60
.10
-5 2
.60
-63
.64
.64
.20
-71
-74
-73
.72
.71
3
.72
-73
.72
,71
.70
.40
.68
.68
-67
.66
-65
PROPELLER DESIGN.
263
TABLE V. GIVING FEET PER MIN. CORRESPONDING TO KNOTS,
OR VALUES OF IOI.3?*.
u
101. 3.
u
ioi.3.
10
1013
25
2533
II
IH5
26
2635
12
1216
27
2736
13
1317
28
2837
14
1419
29
2939
15
1520
30
3040
16
1621
31
3HI
17
1723
32
3243
18
1824
33
3344
19
1925
34
3445
20
2027
35
3547
21
2128
36
3648
22
2229
37
3749
23
2331
38
3851
24
2432
39
3952
EXAMPLES.*
(a) As the first example let us take
I.H.P. =4600;
speed = u = 1 6 knots;
friction power = 13 per cent = 598 H.P. (AB, Fig. 75).
Hence
propeller power = 4002 H.P. (BE, Fig. 75).
Next from the data given suppose w found or estimated
as .14. We will also select .26 as the proposed value of the
true slip or s^ We then find from (6)
I 5, = .844 and j, = .156.
Hence
and
16 X 101.3
pN = - Q - = 1922 ;
.044
* In these examples we shall assume that the influence of number of blades
as such may be neglected and that the influence of surface is sufficiently taken
care of by the factor m.
204 RESISTANCE AND PROPULSION OF SHIPS.
Mow supposing neither p nor N fixed, we may proceed as
follows: We first select a pitch-ratio of, let us say, = 1.20. We
will also assume in this problem that the blades are elliptical,
with area-ratio .40, so that m = i. From the tables we then find
= .181;
/ = .8o6;
e 2 = .6S.
Hence
useful work # = .68X4002 = 2721 (CiE, Fig. 75).
Then, substituting, we have
2721 = (i9.22)% 2 X.iSi X.8o6;
whence di 2 = 2.62 t j-
and d = i6.2i;
^ = I 9-45;
N = 98.8.
(b) With the same otherwise as in (a), let the area of blade
be increased to area-ratio .48. Then from Table III we may
take w = i.o4. This will produce a 4 per cent decrease in di 2 ,
and approximately a 2 per cent decrease in d. We shall have
therefore
^ = 15.89;
# = 19.07;
TV =100.8.
(;) Suppose with the same data as in (a), we fix both pitch
nd diameter and propose to find the necessary amount of area.
1 hus let p = 24 feet and d = iS feet. Then with the same value
.of pN we find AT = 8o.i. Also ^ = 1.33, and we have
PROPELLER DESIGN. 26$
= .181, as before;
2 = -75;
e 2 = .6() (for probable area-ratio);
and 17 = 2761.
Substituting and solving for the factor w, we find
2761
This may be realized with an elliptical blade of area-ratio about
35-
(d) Given the same data as in (a), let us fix 2V at 130. Then
with the same value of pN we have
^=1922-^130=14.8.
Let us now assume for trial a diameter of 14 feet. Then
c = 14.8 -7-14 = 1.06.
and we have
ft .181;
/ = .94;
e 2 = .67 (minimum desired) ;
and ^7 = 2682.
Then solving for m as in (c), we have
__ 2682 _ =
W = 7iooXi.Q6X.i8iX.94~
This result is impracticable and shows that we must go back
and modify the other conditions. We will therefore assume a
true slip of .28 giving
1974;
2 66 RESISTANCE AND PROPULSION OF SHIPS.
Then assume d = 14.50;
then c = 1.05, and we have
= .186;
^=95;
2 = .67, as before;
and 7 = 2682.
Then solving for m, we find
which is easily realized by a slight increase in area-ratio above .40.
The propellers thus far considered have been implicitly
supposed to have 'four blades. To illustrate the application
of the equations to propellers of two or three blades we may
take the following:
(e) Let LH.P.=5ooo;
u = 20;
JF P = . 87X5000 =4350;
w=.o8;
s 2 = .24.
Then (i-,si)=.82i,
2027
and #^ = ^7
whence p\N\ = 24.69.
Take, also, pitch-ratio = 1.4;
number of blades = 3 ;
area-ratio = .30.
PROPELLER DESIGN. 26?
Then we find
= 175;
' = 655;
^=.89;
^ = .73;
17=3176.
Then, substituting, we have
3*76
dl = 15051 X.I75X.655X.89" 2 - *'
Whence we find
d= 14.38;
p = 20.13;
N=I22.J.
As an example of a two-bladed propeller we may take the
following:
*2 = -30.
Then (i-si)=-77>
and pN = g2i\
whence piNi = g.2i
Take also pitch-ratio = i . i ;
number of blades = 2,
area-ratio = .20.
2 68 RESISTANCE AND PROPULSION OF SHIPS.
Then we find
= .191;
/ = .8 99 ;
^=.755;
Then, substituting, we have
whence ^ = 2.14;
^ = 2.35;
(g) As a further illustration of special design let us take
the case of a tugboat. If the design is based on the conditions
of use we must naturally take the speed low and the slip high.
Let I.H.P.= 3 oo;
w = 7 ;
^ = .83X300 = 249;
1V = .I2',
5! = .40.
Then i-5i=.6
and ^i7Vi = 10.59.
Take also pitch-ratio = i . i ;
number of blades = 4 ;
area-ratio = .5 5.
PROPELLER DESIGN. 269
Then we find
= 205;
.59;
Then, substituting, we have
; 2 = _ 147 _ = fi
1 188 X. 205 X. 903X1.05 37 '
whence ^ = 7.98;
TV =120.6.
Propellers with Steam-turbines The design of a propeller to
operate with a steam-turbine presents no new fundamental
problems. The methods of treatment are the same as in any
case of design where similar data are at hand. With turbine
design, however, the I. H. P. is not one of the items of funda-
mental data, but is rather inferred from the E.H.P. and the
general conditions of operation. Most naturally, therefore, the
E.H.P., represented by CF, Fig. 70, will be taken as the point of
departure, and from which C\E directly follows on an assumption
of hull efficiency. From this point the method of procedure will
follow along lines similar to those noted above.
The special feature of propellers used with turbines is the low
pitch ratios necessitated by the comparatively high revolutions.
Propellers with turbines have as a rule pitch ratios found
between .7 and .9, and this low value should lead to caution
with regard to the area-ratio employed.
270 RESISTANCE AND PROPULSION OF SHIPS.
In general, experiment has shown that the best propeller
efficiencies cannot be realized with extremely low values of the
pitch-ratio, and furthermore that with such values of the pitch-
ratio the efficiency is more dependent on area than with higher
values. For this reason, there is danger of running into low
values of the efficiency with increase in area beyond moderate
proportions. While exact limits cannot be stated, it seems
likely that having regard to efficiency the area ratio with such
propellers should not exceed .40 to .45.
On the other hand with the small propellers to which such
proportions lead, there is likely to be trouble, especially at
high speeds, with too little area and consequent cavitation
see remarks on this subject in 52. In the face of these con-
flicting tendencies, some compromise must be made, and this
has often lead to the adoption of higher area-ratios than would
be warranted by considerations of efficiency alone.
Values of the constants in Tables I, II, III, are not extended
below a value of pitch-ratio = .8. For carrying out design with
still lower values of the pitch-ratio recourse may be had to
methods by comparison, as referred to later, and also to the
data derived by experiments on propellers of low pitch-ratio at
the U. S. Naval Tank at Washington.*
The usual operations involved in the preceding problems may
be somewhat generalized as follows :
Given I.H.P., speed u, friction, wake factor w y and true
slip s 2 .
Whence W P or BE, Fig, 75, si, pN, and k.
Then (i) Given neither p nor N.
Assume pitch-ratio c and area factor m.
* Transactions Society of Naval Architects and Marine Engineers, 1905, 1906.
PROPELLER DESIGN. 2 *Jl
Thence find e 2 , U, and /.
Thence by (i) find d.
Thence p and N.
(2) Given either p or N.
Thence the other N or p.
Assume d and the efficiency 62.
Thence find c, U, and /.
Thence by (i) find m and determine by judgment
as to the practicability and consistency of the re-
sulting value.
(3) Given d.
Assume c or p and 62.
Thence find p or c and N, also U and I.
Thence as in (2).
By an inversion of the above processes these equations may
be used for the analysis of a given set of trial data, and thus
for the determination of the values of s 2 and w in the given
case. The use of the equations for this purpose assumes them
as applicable to the given propeller, and hence we should
expect the results to be generally more reliable the more closely
the actual propeller approaches to the standard form and pro-
portions of blade. The operation is as follows:
(h) Given I.H.P., p, N, d, and si, and assume engine-friction.
(i) Thence find W P and pitch-ratio.
('2) Then assume a value of $2 and find resulting values of
m, I, and e 2 .
(3) From equation (i) find value of k.
(4) By means of Table I compare the corresponding value
of 52 with the value assumed in (2).
(5) Correct the assumption of s 2 until by process as above
a consistent set of results is obtained. The final result will thus
determine s 2 and thence w by equation (5).
272 RESISTANCE AND PROPULSION OF SHIPS.
Thus for example, taking the data and results of (a) except
52 and w, let us first assume $2 = .20.
Then we have /= .795;
w = i.oo;
e 2 = .69.
Then
.69X4002
2.627X7IOOX-795
.i86.
But .186 for k corresponds to s 2 =.2&, which shows that the
first assumption was too low. The real value will lie between
.20 and .28, and a second or third trial would locate it at .26,
which is, of course, the correct value for this case, and from
which w may be determined as in (a).
As a further example we may take the following data
from the U. S. cruiser Detroit:
We have given as mean results for the two propellers:
d= 1 1 ;
/= is;
pitch-ratio = 1 . 1 8 ;
I.H.P. =2577;
N = 1 70. 1 ;
u = 18.71 ;
s i = -I4J
area-ratio = .305 ;
Engine-friction is taken at 12 per cent.
Hence ^-2270.
Assume for trial s% = .25.
Then/ = .822;
^ = .906;
PROPELLER DESIGN. 273
Then
69X2270
!.2iXio8o8X.822X.9o6~
For k = .i6i, s z = .20 (see Table I).
Hence correct value will lie near .21 or .22.
Further trial will show that $2 = .22 will give consistent values
for k. Hence we have, from equation (5), ^ = .077.
The use of the equations in this manner in a large number
of cases carried out under the direction of the author showed
that it is somewhat difficult to satisfactorily determine the
value of w from such analysis alone. This is not so much
due to inaccuracy in the equations as to the fact that a
slight change or error in the data will make a relatively much
larger error in the value of w. This arises from the fact that
the value of w is determined from the quantity (i -f- w).
Now an error of moderate amount in the data or in the suit-
ability of the equations to the data might give, for example,
an error of 2 or 3 per cent in the value of (i + w )i making,
for example, its value 1.12 when it should be i.io. This
would make a difference of .02 in w t or an error of 20 per
cent in excess of the true value. So far as ordinary applica-
tions are concerned, however, and especially so far as the
problem of design is involved, it is the factor (i -|- w) and
not w which is used, and hence changes of relatively large
percentage amount in w will have comparatively small influ-
ence on the solution of such problems. It may be also men-
tioned that a comparison of the values of w determined in
this manner with those given by equations (9) and (10) and
by general estimate showed in nearly all cases that a differ-
ence of a few per cent in the friction of the engine, or more
274 RESISTANCE AND PROPULSION OF SHIPS.
especially of a very small amount in the pitch of the pro-
peller, would be sufficient to account for the difference in the
values of w. The uncertainty as to the real value of the
pitch in cases of variable pitch has been considered in 42 ;
and bearing these facts in mind as well as the possibilities of
slight errors qf measurement in all cases, and the further
uncertainty regarding engine-friction, it would appear that
the degree of fulfilment is quite as close as might naturally be
expected, and that for problems of design such equations and
methods may be used with a high degree of confidence in the
results.
The influence of an incorrect estimate of w in a problem of
design may be illustrated by the following example :
(*) In problem (a) above, suppose .14 to be the correct
value of w, but let its estimated value be taken as .09. Then
with the same assumed value of s^ we shall have
I-^= .807;
pN 2009.
Following the work through as in (a)\ we should find the
following results:
d= 15.17,
/ = 18.20,
instead of those found in (a).
Now suppose this design accepted and the propeller fitted
to the ship. We may then ask two questions: (i) What will
be the change in efficiency and what the power necessary at
the desired speed of 16 knots? or, (2) What will be the re-
PROPELLER DESIGN. 2*J$
sultant speed if the power 4002 is delivered to the propeller as
in (a)?
We first assume the speed of 16 knots attained with a value
of CiE = 2'j2ij as in (a). We then have, by substitution in (i),
2721 = OiNi) 3 (i.52) 2 .8o6.
In this equation -(piNi) and k are both unknown, but both
dependent on s 2 . Hence solving, we have
We must now find by a process of trial a value of $2 such
that the resulting values of k and p\N\ (using of course the
true value .14 for w) will give for (piNi) 3 k the value 1461, or
a sufficiently close approximation to such value. For the value
S2 = .26 assumed in (a) we have k = .iSi and (piNi) 3 = jioo.
Whence (piNi] 3 k = i2S^. The value 1461 required, indicates
a value of 50 larger than .26, and one or two trials are suf-
ficient to show that the desired result is very near to .283 for
which (/>iJVi) 3 = i459. We may take, therefore, as the result
= 283;
= 1983;
N= 109;
62 will be slightly decreased;
W P will be correspondingly increased.
According to these results, therefore, the revolutions would
be slightly less than the 110.5 expected, the efficiency would be
slightly decreased, and there would be required a correspond-
ing small addition to the power.
276 RESISTANCE AND PROPULSION OF SHIPS.
If we next suppose but 4002 H.P. delivered to this pro-
peller and inquire as to the resulting speed, we have a
problem requiring for its solution a resistance-curve for the
ship. For all practical purposes, however, we may assume
a loss of efficiency of, say .007, and that power varies as the
cube of the speed. This would give for the ratio of the power
in the two cases the value
.68 -T- .673 = 1. 01,
and therefore
Ratio of speeds = t'l.oi = 1.003.
Hence with 4002 H.P. at the propeller in the case assumed
we should have a speed of
u = 16 -f- 1.003 = 1 S-9St
or a resulting loss of .05 knot.
In a similar fashion if we suppose w overestimated by .05,
or taken at . 19 instead of .14, we may carry the work through,
finding for the propeller the following :
pN ' 1840;
d = 17-31;
/= 20.77;
Then, as before, we should find that such a propeller ap-
plied to the ship in (a) with w = .14 instead of .19 would give
rise to the following values :
s t = .24 (app.) ;
= 1872;
PROPELLER DESIGN.
JV = 90.i;
e 2 will be sEghtly increased j
W P will be correspondingly decreased.
In the same manner as above we should also find that with
4002 H.P. delivered to this propeller the speed would be in-
creased by about .04 knot or to 16.04 knots.
These illustrations of the influence of an error in 'the esti-
mation of w may perhaps be taken as extreme cases, for we
should under all usual conditions be able to estimate w within
.05 , and it thus appears that the errors resulting from an incorrect
estimate of w are in themselues not likely to be serious, espe-
cially if the propeller is designed to work at or near its maxi-
mum efficiency. Some further discussion of these points will
be found in the following section.
Additional Methods of Propeller Design. The formulae and
methods of design whose applications are shown in the present
section are such as naturally result from the general method
of treatment developed in this work. Similar data may be used
in a variety of other ways as illustrated by Froude,* Barnaby,t
Caird,J and McDermott. There have been long in use also
briefer and less accurate methods of screw-propeller design than
those involving the detailed considerations discussed in the
present work. A brief reference to such methods may here be
made.
The value of pN is found from the given speed u and an
* Transactions Institute of Naval Architects, vol. xxvu. p. 250.
| Transactions Institution of Civil Engineers, vol. en. p. 74.
J Transactions Institute of Engineers and Shipbuilders in Scotland,
1895-96, p. 21.
ij Transactions Society of Naval Architects and Marine Engineers, vol.
IV. p. 159-
278 RESISTANCE AND PROPULSION OF SHIPS.
assumed value of the apparent slip s l (see (4) ). The values of
/and TV are then selected by judgment.
The value of d may then be found from the formula
=^y
I.H.P.
in which K l is usually taken between 18000 and 25000.
It is readily seen that this is equivalent to taking the value
of the product k, /, m, (i) as constant, or assuming its value
by judgment between limits corresponding to those specified
for K v
The helicoidal area A is sometimes determined by the
formula
'I.H.P.
A =
in which K^ may vary from 8 to 15.
The helicoidal area may also be determined by the formula
Indicated thrust I.H.P. X 33000 ,
A = ~ - - ~ (S6e 47)>
in which K t may vary from 1000 to 1500.
Having thus determined A, the resulting values of the
disk area and of the diameter d may readily be found by the
use of standard proportions.
Since helicoidal area should vary nearly as <^*, it is evident
that the three equations above are mutually inconsistent in
form, though when used with judgment and with abundance
of data for comparison they may be made to yield good re-
PROPELLER DESIGN. 279
suits. The third, relating helicoidal area to indicated thrust,
is probably the most satisfactory of the three.
The disk area may also be related to the wetted surface of
the ship by the formula
Wetted surface
Disk area = - ,
in which K^ is usually found between 70 and 90, while for
specially high-powered craft it may fall to from 50 to 70.
The disk area may also be related to the area of midship
section by the formula
Area of midship section
Disk area = f
in which K^ is usually found between 2 and 2.5, while for
specially high-powered craft it may fall to from 1.2 to 2.
The disk area may also be related to the displacement D
by the formula
Disk area = ,
in which K t is usually found between .8 and i.i, while for
specially high-powered craft it may fall to from .6 to .8.
In formulae relating the disk area to the ship the result
found is considered as the area for all the propellers, one,
two, or three, as fitted. The value may then be divided as
required, and the diameter immediately found.
Normand has proposed * a formula for the area of the pro
pulsive surface, the constants of which he has checked by appli-
cation to a large number of actual cases, and which he con-
* Bulletin de 1' Association Technique Maritime, 1899.
280 RESISTANCE AND PROPULSION OF SHIPS.
siders may be taken as a safe guide for the determination of the
necessary helicoidal area, in particular when the speed of the ship
is not far from the normal maximum speed as discussed in 25.
In English units this formula is
nd 2 u 2
where r = area ratio as defined in 34;
# = I.H.P.;
n = number of propellers;
d= diameter in feet;
u = speed in knots;
J = constant with values from 6 to 8.
The values of the constant are to be selected according to cir-
cumstances, less as the immersion is greater and the general con-
ditions are favorable, and greater as the immersion is less and
the conditions of operation are less favorable.*
The methods and formulae briefly referred to in the various
equations above involve necessarily a large element of judgment,
with but slight opportunity for its orderly exercise. They
have, furthermore, no provision for estimating in detail the
probable results due to variations in the different charac-
teristics of the propeller. With the more detailed methods
presented and illustrated in the present chapter, the range of
judgment is somewhat narrowed, its application is simpler,
and full provision is made for the orderly estimate of the
probable results due to changes in the more important char-
acteristics.
* In addition to the above formula for propulsive surface, the reader will find
an excedingly interesting and highly suggestive series of papers on propeller theory
and design under the following references: Rateau, Bulletin de 1' Association Tech
nique Maritime, 1900; Drzewiecki, ibid., 1900-1901; Doyere, ibid. t 1901-1902
Brosser, ibid., 1903; Bassetti, ibid., 1906-1707.
PROPELLER DESIGN. 2 8l
51. GENERAL SUGGESTIONS RELATING TO THE CHOICE OF
VALUES IN THE EQUATIONS FOR PROPELLER DESIGN,
AND TO THE QUESTION OF NUMBER AND LOCATION OF
PROPELLERS.
In the method of the preceding section the I.H.P. is taken as
a part of the fundamental data. This implicitly involves values
of the augmentation of resistance due to the propeller and of the
wake factor. While, therefore, the augmentation of resistance is
not explicitly involved in our equations, it is really represented
by the assumed or determined I.H.P. The wake factor w is not
only involved in the I.H.P., but we are also called on to assign to
it a definite value.
The principal source of information relative to both the
wake factor and the factor of augmentation (or its reciprocal
"thrust deduction," as termed by Froude) is in the experi-
ments on models. These points are discussed by Mr. R. E.
Froude in the paper* to which previous reference has been
made, and to which the reader may be referred for details.
The conclusions were that for both primary uses of such data,
viz., the determination of the standing of a propeller on the
scale of efficiency and the relation of its thrust to revolutions,
such sources of error as may properly seem to exist appar-
ently tend to neutralize each other, so that we have to deal
not with a single error, nor with several tending in one direc-
tion, but with a balance of errors in which, though the indi-
vidual values may be sensible, it is quite likely that their
combination will not form an error of serious amount. Com-
paring the numbers of revolutions for the models with those
for the full-sized ships, Mr. Froude states that the model
data on the average were found to overestimate the revolu-
* Transactions Institute of Naval Architects, vol. xxvii. p. 250.
282 RESISTANCE AND PROPULSION OF SHIPS.
tions of twin-screws by about 2 or 3 per cent, and to under-
estimate those of single-screws by about the same amount.
As shown by example (i), 50, the result of an under- estimate
of w when the design is carried through for a given pitch-ratio,
as in (<z), will be to obtain a propeller too small and with a
larger true slip than that assumed, while an over-rated w will
give rise vice versa to a propeller too large and with a true slip
less than that assumed. If the assumed conditions correspond
nearly to the maximum efficiency, as indicated by Table IV, then
it is probably better that w should be underrated rather than the
reverse. This is because the efficiency falls off from the
maximum much more slowly for an increasing than for a decreas-
ing true slip. If, on the other hand, the conditions fixing the
efficiency are such as to place the propeller well over beyond the
maximum in the direction of increasing slip, then it is better to
overrate the value of w rather than the reverse. This is because
the consequences will be a decreased true slip and a better
efficiency than that assumed. Bearing these points in mind we
may usually take the values of w and s 2 in such way that the
probable error will either increase the efficiency or insure the
minimum amount of decrease. These remarks relate simply, of
course, to consideration of efficiency. It will very commonly be
found that the final results adopted are largely dependent on
structural and other considerations, aside from efficiency.
According to Blechynden, whose experiments on propellers
have been mentioned previously, the best results will generally
be obtained by using values which would place the propeller on
the efficiency curve somewhat beyond the crest in the direction
of increasing slip. According to the same authority propellers
of low pitch-ratio, as for example i.o or i.i, are not efficient
when used for the propulsion of ships of full form, such as
cargo-vessels of block coefficient from .7 to .8. On this
PROPELLER DESIGN. 283
point, however, there is much conflicting testimony, there
being considerable data indicating in a general way the suit-
ability of moderate pitch-ratios for ships of full form, and of
higher pitch-ratios for ships of moderate and fine forms. The
question seems to turn principally on the effect due to the
augmentation, this being more pronounced on a full than on
a fine after body. Now the augmentation increases in
general with the diameter and with the pitch-ratio. Hence,
so far as this effect is concerned, we should expect the best
results from moderate pitch-ratio and small diameter, and
these two conditions, by equation (i), 50, are seen to be
mutually consistent. On the other hand, with very full ships
there will be at the stern a certain amount of dead or eddy-
ing water. If the propeller is so small and so near the stern
as to work for the most part in this eddy, its supply of water
will be incomplete and irregular, and the efficiency will be
correspondingly decreased. Judged, therefore, from this
standpoint we should expect the best results from a propeller
placed as far aft as the structural arrangements will allow,
and of a diameter sufficient to reach out through the eddying
water into that possessing regular stream-line motion. Such
a value of the diameter would naturally go with a relatively
large pitch-ratio. We have therefore in this case, as in
nearly all others with which we have to deal, opposing con-
siderations, one indicating a small diameter and the other a
large. The actual best value in any given case will neces-
sarily depend on the balance of these considerations, and for
this, in the present state of our information, no general rule
can be framed.
In connection with the influence of a very full stern,
reference may be made to an extreme case, instances of which
284 RESISTANCE AND PROPULSION OF SHIPS.
have been said to occur. In cases where the inflow of water
is more or less hindered, the action of the propeller may
approach more and more nearly to that of a centrifugal
pump, delivering the water with a large radial or transverse
velocity and with slight longitudinal acceleration, and thus
absorbing energy with but slight gain of thrust. The defect
of pressure forward of the propeller will, however, produce in
full measure its effect on the hull, and in the extreme case
the thrust gained may be less than the sternward resultant
due to this defect, and thus the final resultant force on the
ship may be aft rather than forward. In such case the ship
would move aft no matter in what direction the propeller
turned. Such cases have been reported, although, of course,
they are not possible with usual forms.
The Number of Propellers. This question is one which
depends chiefly on the desired subdivision of the power, the
available draft, and the desired number of revolutions of the
engines.
In many cases the total power is more than it seems desir-
able to transmit with one shaft, and thus two or more shafts
and propellers are fitted. This was the case with the U. S.
triple-screw cruisers Columbia and Minneapolis, in which it
seemed desirable at the time of their design to subdivide the
total power into three units for transmission to the propellers.
This was only one of several reasons for adopting this
arrangement; but as such it occupied a prominent place in
the discussion of the general problem. Similar considera-
tions hold for modern ocean-liners developing from 20,000 to
30,000 I.H.P. In these cases it seems desirable to transmit
the power to the propellers in two units, and if there were no
other reasons for subdivision, it is doubtful if under present
PROPELLER DESIGN. 285
conditions engineers would wish to transmit the entire
amount by one shaft. Then, aside from reasons of this char-
acter, we have the more important considerations of subdivi-
sion for safety and for manoeuvring power. The duplication
of propelling machinery gives vastly greater security against
total breakdown or disablement, and this, as well as the added
manoeuvring power for war-ships, furnishes considerations of
sufficient importance to justify or rather demand the fitting
of twin-screws in all such cases.
Again, the average draft may be so limited or the power
so great that a single screw would be of too large diameter
for the necessary immersion of its blades. This would be the
case with most of the fast liners and war-ships, and with all
moderately fast vessels of light draft. In extreme cases the
draft may be so small that three or more propellers might be
needed in order to absorb the necessary power with the
limited diameter permissible. There seems to be no reason,
so far as efficiency of operation is concerned, why such sub-
division, if needed, should not be made. For such cases,
however, attention may be called to the peculiar advantages
of the turbine propeller as described in 40.
With regard to the question of revolutions, it is obvious
that the smaller the propeller, pitch-ratio and speed being
the same, the higher will be the revolutions. It may arise,
therefore, that the desired number of revolutions, if high, can
only be attained with twin- or multiple-screws.
In the question of the applicability of triple-screws to
cruising war-ships, one of the chief considerations not already
mentioned relates to the resultant possibility of cruising at
low speeds with one or two engines working at nearly full
power, rather than with all engines working at a small frac-
286 RESISTANCE AND PROPULSION OF SHIPS.
tion of full power. In such cases the thermodynamic and
mechanical efficiencies should be improved. The screws not
in operation, however, are dragged, and this increases the
resistance. The slip also usually increases, especially with
one screw, and this may result in a loss of propulsive effi-
ciency. In practice, therefore, it has been found, both in this
country and in Europe, that the use of one or two screws at
low speeds does not result in increased efficiency. In a
number of cases the coal required for a given speed with one,
two, and three screws showed but slight variation, and the
trials seemed to indicate an approximate equivalence in
general efficiency. Notwithstanding this, triple-screws have
been extensively introduced into the practice of the German
naval authorities as well as elsewhere, the chief reasons being
the possibility of smaller and lower engines, smaller castings
and easier construction, better stowage of the engines in the
lean after body of fast ships, and advantages arising in time
of actidn from a triple subdivision of the power and engine-
room personnel.
In the general question of single- or multiple-screws there
is probably no necessary difference in propeller efficiency alone,
except such as may arise from a better possible selection of
characteristics in one case or another. With the proper
estimate of wake and proper design throughout there is
reason to believe that no especial advantage in propeller
efficiency can be expected on either hand, and that the choice
must be made rather upon the considerations discussed above.
Location of Propellers. The location of a propeller
depends on the question of augmentation of resistance, on its
relation to the wake, and on various structural considerations.
As already seen, the augmentation of resistance is more pro-
PROPELLER DESIGN. 287
nounced with a full than with a fine form, and with the
former especially will vary in marked degree with the dis-
tance of the propeller from the stern-post. In such cases it
is believed that the propeller, if single, should be at a distance
from the stern-post not less than one half or two thirds its
diameter. As the distance is decreased from about this
amount the flow of water to the propeller becomes more and
more indirect and incomplete, the thrust developed becomes
less than it should be, and the augmentation effect becomes
more and more pronounced. With vessels of moderate and
pronounced fineness these effects are less notable in character,
and the propeller if single is usually placed directly aft of the
stern-post, at such a distance as the necessary structural .
arrangements make convenient. In torpedo-boats, launches,
and yachts the propeller is frequently dropped below the keel
line and is sometimes placed aft of the rudder. This favors
a full and free flow of water to the propeller, and tends to
decrease augmentation of resistance.
Twin-screws are usually placed in the same plane and
slightly forward of the stern-post. In such case the average
distance from the disk of the screw forward to the ship's sur-
face is much greater than with a single-screw, and the aug-
mentation of resistance is correspondingly less. In some
cases an aperture has been cut through the dead-wood oppo-
site the propellers in order to give an outlet at this point, and
thus relieve the stern from the periodic shock to which it
might be subject with the blades passing very near the sur-
face. In other cases the screws have been located in different
transverse planes, each with apertures, and with their disks
overlapping. The advantages claimed are decreased distance
between the shafts with a given diameter of propeller, or
288 RESISTANCE AND PROPULSION OF SHIPS.
greater diameter with given distance between the shafts. It
seems doubtful, however, whether the final results are such
as to recommend this arrangement for general adoption.
With three propellers the center one is usually more
deeply immersed than those at the sides, the three shaft-
centers projected on a transverse plane forming a triangle
with apex downward. The center propeller is also naturally
farther aft than those at the side, and with this distribution
there seems to be little interference between the propellers,
and no apparent loss of efficiency in action.
In general it should be borne in mind that the farther a
propeller is located from the stern of the ship the less in
general will be the resultant augmentation of resistance, and
the less also the valuable return from the wake. It is evident,
therefore, that so long as the propeller is at a sufficient dis-
tance from the stern to obtain a good flow 'of water and to
avoid an abnormal degree of augmentation there will be in
general no advantage in its farther removal, for the loss in
wake return will offset any further gain by way of a decrease
of augmentation.
52. SPECIAL CONDITIONS AFFECTING THE OPERATION OF
SCREW PROPELLERS.
Indraught of Air. The application of the equations of
the preceding sections presupposes that the propeller has
available a full supply of what is sometimes termed " solid
water"; that is, water without sensible admixture of air. It
is found as a result of the defect of pressure which exists in
the interior of the stream flowing to and through the pro-
peller, that if the blades in the upper part of their path
PROPELLER DESIGN. 289
approach near the surface of the water, and more especially if
they cut or break the surface, that air is drawn down and
mixed with the water. The result is a more or less frothy or
foamy mixture, the density of which is much less than that of
water. In consequence the thrust for given revolutions and
slip is decreased, or for given thrust the revolutions and slip
must be increased. In either case the efficiency is generally
decreased. If the tips of the blades come very near the sur-
face or actually cut it, the loss in thrust and efficiency may
become serious, especially with large pitch-ratio and high
peripheral velocity.
It should be clearly understood that a loss of efficiency is
by no means necessarily attendant on the working of a pro-
peller in foam, or on the mere fact that the tips of the blades
come near the surface or actually cut it. The whole ques-
tion, so far as the operation of the propeller is concerned, is
determined by the resultant true slip at which it works. It
would be perfectly possible to design a propeller with blades
cutting the surface but with sufficient diameter so that even
with the indraught and admixture of air the true slip would
be within the proper range for good efficiency.
In order, however, to keep down the size of the propeller
and to insure its working in water prac'.icaily free from air, its
diameter should be so chosen with reference to the draft of
water at the stern, as modified by the possible change of
trim and stern-wave, that the tips of the blades will always
be well covered. Experience seems to indicate that the
desirable immersion of the tips of the blades when nearest
the surface should not be less than about .2 or .3 the
diameter of the propeller. In addition to the indraught of
air from the surface there is always under normal conditions
290 RESISTANCE AND PROPULSION OF SHIPS.
a certain amount of air held in absorption by the water.
This air is given off more or less completely about the edges
and backs of the blades in consequence of the partial vacua
which exist at these points. This will give rise unavoidably
to a small amount of foam in the wake, the consequences of
which, however, are believed to be insensibly small.
Very Full Form at the Stern. Mention has been made of
this feature in the preceding section.
Racing. When a ship is in a seaway the blades are con-
tinually varying their distances from the surface or actually
emerging from it, and the revolutions undergo correspond-
ingly sudden and perhaps extreme variations, unless by some
form of governing device the supply of steam to the engine
is suitably controlled. Even at the best the revolutions, and
hence the slip, will frequently vary through very wide ranges.
Such variations may also be due partly to the effect of the
internal motion of the water composing a wave, the move-
ment being forward in the crest and aft in the hollow. Such
variations in the speed of the propeller are favorable to the
production of eddies, and thus to a waste of energy and loss
of efficiency. There is in addition a further loss in efficiency
due to the wide range of slip through which the propeller
works. We know that for the best results the propeller should
work at or close about a single value of the slip. But in violent
racing the slip for the propeller as a whole may vary from per-
haps a negative value to 50 or 75 per cent or even more.
It is readily seen that such wide variation in the value of
the slip must inevitably cause a serious falling off in the aver-
age efficiency. Other things being equal, these effects will
be more pronounced the more readily the propeller-blades are
PROPELLER DESIGN:
291
partially uncovered, and hence the nearer they are to the sur-
face of the water in normal trim. We have here, therefore,
an additional reason for good immersion of the propellers. It
is here that twin-screws gain one of their advantages, their
diameter being less and possible immersion greater than for a
single-screw to absorb the same total power.
Cavitation or the Effect due to Very High Propeller Veloci-
ties. If a blade such as A&, Fig. 76, is drawn through the
FIG. 76.
water as indicated, the space immediately in its rear is ren
dered more or less empty on account of the time required for
the water to close in behind the moving blade. At slow
speeds this open space will only extend slightly below the
surface of the water, while if the speed is sufficiently great it
may reach to the bottom of the blade, being shaped in con-
tour somewhat as shown by BCD. The velocity in feet per
second with which water tends to flow into an open space is
given by the general formula v = V2gh. Where the space
is open to the air, as in Fig. 76, h = simply the head of
water. Thus, for example, at the bottom of a board immersed
to a depth of 4 feet v = 16, while if the depth is I foot
7- = 8. Where the space is shut off from the air and is a
perfect vacuum, h = the head due to the atmospheric pressure
plus that due to the water, or h = 34 feet plus depth of
2Q2 RESISTANCE AND PROPULSION OF SHIPS.
water. In any actual case the vacuum cannot be perfect, but
will contain water-vapor and some air given off from the
water. In consequence the values for such case will be less
than those resulting from the head given above, which must
be considered rather as an upper or limiting value.
Now if the velocity of the blades of a propeller is suffi-
ciently high it is evident that such spaces will be formed at
the back in this case of course entirely under water. When
this phenomenon is present in any marked degree the water
acted on by the propeller is found to become turbulent
instead of measurably continuous, and with further increase
of revolutions the gain in thrust is very slight.
This phenomenon has been investigated by M. Normand *
for a torpedo-boat fixed in location instead of in free route.
The diameter of propeller was about 6.5 feet and mean pitch
7.7 feet. The center of the shaft was maintained at four
immersions varying from 4.8 to 3.85 feet. The curves con-
necting thrust with revolutions are as shown in Fig. 77, in
which the ordinate is the square of the revolutions N, and
the abscissa is the resultant thrust. The regular increase of
thrust with TV 2 for the deepest immersion is shown by the
approximately straight line OD. For the lightest immersion
and moderate revolutions the thrust was slightly greater than
for the deepest immersion, but the difference was not large,
and the increase with the square of the revolutions was
regular. After reaching about 135 revolutions, however, the
rate of increase of thrust with revolutions began quickly to
decrease; and from this point on there was but slight increase
of thrust for increase of revolutions up to about 230, the
* Bulletin de 1'Association Technique Maritime, vol. iv. p. 68.
PROPELLER DESIGN.
293
maximum number obtained. Very near the end, however,
as shown by the curve, the rate of increase was slightly
greater than at intermediate points. M. Normand suggests
that the curve between B and C corresponds to a period of
1000 2000 3000 4000 5000
THRUST IN KILOGRAMMES.
FIG. 77.
6000 7000
instability of the open spaces, during which they are being
formed and again filled with more or less irregularity, thus
increasing the turbulence of the water, while beyond C the
condition is more permanent, and the increase of thrust with
N* is somewhat more rapid. The remaining curves give
similarly the thrusts for intermediate immersions.
The direct application of this to the case of vessels in
free route is somewhat uncertain, because the conditions are
quite different. The slip is much less, the amount of water
handled much more, and the actual velocity of the blades
2Q4 RESISTANCE AND PROPULSION OF SHIPS.
through the water greater for the same revolutions. It
seems quite certain that in free route the revolutions might
be considerably greater without rupture of the column than
for a boat fixed in location.
These phenomena have also been independently examined by
Parsons* and Barnabyf in England. According to the latter,
the limiting conditions when cavitation is about to set in are
more readily expressed in terms of the average thrust developed
per unit of projected blade area than in terms of velocity, and
both investigations furnished for this limiting thrust substantially
the same value of 1 1 J- pounds per square inch, the immersion of
the tips of the blades being n inches. At the point where this
average thrust per square inch of projected blade area was reached
signs of cavitation began to appear, and on increasing the
revolutions the phenomenon became more or less pronounced,
and but little additional thrust was gained. Barnaby states
also that for every additional foot of immersion the thrust per
square inch may be increased about f pound. The value
should also vary slightly with pitch-ratio, being less if the
latter is large; but the influence due to this feature is so small
as to be practically negligible.
It follows, therefore, according to these investigations,
that from n to 12 pounds per square inch of projected blade
area may be considered as the maximum thrust which can be
efficiently developed, and if more than this is required it can
only be attained with difficulty and at a loss of efficiency.
The phenomenon of cavitation has presented itself more or
less prominently in connection with the modern torpedo-
* Transactions Institute of Naval Architects, vol. xxxvin.
Ibid.
PROPELLER DESIGN 295
boat destroyer and boats of similar type or condition. In such
cases 3000 to 4000 I.H.P. provided for each of two pro-
pellers 7 to 8 feet in diameter at 300 to 400 revolutions
develop a speed of 33 knots and upward. It seems probable
that this phenomenon may cause trouble with further increase
of speed, especially in boats of this character; and the all-
important question in such cases is therefore as to the best
disposition of proportions and dimensions for the develop-
ment of the maximum thrust per unit area without undue
sacrifice of efficiency due to this cause. The obvious way to
avoid the trouble, where possible, is to provide a sufficient
projected area of blade to reduce the average thrust below
the limiting value.
Springing of Propeller-blades. From the laws regulating
the distribution of pressure over a plane moving through the
water as explained in 6, it .follows that the outer portions
of propeller- blades, or such portions as may be considered
substantially equivalent to thin flat surfaces, will tend to place
themselves normal to the stream or line of flow. This would
result in a tendency to increase
the pitch and slip. The natural
tendency of the inner portions or
those with a sensibly rounded
back will be quite different,
readily shown by experiment that
when placed slightly oblique the
tendency of such sections is indeed
to place themselves at right angles to the stream, but, as
illustrated in Fig. 785 the incipient movement is clockwise
instead of the reverse, the final position toward which the
296 RESISTANCE AND PROPULSION OF SHIPS.
section tends being as shown in dotted lines, with convex
side toward the stream. This will result in a tendency to
decrease the pitch and slip. (Compare also 42, Fig. 70.)
In addition to the tendency to spring due to the irregu-
larity of the distribution of pressure, the blade as a whole
will bend more or less under the influence of the thrust.
This bending will be accompanied by a slight untwisting of
the blade, thus tending toward an increase of pitch and slip.
This influence is relatively more important than the others
mentioned above, and hence in the complex effect the result
is usually a slight untwisting of the blade with increase of
pitch and slip. With blades of the usual stiffness it does not
seem likely that this effect will be sufficient to produce any
sensible influence on the performance of the propeller. In
extreme cases, however, where the blades have been thinned
beyond the proper limit, the effective pitch and slip might
be so increased as to sensibly affect the efficiency usually
for the worse.
53. THE DIRECT APPLICATION OF THE LAW OF COM-
PARISON TO PROPELLER DESIGN.
Instead of the method of propeller design indicated in the
preceding sections, it is possible to apply directly the laws of
comparison as explained in 26. As required by the nature
of the law, we assume :
(1) Similarity in geometrical form and situation with a
linear ratio A, the same as that which exists between the
ships themselves.
(2) Corresponding speeds as defined for ships, or linear
speeds in the ratio of A*. This condition will evidently apply
PROPELLER DESIGN. 297
only to corresponding points on the two propellers. Let r l
and r^ be the radii of such points on the two propellers, and
N l , N^ the revolutions. Then for equal slips the actual or
linear speeds of these points will be proportional to r^N^ and
rN t . Hence we shall have for corresponding speeds
But -' = A.
r,
Hence W = tf'
Hence at corresponding speeds the revolutions will be
inversely as the linear speeds, or inversely as the square root
of the linear dimension-ratio.
It must be remembered, further, that these assumptions
implicitly involve equal percentages of slip, equal values of
the wake factor, equal values of the factor of augmentation
due to the propeller, and equal mechanical efficiencies for the
engines.
It then results, on the assumptions of 26, that the ratio
between the powers necessary to drive the two ships is A*.
With the propellers thus assumed, it is seen that they fulfil
in all points the conditions of the law of comparison, consider-
ing it applicable to the whole resistance. Hence the ratio of
the resistances to transverse motion, the ratio of the thrusts,
and in fact the ratio of all similar components of the forces
acting on them, will be A 3 . The speed-ratio is A*. Hence the
ratio between the amounts of useful work will be A 5 , the
same as that for the total amounts; or, in other words, the
298 RESISTANCE AND PROPULSION OF SHIPS.
two propellers thus situated will give similar results with
equal efficiencies.
We have therefore
For example, let A, = 1.44, z^ = 16 knots, ^V, = 100.
Then A* = 1.2 . A.* = displacement-ratio = 2.986, and A* =
power-ratio = 3.58. Then u^ = 19.2 and N % = 83.3. It
follows, therefore, that the larger ship would require a propeller
1.2 times the size for the smaller, and with an expenditure of
3.58 times the power at 83.3 revolutions would drive it at a
speed of 19.2 knots, with the same efficiency as for the
smaller.
This method by comparison is really a special case of that
discussed in the preceding section. The method by com-
parison is a restricted mode of applying the results of a given
propeller to the design of one of similar form under similar
conditions and at corresponding speeds, while the method of
50 assumes such laws and relations as will provide for the
application of the results of a given propeller to the design
of one of similar form under similar conditions but at any
speeds. The relation between 50, (i), and the law of com-
parison is readily seen by applying the former to the design
of a propeller using the data of a similar case at correspond-
ing speed. Taking first the latter, we have
PROPELLER DESIGN. 299
We suppose in this equation /, d, /, and N as well as the
speed and all characteristics of the propeller to be the given
data. Then the resulting combined factor (klm) is readily
found. We next propose, by means of this general equation,
to use this data for the design of a propeller for a similar ship
at corresponding speed. Assuming the same slip, pitch-ratio,
and other proportions for the propeller, the factor (klm) will
be the same in both cases. /, from the law of comparison
for power will be A* /; , and since the speeds are correspond-
ing, (piNi)2 will equal fi(piNi)i. Hence for the second case
we shall have
^U l = ^(p l N l } l \d l ^(klm) l ...... (2)
Comparing this with (i), it is evident that
or < a = \ l ;
whence /, = Xp lt
and N t = J-JV;.
Hence the propeller will have the same factor of similitude
as the ship, and will have its revolutions in the inverse ratio of
the square root of this factor, the same as if determined by
the law of comparison discussed above. In other words, the
application of 50, (i), to such a case in the manner above
shown would give exactly the same results as would be given
by the direct application of the law of comparison itself.
54. THE STRENGTH OF PROPELLER-BLADES.
The blade of a propeller may be considered as a beam
supported at one end, the root, and loaded in a variable and
300 RESISTANCE AND PROPULSION OF SHIPS.
complex manner according to the distribution of pressure
over its surface. As in 35, the amount of this pressure may
be approximated to, and by the proper treatment as explained
in mechanics, and by approximate integration, the amount
of the resultant bending moment at the root may be de-
termined.*
In such case, however, we must not forget that the as-
sumptions are necessarily somewhat inexact, that the actual
pressure will undergo wide variations due to irregularities in
the wake, and that as in all structural design a considerable
factor of safety must be introduced in order to allow for the
unknown and uncertain elements necessarily involved in the
operation of the propeller. We propose, therefore, to sub-
stitute for the actual blade a rectangular plane area set at an
angle ot to the transverse. The value of this angle will be
considered at a later point.
Let n be the number of blades, and T the actual thrust.
Then T -T- n thrust for one blade, and (T -=- n) sec a will
be the normal pressure on the blade. The center of this
pressure will correspond to the center of a system of forces
varying as the square of their distance from the axis. Hence
denoting the outer radius and that of the hub by r l and r a we
have, as the distance from the axis to this center of pressure,
_
~'
If we take approximately r a = .2r lt we find
r = .756^,, or sensibly .75*V
We shall also omit all account of bending moment due
to the tangential forces, as they will in any case be small in
* See Taylor, " Resistance of Ships and Screw Propulsion," p. 208.
PROPELLER DESIGN. 301
amount, and will be sufficiently provided for by the factor of
safety, and by the reference of our final equation to actual
results for the determination of the value of the empirical
factor which is to be introduced. The bending moment at
the root reckoned normal to the face is therefore
T
M= (.75^1 r t )- sec a (i)
Now let b denote the length of the section where the
blade joins the hub, and / the maximum thickness. We may
safely consider that the geometrical characteristics of this
section will not widely vary. In actual form it closely re-
sembles a circular or parabolic segment. The general equation
for the beam gives us
KI
M= 9
y*
where K = stress at outer fiber ;
j = distance from neutral axis of section to such fiber;
/ = moment of inertia of section about neutral axis.
With the suppositions above made regarding the nature of
the section we should have
proportional to / a .
Hence we may put
/
(2)
and (.75r, - rj sec a = KQbf. ... (3)
302 RESISTANCE AND PROPULSION OF SHIPS.
Now, turning to Fig. 72, we readily see that
a(I.H.P.)'x 33QOQ
-
where a represents the ratio (Thrust H.P.) -r- (I.H.P.).
For OL we shall take the inclination at the assumed center
of pressure, or rather at ./r l in order to allow somewhat for
the rounding off of the blades at the outer end. We have,
then,
/ P
tan a = L - - r.
Then if p -f- d c as heretofore, we have
tan a = and sec a = r I -I
2.2 4.84
Collecting and substituting in (3) we have, finally,
, , _ 33QQo(.75r. - r.XI.H.P.) 4/i + c* + 4.84
(i - s^pNKQn
This may be further simplified by making the following
assumptions, which are quite admissible, considering the un-
certainties necessarily involved :
r 2 = .2r, , as above,
(i j,) and a sensibly constant.
Then uniting all constants in one, which we denote by A*,
we have
. H.P.) 4/i +t-H- 4.84
- (6)
Also, let us put ' = < and I. H.P. =
PROPELLER DESIGN.
33
Then
Nn
or
/ =
He
bNn
(7)
(8)
In this it must be remembered that A includes the
strength of the material, and hence its value must be taken
accordingly. This result, it must be remembered, gives the
thickness at the actual root of the blade. The fillet by which
it is connected to the hub is of course extra, and is not here
considered. The values of e may be taken from the following
table :
c = p H- d
I
I.I ............ 1.02
1.2
1-3
1.4
1.5
1.6
e
. 10
1.7 . .
d
.02
1.8 .
.Q^
I.Q .
.7 J
.89
2.O. .
.8<;
2.1 ..
.81
2.2 . .
.77
2.3 ,
e
74
.72
.70
.68
.66
.64
.63
Now comparing this equation with a large number of
propellers which have shown sufficient strength, it is found
that for bronze or steel A may be taken at from 9 to 12, while
for cast iron its value should be increased to from 14 to 17.
Collecting for convenience these various items, we have
therefore
304 RESISTANCE AND PROPULSION OF SHIPS.
where / = thickness in inches at root of blade;
/f=I.H.P.; '
e factor taken from the table above ;
b length in inches of section at root of blade;
N = revolutions per min. ;
n = number of blades ;
9 to 12 for bronze or steel;
A =
^ 14 to 17 for cast iron.
MATERIALS SUITABLE FOR SCREW-PROPELLERS.
Though the fundamental purpose in the present volume is
the design of the dimensions, and form of propellers, yet some
brief notice of the materials suitable may not be out of place.
Cast iron, cast steel, brass, gun-metal, and the various
bronzes are used. Cast iron is the cheapest, but being rela-
tively weak and brittle the blades are necessarily thicker and
less efficient than with steel or bronze. The strength avail-
able is usually from 20000 to 25 ooo Ibs. per square inch of
section. Cast-iron blades may be broken short off by striking
logs or obstacles, and if such collisions are likely to occur it
might be better to have cast-iron blades than steel or bronze,
as the rupture of a blade might spare more serious results
arising from strains on the engine and stern of the ship.
Usually, however, we wish the blades to hold and not to
break, and for this purpose cast iron is at a relative disad-
vantage.
Cast steel is stronger than cast iron, its ultimate strength
in castings suitable for propeller-blades ranging from 50 ooo
to 60 ooo Ibs. per square inch of section. The sections may
therefore be made thinner, and a better efficiency obtained in
so far as dependent on this feature. The surface is naturally
PROPELLER DESIGN. 305
not as smooth as that of cast iron, but with improved
methods of production the difference in this feature is insig-
nificant.
Bronzes have naturally a smoother surface, and seem,
furthermore, to have a lower coefficient of skin-resistance.
This added to their strength and good casting qualities makes
possible a relatively smooth thin blade with sharp edges, all
of which are features favorable to good efficiency. The
strength available with the best bronzes varies from 40 ooo to
60 ooo Ibs. per square inch of section. With ordinary gun-
metal from 25 ooo to 35 ooo Ibs. per square inch of section
may be allowed, while with common brass not more than
20 ooo to 25000 Ibs. should be depended on. Of these
various alloys, manganese-bronze is probably more used than
any other, due to its better combination of desirable qualities
such as strength and stiffness, good casting qualities, resist-
ance to corrosion, etc. Care is needed in the manipulation of
the bronzes in melting, pouring and cooling in order to obtain
the full benefit, but with such care the product is homogene-
ous and reliable. Its greater relative cost restricts its use,
however, to war-ships, yachts, and launches, ocean-liners, and
other cases where the importance of a saving in propulsive
efficiency is considered worth obtaining at a slight increase in
first cost.
The durability of propeller-blades is in the order: bronze,
cast iron, cast steel. The two latter usually deteriorate by
general corrosion and local pitting, the average life being
usually from five to ten years. The life of bronze blades is
practically indefinite, or at least as great as that of the ship
itself.
306 RESISTANCE AND PROPULSION OF SHIPS.
55. GEOMETRY OF THE SCREW PROPELLER.
Some general notions relating to the geometrical form of
the screw propeller have been given in 34. We have now
to give a more detailed account of the principal forms which
may be produced.
We may define the surface of a screw propeller in general
as a surface generated by a line / moving on two other lines as
guides, of which one, a, is straight, forming the axis, and the
other, b, lies in the surface of a cylinder of which a is the axis.
This surface being developed, we have the actual form of b.
For the common uniform-pitch propeller with blades at right
angles to the axis it is readily seen that / is straight and at
right angles to a, and that the developed b is straight and at
an angle OL with the transverse plane such that we shall have
pitch 2nr tan a, (i)
where r is the radius of the cylinder in which b is supposed
to lie.
The line / is the generatrix ;
" "a" " axis;
" " b " " guide, or guide-iron.
We will denote b when developed by b.
The variations usually found in / are as follows :
(1) Straight and at right angles to a\
(2) " " inclined to a\
(3) Bent or curved in an axial plane ;
(4) " " " "a transverse plane.
The axis a .is invariable in character.
PROPELLER DESIGN. 307
The guide b may vary as follows :
(1) Straight;
(2) Curved.
The curvature under (2) is usually such that the angle a f
and hence the pitch, increases from the forward to the after
edge of the blade.
In the actual generation of the surface we may also have
variations according as to whether / changes its angular position
relative to a or not. If the longitudinal component of the
motion of the point of contact of / with b is greater or less than
that of the point of contact of / with a, the pitch will vary
radially, or from the hub outward. In such a case practically
two guides would be required one near the hub to suit the
pitch at that radius, and the other as usual at the outer end
and suited to the pitch at that point.
Let c denote the ratio of the longitudinal velocities of the
points of contact of / with a and with b. Then we may have
three cases:
(1) c equal to I ;
(2) c greater than I ;
(3) c less than i.
All common varieties of helicoidal surfaces may be gener-
ated by giving to /, ~b, and c the different variations as above
enumerated. We may have, however, other forms of heli-
coidal surface which cannot be generated by a rigid line in any
of the ways above mentioned. We may most readily conceive
of such a surface as made up of the summation of an indefinite
number of guides b, and thus as generated by a variable guide
moving radially, and perhaps angularly and longitudinally as
well, and taking at the same time the successive shapes and
inclinations as determined by the nature of the pitch at the
308 RESISTANCE AND PROPULSION OF SHIPS.
successive radial locations. In this way any form of blade
and any distribution of pitch, no matter how complex, may
be geometrically determined, and by the use of a reasonable
number of guides actually produced in the form of a pattern,
or moulded direct in the foundry.
Usually, however, the surfaces of propellers are such as
can be generated by a rigid line, and of the great variety
which may thus be formed but few are commonly found.
These are as follows :
IJf^ gives the common propeller of uniform pitch ;
IJ>\ C \ gi ves tne propeller of uniform pitch with straight ele-
ments bent back from the plane of revolution ;
IJ>\ C \ gi ves the propeller of uniform pitch as in the last, but
with the elements curved away from the plane of revo-
lution ;
ljb l c l gives the propeller of uniform pitch with elements bent
or curved in the plane of revolution ;
/i# t , gives the common expanding-pitch propeller, the pitch
increasing from forward to after edge.
We may also form these various combinations with c t or c
and thus introduce radial variation of pitch in any way
desired.
It is also possible to greatly vary the appearance of a pro-
peller-blade by varying the location of the part actually taken
for the blade. Thus from a helicoidal surface of uniform
pitch, by appropriately locating and shaping the contour, we
may produce a blade which has the appearance of being bent
back from the plane of revolution, and which on casual exam-
ination may seem to be the same as that given by l, t b l c l or
IJb^c^ though such is by no means the case.
PROPELLER DESIGN. 309
While thus an indefinite variety of propellers are possible
as regards shape of blade and distribution of pitch, it must be
clearly understood that so far as experimental or actual results
are concerned we are not in a position to select among them
any one set of characteristics as the best, and especially any
one distribution of pitch, or indeed to say that any distribu-
tion of pitch will produce a propeller superior to that of uni-
form pitch. It is possible that some distribution may be
superior to the uniform, but we are not at present in a posi-
tion to either confirm or disprove this by experimental data.
We find therefore a very general tendency to consider the
propeller of uniform pitch as the standard in type. The bent-
back blades as given by /?,, has been much used on tor-
pedo-boats, yachts, and small craft, the presumable reasons
being a possibly better hold on the water, and a decrease in
the so-called short-circuiting around the tips of the blades.
In regard to the influence of any fanciful variation of pitch,
we have but to remember the nature of the wake as discussed
in 41. It is there shown that relative to a uniform pitch
any propeller-blade must necessarily work with an excessively
varying distribution of slip, both over its surface and during
a revolution, and hence relative to a uniform slip it must act
as though the pitch were correspondingly variable. A pro-
peller of pitch variable according to some definite law will
therefore not give a corresponding distributed variation of
slip, and its actual variation of pitch is so small as to be en-
tirely swallowed up in the greater wake variation of slip, so
that its whole relation to the wake will be but slightly differ-
ent from that of a uniform-pitch propeller, and in any event
entirely different from that in a uniform stream.
Any benefit supposed to arise from some particular distri-
310 RESISTANCE AND PROPULSION OF SHIPS.
bution of pitch and slip will therefore be wholly fanciful or
accidental, since in the actual case the variation of slip corre-
sponding to that of pitch will not be even approximated to.
See also 42 and 44.
The Laying Down of a Screw Propeller. We take as
the simple and standard case a propeller of uniform pitch
as given by A?,^. We must have the following details given
or assumed :
Diameter.
Pitch.
Area.
Diameter and Shape of Hub.
Shape of Blade.
We may also assume for simplicity that the blades are to
be cast with the hub, and that they are to be symmetrical
about a radial center line.
Let XX, Fig. 79, be the axis and OY a perpendicular,
on which the center line OB is taken equal to the radius of
the propeller. The hub is then laid off as in the figure. The
blade area being known, we readily find the mean half-width,
and using this as a general guide sketch in a trial half-
contour AGB. This area may then be checked by planimeter
or otherwise, and adjusted until it is correct in amount and
the contour is satisfactory. The length of the root-section is
usually taken about .9 that of the maximum width.
The distance CB is then divided into any convenient
number of parts, measuring from either B, C, or , as may
be most convenient.
From (i) the value of tan a for any point F is
tan a = = ~- r -2- -^ OF.
2nr 2n 2n
PROPELLER DESIGN.
If, therefore, we take a point E such that OE = / ~- 2n,
the angle EFO will give the corresponding value of a. A
similar relation holds for all other points, so that if from E a
Y
FIG. 79.
bundle of lines be drawn to the several points on the radius,
the inclinations of these lines to OB will give the correspond-
ing values of a.
312 RESISTANCE AND PROPULSION OF SHIPS.
Hence if EH is taken equal to FG, it is evident that El
is the projection of FG on a longitudinal plane, and HI its
projection on a transverse plane. If, therefore, FK is made
equal to El, .AT will be the projected view of G, and making
FL = FK, we have two points on the projected contour of
the blade. In this way we may find the entire projected
view of the blade as shown. The other projections are read-
ily found in a similar manner; but as they involve only an
exercise in descriptive geometry, the details need not be here
explained. It is readily seen that, so far as the contour is
concerned, this whole construction is approximate, and not
accurate; and it may be mentioned that in the projection on
a transverse plane the lines such as GG l may be taken as
arcs of circles with OF as radius, and the distances such as
HI would then be laid off along these arcs, or practically as
chords.
If the blade is not symmetrical about OB, it may be laid
off in any form desired, and then the two parts must be pro-
jected separately, but otherwise in the same general manner.
For detachable blades the same general method is used,
the form at the root being appropriately modified to suit the
means of attachment to the hub. In any case the blade joins
the hub, or, if detachable, its circular boss, by a well-rounded
fillet.
For blades of more complicated form and distribution of
pitch the same general principles may be applied to find the
approximate appearance of the projections. It may be ob-
served, however, that at best these are but approximate, and
moreover are of no practical value to the pattern-maker or
moulder. The information of which he makes direct use is
PROPELLER DESIGN. 313
much more restricted in amount, as will be explained at a
later point.
Returning to Fig. 79> the next step is to lay down the
thickness CM at the root of the blade. A straight line, MB,
is then drawn to the tip. This will give the middle-line
thickness of the various sections. At the tip such line must
be run into a curve as shown, in order to insure good casting.
For bronze propellers such thickness may be taken at about ^
inch per foot of diameter. For cast iron it must be slightly
thicker. The varying sections or thickness strips may now
be put in. Thus SQ is made equal to ST, and a circular or
parabolic arc is run through P, Q, and R. When the outer
sections are reached, as in Fig. 80, such an arc would give too
FIG. 80.
thin an edge for reliable casting. We may, therefore, draw
lines at the ends as shown, making the edge of the requisite
angle, and then run in a smooth curve tangent to these lines
at the ends, and parallel to AB at the centre D. This mini-
mum angle may vary from, as little as 5 for small bronze
propellers, such as are used on torpedo-boats and fast yachts,
to 15 or 20 for cast-iron propellers of large size, such as are
used in ordinary mercantile practice. The smaller angles are
to be recommended.
The thickness is usually placed on the back of the blade;
that is, forward of the helicoidal surface which has been
laid out as the driving-face. For the sections near the
root, however, certain modifications are sometimes made,
based on the following considerations: AB, Fig. 8i; denot-
3*4
RESISTANCE AND PROPULSION OF SHIPS.
ing the face, we have by the usual construction EC = pitch
and DC = slip. Then, considering the water as undisturbed
and flowing axially to the propeller, we have DO as the direc-
tion of relative motion. Draw HJ parallel to DO. Then,
with the usual form of section, the back of the blade from B
to J will be met by this stream, and the resultant pressure
will have a component from forward aft, thus reducing the
FIG. 81.
effective thrust. If, however, the section is made as shown
by the full line, we have the back merely exposed to frictional
resistance, while the direct flow meets the face only, and may
thus give a slightly increased thrust. This change in section
amounts to a great increase of pitch on the driving-face from
B' to K. The simplicity of this construction, however, is
doubtless far from representing the path of the water in the
PROPELLER DESIGN. 3 J 5
actual case, although experimental investigation as referred to in
49 seems to indicate some advantage from this form of
section.
We have above referred to the fact that the information
actually needed by the pattern-maker or moulder does not
include the various geometrical projections there referred to.
The information strictly necessary is simply the following:
The principal dimensions and characteristics of the propeller
and hub; the expanded form or contour of the blade, as in
AGBG V A^ Fig. 79; the shape and angular position of the
generatrix, or else the form of a series of guides at successive
radial distances; the distribution of the thickness. The
necessary information may therefore be classified under these
four heads :
(1) The chief dimensions ;
(2) The contour of the blade ;
(3) The law of the pitch ;
(4) The distribution of the thickness ;
Thus for a uniform pitch propeller, the information given
in Fig. 79 aside from the projection AKB is all that is needed
to prepare a pattern in wood or sweep up the form in loam.
Instead of the above method some designers prefer to lay
down first the projection on a transverse plane, making the
projected area a certain fraction of the disk area. See, for
example, Fig. 53. The expanded form and the other projec-
tion may then be found by the inverse of the methods given
above.
In Plate A is given a copy of the working drawing for
the propellers of the U. S. battleships Indiana and Massa-
316 RESISTANCE AND PROPULSION OF SHIPS.
chusetts, showing the information provided in these cases of
naval war-ship design.
Measuring the Pitch of a Screw Propeller. From (i) we
have/ 2nr tan a. If therefore at a given value of r the
SCREW PROPELLER
BATTLE'S HIP NO 2,
3CALE
PLATE A.
value of tan a can be determined, the pitch is known. Such
measurement requires necessarily the establishment of a
transverse plane, or plane perpendicular to the axis of the
propeller. If the propeller is lying on a floor, the latter may
be used, care being taken that the propeller is so blocked up
that its axis shall be perpendicular to the floor. Otherwise a
plug carrying perpendicular to the axis a swinging arm may
be fitted to the shaft hole, thus giving a movable line of
PROPELLER DESIGN.
317
reference. If the propeller is in place on the ship, the arm
may be attached to the hub or to the shaft nut, or so
arranged in any way as to give a line of reference movable
or adjustable in the transverse plane.
We will first note an approximate method which consists
in determining at what value of r we have a = 45, tan
a = i, and hence / = 2nr. The transverse plane being
established as above, a bevel or triangle with angle 45 or
even an improvised sheet of metal may be applied, and the
corresponding location of r approximately found.
For more accurate measures we have to determine at a
selected value of r two sides of a triangle, as in Fig. 82. AB
FIG. 82.
lies in a transverse plane and on the surface of a cylinder of
radius r. AC lies on the face of the blade and on the same
cylinder, being in fact their intersection. BC is parallel
to the axis. The triangle ABC developed is evidently
similar to ADC of Fig. 55 for the same radius and pitch, the
relation being as shown in Fig. 83. If, therefore, we have an
arm swinging about the axis, we may by running a line per-
pendicular to this arm at a fixed value of r determine any
desired number of points on the line AC. The length of AC
may then be measured by a flexible rule or batten, or, if it is
short, it will not differ sensibly from the straight line joining
its extremities. BC is then determined as the difference
i8
RESISTANCE AND PROPULSION OF SHIPS.
between the axial distances from the points A and C to the
transverse plane or swinging arm. We have then
sm a =
BC
AC
and
and tan a =
2nrBC
BC
VAC* -- BC*
.
VAC 4 - BC
FIG. 83.
Still otherwise, we may measure the angle ft subtended at
the axis by the arc AB. We then have
AB = rft, tan a = - = -,
and
If the distance AC is small relative to the dimensions of
the propeller it gives the pitch simply for that particular
small part of the surface, and if the pitch is thus determined
in various parts of the surface it will usually be found to vary
very considerably even in a propeller intended to be of
uniform pitch. This arises from minor departures from
exactness in the pattern, mould, and casting. In rough
work such variation may be as great as 10 per cent on either
PROPELLER DESIGN. 319
side of the intended value, though with proper care the limit
of variation should be much less.
The details of the operation of measuring the pitch will
frequently vary with the means at hand and the particular
circumstances of the case ; but if the fundamental geometrical
problem to be solved is kept clearly in mind, such necessary
variations will present no difficulty.
Table for connecting the Slip A ngle with the Slip Ratio s.
This is deduced as follows: Referring to Fig. 55, we have
tan a = and tan (a 0) = ,
2nr 2nr
whence, with the nomenclature there used, we readily find
nys
tan = a a , r .
Table I gives the values of for varying values of y
and s.
Table for Determining the Modification of Pitch due to the
Twisting of a Blade. In propellers with detachable blades it
is customary to provide for twisting the blade slightly on the
hub, and thus modifying the pitch. The amount of modifi-
cation at any given radial distance or value of y, is found as
follows:
Referring to Fig. 55, we have in general
p = 2nr tan a = n\.w\ a
P !
whence tan a = --3 =
nd ny
320 RESISTANCE AND PROPULSION OF SHIPS.
Let ft be the amount of twist. Then, evidently,
p, = 7T</tan(> /?),
and A =
/ tan a
Since ft is always quite small, this may be put in the form
/, _ i /? cot __ i ftny
A "~ I =F ft tan " __ '
*r
Table II gives values of/, -T-/ O for varying values of ft
and j/.
It is thus seen that the amount of change varies consider-
ably for different portions of the blade. Hence if the pitch
were uniform before twisting, it cannot be so afterward.
The twist will have, in fact, transformed the surface from one
of uniform pitch, to one with a more or less irregular distri-
bution from root to tip of blade. This, of course, is by no
means necessarily prejudicial, and so long as the angle of twist
is small we have no reason experimentally to expect any
particular effect either good or bad due merely to the irregu-
larity of the pitch. The twisting thus provided for is for the
purpose of changing the effective or mean pitch of the pro-
peller as a whole, and the above considerations indicate that
so long as the amount of twist is small, we have no experi-
mental basis for objecting to this mode of obtaining such
modification.
As a final question we may ask what will be the mean
pitch of the blade thus modified. This will depend on the
definition of mean pitch as discussed in 42, to which refer-
ence may be made.
PROPELLER DESIGN.
321
TABLE I.
VALUE OF THE SLIP ANGLE 0.
Slip.
05
.10
15
.20
25
.30
35
.40
Diameter
Ratio.
. I
o 51
I 48
2 51
4 oo
5 17
6 44
8 21
10 12
.2
21
2 47
4 20
6 oo
7 49
9 46
ii 53
14 II
3
28
3 or
4 39
6 22
8 n
10 06
12 O6
14 13
4
25
2 54
4 26
6 02
7 41
9 24
II IO
12 59
5
19
2 40
4 04
5 30
6 58
8 28
10 00
ii 35
.6
12
2 25
3 41
4 57
6 15
7 34
8 56
10 18
7
5
2 12
3 19
4 28
5 38
6 48
7 59
9 12
.8
00
2 00
3 oi
4 03
5 05
6 08
7 12
8 16
9
o 54
I 49
2 45
3 41
4 37
5 34
6 32
7 30
I.O
o 50
I 40
2 31
3 22
4 14
5 06
5 58
6 5i
TABLE II.*
FACTOR FOR MODIFICATION OF PITCH DUE TO TWISTING OF
BLADE.
Twist
of
Blade.
x-
2
3
4
5
6
Diameter
Ratio.
Factor for
Factor for
Factor for
Factor for
Factor for
Factor for
Increase.
Decrease.
Increase.
Decrease.
Increase.
Decrease.
1
Decrease.
Increase.
Decrease.
Increase.
Decrease.
.1
.2
3
4
5
.6
'.S
9
I.O
.065
.040
.036
037
039
043
.047
.051
1-056
1. 060
943
963
.966
.965
.962
958
954
950
945
940
1.138
1.083
1.073
1.074
1.079
1. 086
1.094
1-103
1.113
1. 122
.891
.927
933
930
925
.918
.909
.900
.891
.880
.221
.127
.III
.112
. I2O
.130
I . 142
I.I56
I.I70
1.184
843
.893
.901
.897
.886
877
.864
.851
837
.821
315
175
152
.152
.162
.176
.192
.210
.229
247
.800
.861
.870
.864
.852
.837
.820
.802
783
763
425
.226
.194
193
.205
.222
.242
.264
.288
. 7 6l
.830
.840
.832
.817
.798
777
754
730
705
553
.281
237
236
.249
.269
293
320
348
376
725
.800
.811
.801
.783
.760
734
.707
.678
.648
* Taylor's " Resistance of Ships and Screw-propulsion," Table XV.
CHAPTER V.
POWERING SHIPS.
56. INTRODUCTORY.
THE various constituents of the power which it is neces-
sary to develop in the cylinders of a marine engine in order to
propel a ship at a given speed have been already discussed in
46. We have now to consider the various suppositions
which will enable us to make an estimate of the total power
thus required in any given case. These are of two general
classes and lead to two general methods for the estimate of
power.
(i) An assumption of the necessary constants which will
enable us to compute the resistance and the power required
to overcome it, together with that required for the various
other constituents going to make up the I.H.P. This also
may be done in either of two ways:
(a) We may make the' various assumptions necessary to
compute the different constituents individually. This in-
volves the following elements: the power for the resistance
proper or the E.H.P. ; the amount involved in the augmen-
tation of resistance; the wake factor; the propeller efficiency;
and the amount absorbed by friction. By a proper combina-
tion of these as indicated by Fig. 72 we readily work back to
the total I.H.P.
322
POWERING SHIPS. 323
(&) We may make the assumptions necessary to compute
the E.H.P. as above, and then assume the propulsive
coefficient or ratio E.H.P. -f- I.H.P., whence the latter fol-
lows directly from the former, and without special assump-
tions as to its other individual elements.
(2) The second general method involves certain assump-
tions which justify the extension of the law of comparison
from resistance to power, and thus enable us to compute the
I.H.P. in any given case from that actually observed for
similar ships at corresponding speeds.
We shall discuss these methods in order.
57. THE COMPUTATION OF W p , OR THE WORK ABSORBED
BY THE PROPELLER.
The necessary information and assumptions requisite for
the computation of R have been discussed in Chapter I and
in 45. The amount of augmentation of resistance must be
taken by judgment guided by such information relating to
similar cases as may be available. Using the E.H.P. as a
base, the additional amount of power thus required will
usually be found between 10 and 25 per cent. Definite
information with regard to the amount of this constituent of
I.H.P. is only generally obtainable by model experiments as
referred to in 30. With such appliances, the true and
augmented resistances are readily found.
Unless there is available some definite information based
on model experiments, relative to the probable amount of
augmentation, there is little use in attempting its independent
estimate.
The determination of the wake factor w has been already
324 RESISTANCE AND PROPULSION OF SHIPS.
referred to in 44 and 49. With the wake factor and
augmentation known or assumed, we readily pass, as ex-
plained in 46, from CF to C t , Fig. 72.
If, instead of estimating augmentation and wake factor
separately we should consider the ship efficiency ( 46) as I,
then we have, in Fig. 72, CF = C^E and propeller power or
BE = CF divided by propeller efficiency.
The considerations relating to propeller efficiency have
been discussed in 49 to 51. Guided by such information
as is available and as seems applicable to the case in hand, its
value is assumed. This, as we have seen, may in usual cases
be expected to lie between 65 and 73 per cent.
58. ENGINE FRICTION.
We come next to the amount of power absorbed by fric-
tion between the propeller and the cylinders.
We may remark at the beginning that there is very little
exact or satisfactory information on these points. Model
experiments are here of no avail, at least for purposes of
actual measurement. Such measurement would require at
the various speeds the simultaneous determination of the
power in the cylinders and of the power received at the pro-
peller. If, for example, just forward of the propeller there
could be inserted a transmission dynamometer which would
measure the turning moment, then the product of this by 2n
times the number of revolutions per minute would give the
power transmitted to the propeller. Such determinations
are obviously difficult and costly, and hence more or less
indirect methods are resorted to.
In the first place it is assumed that the total power
POWERING SHIPS. 32$
absorbed by friction may be expressed by an equation of the
form
W f = /iN+lW, (i)
where // is a constant for any given engine, N is the number
of revolutions, and hN is the amount due to what is termed
the initial friction, while W is the I.H.P. and /is likewise a
constant for the given engine.
The supposition that the total frictional power may be
thus expressed is quite arbitrary, but seems to answer the
purpose required as closely as present data can determine.
It assumes simply that there will exist at the various joints
and rubbing-surfaces a certain tangential resistance of which
a part proportional to h is sensibly independent of revolu-
tions or load, and represents what is termed the initial fric-
tion. The existence of this factor is chiefly due to the
pressure at the various stuffing-boxes, piston-rings, etc., and
to the weight of the shafting and other members supported
in bearings. This part of the total friction is also sometimes
known as that due to the dead load. The work necessary to
overcome such a constant resistance will vary directly as the
revolutions, and hence will be represented by a term of the
form JiN. It is further assumed that the remainder of the
tangential resistance at the rubbing-surfaces will vary directly
as the load on the engine, or as the mean effective pressure,
and hence the work necessary to overcome this part will be
proportional to the product of mean effective pressure and
revolutions, or to total power developed. Hence this part
of the frictional work will be represented by a term of the
form IW.
Both of these terms may be expected to vary with the type
and condition of the engine, and with the presence or absence
326 RESISTANCE AND PROPULSION OF SHIPS.
of attached pumps. In general the friction of horizontal en-
gines is greater than that of vertical. The power absorbed
in friction also increases in general with the number of cylin-
ders and with the multiplication of rubbing-joints. The fric-
tion of new engines is also naturally greater than when they
have become somewhat worn. Again, incorrect adjustment
or alignment either at the beginning or due to wear or work-
ing of the ship may give rise to excessive frictional loss. In
such cases the condition is likely to manifest itself by hot
bearings and other signs that something is wrong at the points
thus affected. The presence of attached air-pumps occasions
an absorption of power not only to overcome the friction of
the pump, but for its regular running, which amount must be
subtracted from the I.H.P. in order to properly obtain that
sent to the propeller. This item is not to be viewed as a
loss, but simply in such cases as a constituent of the total
power which must be eliminated in order to determine the
net amount delivered to the propeller. With independent
pumps this item, of course, does not exist as a constituent of
the power of the main engines.
The power required to operate pumps has been deter-
mined by various special and individual trials. The power
absorbed by the initial friction has been determined by slowly
running the unloaded main engine by the turning engine,
or by derivation from curves of revolution and power as
described under the next sub-head. The power absorbed by
the load friction has been estimated from various coefficients
derived from stationary engines in which the different quan-
tities are susceptible of measurement, and from analyses of
the total I.H.P. in which the various efficiencies and ratios
are derived by comparison from model experiments.
POWERIXG SHIPS. 327
Determination of Initial Friction from Speed-trial Data.
In 47 we have defined the reduced mean effective pressure,
and we may evidently consider this as balanced against the
mean total resistance to the movement of the engine. This
total resistance may be considered as the sum of two parts,
one due to the useful load and the other due to the friction
load.
We have already referred to the frictional or tangential
resistance at the joints and rubbing-surfaces under two heads:
that due to initial conditions and that due to the load, corre-
sponding to initial and load friction. Evidently the frictional
work will require an additional pressure on the pistons or an
additional amount of reduced mean effective pressure as
above defined, and such additional amount may naturally be
considered in two parts, one due to initial friction and the
other to the load friction. We may thus consider the entire
reduced mean effective pressure as made up of three parts,
one corresponding to the net or delivered power and two due
to friction as above. Denoting these respectively by /,/,,
/,, we have
(2)
From the nature of initial friction as above defined it
follows that/j is constant and is in fact proportional to the
factor h there used, while / 2 and also / will vary with the
load.
If now we have a series of corresponding values of /fand
Nj we may derive by 47 (3) the values of/. Plotting these
on ^Vas an abscissa, we shall have a curve as in Fig. 84. Now
when N=o, /, and/ must necessarily be O, and hence/ will
be equal to/j. If therefore the curve be extended back by
328
RESISTANCE AND PROPULSION OF SHIPS.
judgment, it will cut the axis of Y at a point A which will
give an approximation OA to the value of/,. Then for the
VALUES OF N
FIG. 84.
general value of the initial friction power at any number of
revolutions N we have
33000
= /iN, as above.
Hence we see that
33000
In order that such an approximation may be of value,
special care should be taken with the determinations relating
to the lower points on the curve, and the revolutions should
be reduced to the lowest number at which the engine can be
kept in continuous movement.
We may also for the same result use the same data some-
what differently as follows:
POWERING SHIPS.
From the common horse-power formula we have
a/7 _ 2pLA
'dN ~ 33000 "
3 2 9
Hence TV
33000
Hence if H be plotted on TV as an abscissa we shall have a
curve as in Fig. 85. If this be extended back through O by
VALUES OF N
FIG. 85.
judgment, it will have at O a certain inclination to OX, and
the tangent of this angle or of BOX will therefore be an
?i //""I
approximation to the value of ^ ; and from this the corre-
sponding value of/, may be found. See also 78 for another
method of finding the initial friction.
The existing data relating to initial friction gathered in
this and other ways indicate that in amount it may be
expected to lie between 5 and 8 per cent of the full power of
330 RESISTANCE AND PROPULSION OF SHIPS.
the engine, or that/, will be from 5 to 8 percent of the/ for
full load. This would correspond to a value of from 2 to 3
pounds per square inch on the low-pressure piston for triple-
expansion engines under usual conditions as above noted.
Let this percentage ratio be denoted by i and denote full-
power conditions by accents. Then we should have
Work of initial friction at full load = hN' = iH',
Jt f
h=ijjr\
and work for initial friction in general = hN= , where,
as above stated, we may expect i to be found between .05
and .08. These figures relate more especially to the main
engines alone.
Value of the Load Friction. We have above stated that
this is taken as proportional to the whole power H. It might
seem more correct to take it as proportional to the total
power minus the initial friction. The difference involved is,
however, quite negligible in view of the general uncertainty
surrounding the whole question, and we naturally prefer,
as the simpler of the two, the method as stated. We have
therefore
Load friction = IH.
The existing data relating to load friction indicate that /
may be expected to vary between nearly the same limits as i t .
or from .05 to .08.
General Equation for Friction. Combining the two parts ?
we have
Total friction = ^7 \-
POWERING SHIPS.
331
If we plot the total power and the power absorbed by
these two components of friction on revolutions as an abscissa,
we shall have a diagram as in Fig. 86, where the ordinates
VALUES OF N
FIG. 86.
between X and OA give the values of the initial friction,
those between OA i.nd OB the values of the load friction, and
those between OB and OC the values of the net or delivered
power. This equation and diagram serve to illustrate the
increasing relative importance of friction at reduced speeds.
Thus suppose at full power the friction is taken at 14 per
cent, equally divided between load and initial friction.
Denoting full power by 100, we should then have for one-
half speed approximately one-eighth power or 12.5. The
load friction would be reduced in the same proportion, and
would therefore become 7 -7- 8 = .875. The initial friction
would be reduced in the ratio of the revolutions or approxi-
332 RESISTANCE AND PROPULSION OF SHIPS.
mately to one half its full-power value, or to 3.5. Hence the
total friction will be 4.375, and the percentage 4.375 -f- 12.5
= 35 per cent. This illustrates forcibly the wastefulness of
running at reduced power, and the impossibility of obtaining
good propulsive efficiencies under these conditions.
At full power, therefore, we may expect that the friction
of a marine engine will absorb an amount lying, perhaps, not
far from 15 per cent as a mean value. With the best modern
design and construction this figure may fall to 12 per cent or
even possibly to 10 per cent, while with older engines or
poorer construction or lack of adjustment it may rise to 1 8 or
20 per cent or even more.
It should be understood that the division of the power
absorbed by the friction as indicated in the present section is
not altogether satisfactory, and the only excuse for its use is
our ignorance of a more correct analysis. It is not to be
expected that /, will be absolutely constant at all revolutions
or that /, will vary exactly with the load. Further experi-
mental investigation is much needed on these points.
*
59. POWER REQUIRED FOR AUXILIARIES.
.
In the present work we are fundamentally concerned, of
course, with the power needed for propulsion. Inasmuch,
' however, as we not infrequently meet with air-pumps attached
to the main engine, as referred to in the last section, and also
with estimates of power or power data, including that required
for auxiliaries, it will be well to note briefly the power which
the usual auxiliaries may be expected to absorb.
There has been a great reduction in this amount within
the past few years. Auxiliary machinery has been so im-
POWERING SHIPS. 333
proved in design and construction that its cost in power
required for operation is much less than it was from ten to
twenty years ago. At the earlier period from 5 to 8 per cent
of the I.H.P. was believed to be absorbed by the resistance
of the various pumps, most of which were attached to the
main engine. Blechynden,* writing with reference to this
period gives the following estimate, based on data derived
mostly from merchant steamers with triple-expansion engines:
Dead load and air-pump. . . 7.8
Circulating-pump 1.5
Feed-pump 6
Bilge-pump $ j
Per cent of I.H.P.
If we take an estimate for the air-pump of about I per
cent, which would not be far in error for the cases involved,
we should have 3.6 per cent required for all auxiliaries.
These figures agree generally with those obtained in this
country some years ago, especially in the early vessels of the
modern steel navy. Taking more recently the results of
the latest practice, especially in war-ship design, we have the
following results:
Air-pump I to .2
Circulating-pump 2 to .5
Feed-pump 5 to .6
Blowers 5 to 1.5
:.H.P.
Omitting the item for blowers, we have for the remaining
three items from .8 to 1.3 per cent, or slightly over I per
* Transactions N. E. Coast Inst. Engineers and Shipbuilders, vol. vn.
p. 194.
334 RESISTANCE AND PROPULSION OF SHIPS.
cent as a mean value. The chief improvement has been in
air- and circulating-pumps, which together a few years ago
absorbed from 2 to 3 per cent, while with the latest practice
this figure has been reduced to about one quarter of this
amount. The item for bilge-pumps in earlier estimates is not
so often met with now, having decreased to almost a negli-
gible amount probably less than .1 per cent. The amount
absorbed by blowers is naturally variable between wide
limits, being dependent on the degree of forced draught and
other circumstances.
It may be doubted whether the power of an attached air-
pump can be brought to as small a figure as for an independ-
ent pump. This is due chiefly to the fact that the attached
pump usually runs faster than the independent, and faster
than is needed to maintain a vacuum. In consequence the
power is necessarily greater, and for such cases probably not
less than .5 per cent will be required.
60. ILLUSTRATIVE EXAMPLE.
We will now illustrate the application of the preceding
sections to the computation of the I.H.P. in a given case,
assuming where necessary the various values involved.
Referring to Fig. 72, let true resistance = R.
Then E.H.P. = CF = Ru.
Let the factor of augmentation be 1.16.
Then actual resistance = I. \6R.
And thrust horse-power = CE = i.i6Ru.
Let true propeller efficiency = .67.
Let wake factor w .15.
Then (i + w] = 1.15.
POWERING SHIPS. 335
And horse-power C^E i.i6Ru -r- 1.15 = i.oiRu.
Then propeller horse - power = BE = i.oiRu -f- .67 =
Next, for the frictional loss let i and / each = .07.
Then I.H.P. = AE = i.$\Ru-t> .86 = 1.756^*.
Ru
Propulsive coefficient == == .57-
l
These suppositions correspond to the following subdivision
of the total I.H.P. in percentages.
Initial friction ...................... = .07
Load friction ........................ == .07
1. 1.6
Thrust horse-power = - ^ .......... = .661
Propeller loss decreased by wake gain. = .199
Total = i .000
If, as in 56 (i) (), the propulsive coefficient were to be
assumed directly, we should in the preceding example divide
Ru by .57, thus giving i.f$6Ru at once.
Unless special data are available for the estimate of the
coefficients and ratios involved in the process above, it will
be found usually quite as satisfactory to assume the propul-
sive coefficient directly. This again can only be done intelli-
gently when experimental data are at hand from somewhat
similar ships under generally like conditions. The range of
values, it will be remembered, is usually from .50 to .60.
336 RESISTANCE AND PROPULSION OF SHIPS.
61. POWERING WITH TURBINE MACHINERY.
As already noted, with turbines the I.H.P. or the power
corresponding to the I.H.P. with a reciprocating engine, can only
be inferred. The power of turbines is usually estimated in terms
of the delivered horse-power or power delivered on the turbine
shaft. This correponds to the Wp or propeller power in the
case with reciprocating engines, and in any event can only differ
from what may be called, the I.H.P. by the friction of the rotor-
shaft bearings, which should be small.
In dealing with the problem of the determination of power
for the case of turbine machinery, two methods are available, one
by comparison, as discussed in 62-67, and the other by the
use of the principles of 60, stoppng the determination at the
propeller power Wp.
If the method by comparison is employed, it will be well to
use only such data as may be drawn from ships with turbine
machinery. This results from the -fact that the general condi-
tions of operation and the distribution of coefficients and
efficiencies may vary considerably between turbine and recipro-
cating machinery. If the method by analysis, as in 60, is
employed, then care must be taken in the assumption of a pro-
peller efficiency. It seems to be generally conceded, with the
pitch-ratios, area- ratios and peripheral speeds which the designer
is sometimes forced to employ with turbine machinery, that some
loss of propeller efficiency is likely to be involved, and as the
characteristics of the design become more widely removed from
those known to give the best efficiency, proper caution must be
exercised in the assumption of the value to be employed.
POWERING SHIPS. 337
62. POWERING BY THE LAW OF COMPARISON.
The assumptions involved are as follows:
(1) It is assumed that the entire resistance is subject to
the law of comparison. The propriety of this assumption has
been considered in 26, to which reference may be made.
As there shown, the error is small, and may perhaps be con-
sidered as within the limit of general uncertainty surround-
ing the entire problem. Further, with a slightly roughened
bottom, as actually exists for most of the time, the index of
the speed in the term for skin-resistance will approach the
value 2.00, and the error will correspondingly decrease.
(2) It is assumed that the term similar is extended to
include ship, propeller, and machinery. This involves the
general geometrical similarity between the ships and between
the propellers, the ratio being, of course, the same for each.
Geometrical similarity is not necessary for the machinery, but
it should be of the same general type in each case, or at
least there should be good ground for assuming an equal
engine friction in each case.
(3) It is assumed that the term corresponding speeds is
extended to cover the speeds of the propellers relative to
their ships and to each other, as well as the speeds of the
ships relative to the water.
Hence, as in 53, we shall have similar points on the two
propellers describing similar paths with velocities in the ratio
A>, and as there shown we shall have also
, _ (A_V /2 _ JL
,''' \Lj ~ V 2 '
(4) It is assumed if (2) and (3) are fulfilled that the
thrusts will bear the same relation to the resistance in each
338 RESISTANCE AND PROPULSION OF SHIPS
case, that the slips true and apparent will be the same per cent
in the two cases, that the efficiencies will be the same, and
likewise the augmentation and wake factors. It is assumed,
in short, that the entire relation of the propeller to the ship
will be the same in the two cases.
The propriety of these assumptions evidently depends on
the same general basis as the application of the law of com-
parison to the resistance of the ship itself.
As a result of the preceding assumptions it follows that
similar ships with similar propellers at corresponding speeds
will have equal propulsive coefficients.
Or otherwise we may less safely consider that indepen-
dent of the fulfillment of (2) and (3), the propulsive efficien-
cies and relations are the same in the two cases.
Let the subscripts I and 2 denote the two cases. Then
we have
R t and ^ are the two resistances;
u^ and u l are the two speeds;
^ and /j are any two similar dimensions;
A, = / a -i- / a is the linear ratio ;
a = the propulsive coefficient;
//, and H l are the two powers.
Then
, * **
and Jr=RJTc
Now we have in 26 a series of expressions for R^ -=- R^
in terms of various functions of the ship and of the speeds.
Substituting these, and remembering that u^^-u l = y\ =
POWERING SHIPS. 339
4// a _:_ i it vve readily find the following series of expressions
for H^ -r- //,, the ratio between the powers necessary to propel
similar ships at corresponding speeds:
_ ('.y _ M.y - (y _ (Ay _ (A) (*y_ ( Ay^y
H\-\IJ \Aj "\Vj \DJ "\Dj\lJ-\Dj\lJ
A, (Ay (.y _ (Ay (,y _ (/,y(.y _ f ^v,y
= A^ =: ^ W "W V "\7j\uJ '-\Aj\uJ
>
It is obvious that the list of expressions in which the law
of comparison may be thus represented is by no means
exhausted, and that all are equivalent and equally correct, and
will give identical results, provided the fundamental condi-
tions are fulfilled; i.e., if the ships are similar and the speeds
corresponding.
If therefore we know the data in any one case it is a sim-
ple matter to derive the value for a similar ship at a corre-
sponding speed.
As an illustration, let
A = 2000;
ff l = 3000;
* t s= 16;
A = 3200.
In 26 various expressions have been given for corre-
sponding speeds. Here, having given only A anc * Ai we
naturally , = **\ = M*Si = 17-4
Then ^ = ^^=3000X^X1.085 =
TT
In a similar way, taking other expressions for ~ t the same
**\
340 RESISTANCE AND PROPULSION OF SHIPS.
value would be found. Hence the similar ship of displace-
ment 3200 at 17.4 knots would require 5230 I.H.P.
Assuming that the similarity between the two ships is
perfect, the two chief sources of error in the use of this
method are in the extension of the law of comparison to skin-
resistance, and in the possibility of a difference in the propul-
sive coefficients in the two cases. The nature of the error
due to the first of these has been discussed above and in 26.
In regard to the latter it may be seen that if the speed of the
first ship, for example, is low or far from the normal speed
the propulsive coefficient will probably be poor, and less than
might be properly expected for the second ship if near her
normal speed The use of the power of the first ship in such
a case would therefore, other things being equal, result in an
overestimate of the power for the second.
In all such cases, therefore, judgment must be used as to
what extent the necessary conditions are fulfilled, and as' to
the amount of reliance to be placed on the values resulting
from any given comparison.
63. THE ADMIRALTY DISPLACEMENT COEFFICIENT.
Among the various formulae for powering which have been
in use for many years, none has obtained so wide and general
acceptance as that involving the use of the so-called
Admiralty displacement constant or coefficient. This formula
dates from a period long anterior to the introduction of the
law of comparison, and we shall find it of interest to examine
it in the light of that law.
The formula is expressed by the equation
POWERING SHIPS.
341
In practice it was found that there was a general tendency
to constancy in the values of K, ranging as they usually did
from 200 to 250. The use of the formula required simply
the estimate of the constant suited to the ship and to the
speed. This, of course, was a matter of considerable uncer-
tainty except as experience in similar cases furnished data as
a basis. The values of K naturally vary for the same ship at
different speeds, or for different ships at the same speed.
These values are also seen to vary directly as the efficiency of
propulsion. That is, in any given case the greater the value
of K the less the value of H, and hence .the greater the pro-
pulsive efficiency. In numerous cases in which progressive
speed trials have been made the results for the value of K
are similar to those shown in Fig. 87. At low speeds the
^ '
_
-
-.
^^
B
) 2 4 6 8 10 12 U 16 18 20
SPEED
FIG. 87.
value is low, due to the general mechanical inefficiency of both
engine and propeller at powers and speeds far below those for
which they were designed. As the speed increases the value
of K increases, indicating that due to an increasing efficiency
the total power is related to the speed by an index of u less
than 3. This continues to some speed at which, on this
basis, the propeller is the most efficient and K has its maxi-
mum value. Then as the speed is increased the resistance
begins to increase excessively, and perhaps the propulsive
342 RESISTANCE AND PROPULSION OF SHIPS,
efficiency begins to decrease. The result is that H becomes
related to u with an index greater than 3, and we have a
corresponding fall in the value of K.
So long as speeds were moderate and abundant informa-
tion was available relating to nearly equal ships at nearly the
same speeds, the use of this coefficient was reasonably satis-
factory. Its assumption, however, was almost entirely a
matter of judgment, and there were few guiding principles by
means of which any definite value might be determined as
suitable in any given case.
The use of the formula in this way has therefore in recent
years fallen into disfavor, and it is well understood that thus
employed it can only be expected to give a rough approxi-
mation to the power suitable in any proposed case.
Let us now examine the relation of this formula to the
law of comparison. We have in one case
and in another :
K,
w , , , , ,
Whence _=_.__ (l)
Now suppose that D l and D 2 are similar ships and u : and
U 9 are corresponding speeds. Then from 61 (i) we have, by
the law of comparison,
Hence, comparing (i) with (2), it follows that K l ~- K^ i
or K l = K y We therefore derive the important result that
POWERING SHIPS. .343
Similar ships at corresponding speeds have the same Ad-
miralty coefficient.
This may be taken as a statement of the law of extended
comparison, and as such it will be equally correct with any of
the other statements implied in the general equation of
61, (i). This statement supplies therefore the principle
necessary to guide in the choice of Admiralty coefficients, and
once such principle is recognized and followed, this formula
is instantly raised from its former condition of relative unre-
liability and placed on the same level, or rather made iden-
tical with the law of extended comparison, as already so
numerously expressed above.
As an illustrative example let us take the following:
D % = 8000 ;
,= 1 8.
Among the available data is a speed-power curve for a
similar ship of displacement 6000. We have now to find the
speed at which the latter ship will correspond with the former.
Without involving any other data we may take the ratio
as (D 9 -i- D$ = 1.049. Hence u, = 18 -=- 1.049= 17.16.
Turning now to the given curve at a speed of 17.16, suppose
we find H 6320. We have then D, = 6000, , =-17.16,
and HI = 6320. Substituting this in the Admiralty formula
we find K = 264. Now by the law of comparison as above
shown, this coefficient is suitable for the similar ship of dis-
placement 8000 at the corresponding speed 18. Hence,
making the substitution in the formula, we find for the pro-
posed case
344 RESISTANCE AND PROPULSION OF SHIPS.
64. THE ADMIRALTY MIDSHIP-SECTION COEFFICIENT.
In addition to the Admiralty displacement formula, the
equation
(area of midship section)?/ 8
has been used to a considerable extent, especially in con-
tinental Europe. Where the value of K^ is selected by
judgment alone, it stands on the same basis as the D% formula,
though it is hardly as satisfactory, since the length of the
ship is entirely unrepresented in the formula, except as its
value is taken into account in the judgment by which the
value of K l is selected. Comparing it, however, with 61
(i), it is seen to be identical with the D% formula in its rela-
tion to the law of extended comparison. If, therefore, it is
used in this way for similar ships at corresponding speeds, it
becomes simply another mode of application of this extended
law. We may thus say, as before:
Similar ships at corresponding speeds have the same midship-
section power coefficient.
65. FORMULA INVOLVING WETTED SURFACE.
Formulae have been proposed and used quite extensively
of the general form
As proposed, these formulae were to be used in a manner
similar to that in which the two Admiralty formulae have
usually been used. That is, the values of K^ were to be
selected by judgment. They are, when thus used, open to
POWERING SHIPS. 345
the same objections and are under the same lack of confi-
dence as the two above described when used in the same way.
In the method known as " Kirk's Analysis " an approxi-
mation to the wetted surface was found by computing that of
a substituted block model with prismatic middle body and
wedge-shaped fore and after bodies. The appropriate value
of K^ was then assumed from the data available in similar
cases. This method was more especially intended for the
powering of cargo steamers of moderate size and speed, with
an abundance of data available to serve as a guide in the
selection of the values of the constant. Under these circum-
stances the method gave fairly satisfactory results, as indeed
would any of the other methods here discussed when used in
the same way.
In Rankine's " augmented surface" method the wetted
surface, or more commonly the reduced surface, 8, was
multiplied by a factor called the coefficient of augmentation.
This coefficient is a function of the form of the ship at the
bow, or more specifically of the angles of maximum inclina-
tion of the various water-lines at the bow. The area thus
obtained was called the " augmented surface. " Its value
was used for 5 in (i) above, and appropriate values of K t
were assumed exactly as in other cases.
The derivation of this formula proceeded on the assump-
tion that the only resistance necessary to consider was the
frictional, or at least that the resistance might be considered
as varying with the square of the speed and hence the power
as the cube of the speed ; and further, that the influence of the
form on the velocity of gliding or the relative velocity of skin
and water could be represented by this function of the angles
of entrance. The method is interesting as containing an
346 RESISTANCE AND PROPULSION OF SHIPS.
attempt to represent the influence of the form at the bow.
It is entirely inadequate, however, to properly provide for
either variation of form in general, or for varying speeds. It
is in fact subject to exactly the same limitations and errors as
the other formulae here discussed, and requires the same kind
of judgment for the proper selection of the coefficient in-
volved.
The relation of such formulae to the law of extended com-
parison is seen to be identical with that of the two previously
discussed. In general it is seen that they are all special cases
of the general term in which the ship is represented by an
area or second-degree function, and the speed has the
index 3. If therefore they are used as expressions of the
laws of extended comparison they will have the same authority
and will give the same results as any of the other expressions
previously given. All such formulae may therefore be made
expressions of the law of extended comparison by the simple
rule:
Similar ships at corresponding speeds will have the same
coefficient in the formula for power.
6(>. OTHER SPECIAL CONSTANTS.
The various forms for the law of extended comparison in
6 1 show that a large variety of " constants " or coefficients
might be found which would be the same for similar ships at
corresponding speeds. The constants considered in 62,
63, and 64 are all derived- from the function / V. Of the
various constants which might be found in a similar manner
we will only refer to those derived from the functions Du and
POWERING SHIPS. 347
Denoting these by K z and K we have for the corre-
sponding formulae for power
and
, Du
whence A 3 = ~jr *
and
H
If for similar ships at corresponding speeds the same
values of the constant K 3 be taken, or the same values of the
constant K^ the law of comparison will be fulfilled and the
formulae may thus be used for the determination of power by
means of this law in the same manner as with those discussed
in 62, 63, and 64.
We will now briefly discuss the use of these various con-
stants in cases where the geometrical similarity is not com-
plete.
67. THE LAW OF COMPARISON WHERE THE SHIPS ARE
NOT EXACTLY SIMILAR.
Strictly speaking, the law of comparison has no significance
where the ships are not similar. Where they are, any and
all of the various expressions for corresponding speeds, for
resistance, and for power will give exactly the same results.
When the similarity is not perfect, however, the various
expressions for corresponding speeds, for resistance and for
power will not give the same results ; and while strictly none
are applicable, it is quite evident that they may all be consid-
348 RESISTANCE AND PROPULSION OF SHIPS.
ered as approximations, more or less near according to the
nature of the similarity and to the particular expression used.
It is evident therefore that in such cases, and hence in most
cases likely to arise in actual practice, it is not altogether a
matter of indifference what expressions are used for the rela-
tions between the speeds and between the powers in the two
cases. We must first note clearly the two questions involved:
(a) The relation between the powers; and (ft) the relation
between the speeds, or the definition of " corresponding
speeds." With regard to the latter, we are free to take any
of the various expressions of 26, (7) and (8). Correspond-
ing speeds are intended to be those for which the liquid sur-
roundings of the two similar ships shall also be similar. These
depend principally on the wave-formation, and this on the
length of the ship more than on any other one feature. It
therefore seems preferable to fix the corresponding speeds by
the ratio of the lengths, and we take therefore
For the power-ratio we will discuss the three functions
Du, %*, D*u', or the functions which give rise to the three
coefficients K K, K^ as above derived.
Taking the first, its use is equivalent to the assumption
that for similar ships at corresponding speeds the power varies
as the function Du, or in symbols
H - Du.
Denoting the chief dimensions of the ship by Z, B, h,
the block coefficient by b, and remembering that u ~ Z* we
have for similar ships at corresponding speeds
R ~ D ~ BhLb ~ -0' ~ Bhbu\
POWERING SHIPS. 349
If therefore we assume more broadly that resistance in
general can be expressed as the product of some function of
the ship by the square of the speed, the use of the constant
K^ is equivalent to the assumption that such function is Bhb.
Similarly we have
Hence, likewise, if we assume that resistance in general
can be expressed as the product of some function of the ship
by the fourth power of the speed, the use of the constant K 3
is equivalent to the assumption that such function is j .
Now taking a general equation of resistance involving the
second and fourth powers of the speed, let us introduce those
functions into the coefficients. We have then
R = P(Bhby + 0' ..... (I)
The equation of resistance in this form is thus seen to
correspond to the use of the coefficient K 3 as above.
In a similar manner we find for the form of the general
equation which corresponds to the use of the coefficient K
the following:
R =
Likewise for that which corresponds to the use of K^
R = P(DLfu* + Q(Bhb)*u* ..... (3)
But in 8 we have seen that approximately (DL)^ ~
wetted surface or (DL)* ~ S. Whence we may write (3) in
the form
R = P(S)u* + Q(Bhb) V ...... (4)
350 RESISTANCE AND PROPULSION OF SHIPS.
We may also show the relation of the three functions to
the dimensions of the ship and the block coefficient as follows:
For comparison the function Du is equivalent to L*Bhb ;
From these last expressions it is readily seen how the
different values would vary in cases where the similarity is not
exact. Thus suppose that the second ship is relatively longer
than the first. Then the length-ratio will be greater than the
others, and the power as derived from the above expressions
will increase in amount from the first to the last. Vice versa,
if the second ship were broader, deeper, or fuller, the first of
the above functions would give the largest value, and the
others successively less.
Looked at from the standpoint of the formula of resist-
ance, it would appear as though (3), and hence the function
Z% 4 , or the constant K t should be the most nearly correct.
Experience seems to indicate, however, that there is no one
form equally applicable to all modes of variation from exact
similarity,* and sufficient data are lacking for the definite rela-
tion of the appropriate form of function to the nature of the
departure from similarity.
Mention may also be made of the function Z>*, which is
sometimes used, especially in connection with a speed-ratio
(A * D$-
Of these various functions, D*u* and D*u* or the constants
K and K 4 may be recommended in preference to the others.
The latter is perhaps likely to be the more correct, though
the former, from the fact that it is the well-known Admiralty
coefficient, has been much used ; and such use is likely to con-
POWERING SHIPS. 351
tinue so long as no more definite information is available
regarding the relative values of the various functions and
coefficients available.
We may illustrate the use of these expressions by an
example as follows. Given the following data:
L B h D u H
(l) 30 40 20 3700 15 3129
(2) 400 48 24 7640
Then (^) =(1.33)*= 1.155.
Hence u z = 15 X I.I55 = 17-32.
We will now find the value of //,, using the following
ratios, the results being as shown :
TT TT
M* H*
H,
D^
~) = 2.32 7260
rZW "' V =-497 78H
= 2.385 7464
TO- W =-'> 8
Where the variation from similarity is marked, the method
of comparison, of course, fails entirely, and it must be
remembered that its reliability rapidly decreases as departures
from similarity increase. The discussion of the present sec-
tion is therefore not intended to show a means of extending
352 RESISTANCE AND PROPULSION OF SHIPS.
the method of comparison to widely dissimilar forms, but
simply to show what forms of expression are most likely to
take rational account of such slight variations as experience
indicates may be admitted without sensibly affecting the
application of the law as stated.
68. ENGLISH'S MODE OF COMPARISON.*
Given two similar ships of displacement D l and Z> 2 at speeds
Vi and V^, not in general " corresponding." Then it is always
possible to make two models of different dimension such that
at the same speed one shall correspond to D l and the other
to Z>,. From the principles of 26 we should then have the
following:
Residual
Resistance.
Skin-
resistance.
Dis-
placement.
Speed.
W,
$
A
f,
w*
s,
A
P.
Let be the ratio of the total resistances of the models
at the speed V\^-) . Then
d^
and (w 9 + j.) = (/, + J,).
Also ^, = w, -^.
"a
* Proceedings Institution of Mechanical Engineers, 1896, p. 79.
POWERING SHIPS. 353
Substituting for these values as above, we find
Then
Total resistance of A = 5 a + W 7 ,.
The values of S lt S t , s lt s t , are to be computed in the
usual way, using appropriate values for the skin-resistance
coefficient. Assuming also for the ship D l the ratio between
thrust horse-power and I.H.P. (CE -r- AE, Fig. 72), we can
find 5, + W lt and hence W^. The value of n is found by
towing the models d^ and d^ simultaneously from the ends of
a bar supported at an intermediate point. The point of sup-
port is then adjusted until the bar remains at right angles to
the direction of motion. In this condition the resistances are
evidently in the inverse ratio of the lever-arms, and hence a
measurement of the latter will give the value of n as desired.
In this way it becomes possible to determine the value of W+
above, and hence of the total resistance 5, -f- W^
As an extension of the above method of treatment, let us
assume the law of comparison to cover the total resistance,
instead of the residual only. Let the experiment be made as
before and the value of n similarly determined. Then denot-
ing total resistances by R lt R r lt r we have
. r A_ A
- r '~ '
354 RESISTANCE AND PROPULSION OF SHIPS
Hence ,
r -R t = n \
TT I T/"\ 7
and therefore = ratio of powers = 72(77)
i \"i
This method being wholly comparative, no dynamometric
apparatus is necessary, and the experimental determinations
are comparatively simple. Since, however, the results are
comparative and not absolute, its field of usefulness will be
much restricted in comparison with the usual method as
described in 26.
69. THE VARIOUS SECTIONS OF A DISPLACEMENT SPEED
POWER SURFACE.
Let us consider a general series of ships, all of the same
type of form. The law of extended comparison furnishes the
connecting law between all the ships of such a series, no
matter what the displacement or speed. It is therefore evi-
dent that any speed-power curve for a given displacement
may be considered as the speed-power curve for the whole
series by means of an appropriate modification of horizontal
and vertical scales. Suppose, for example, that we have a
curve for displacement D l plotted with scales as follows:
Abscissa, I linear unit =/ knots. Ordinate, I linear unit =
q I.H.P. Then the same curve will also represent the speed-
power relation for D^ with the following scales :
//M*
Abscissa, I linear unit = /\TT~) knots ;
M*V
Ordinate, i linear unit = q\jy} I.H.P.
POWERING SHIPS.
355
With such scale modifications, therefore, one curve would
represent the whole series of values of power for the various
values of displacement and speed.
8COO-
6000
4000
2000
10
11
12
11
10000
8000
6000
4000
2000
15
1(5
13
* SPEED
FIG. 88. Relation between Speed and Power. Displacement constant for
each curve, as shown on the right.
aooo
c.
u
1
4000 6000
DISPLACEMENT
10,000
FIG. 89. Relation between Displacement and Power. Speed constant for
each curve, as shown on the right.
With a uniform scale such a series of values would require
for their complete representation a surface of which the
356
RESISTANCE AND PROPULSION OF SHIPS
ordinate is the value of //, located by abscissae corresponding
to the given values of D and u. The various sections of this
surface by planes parallel to the axes of H and u would repre-
sent each a speed-power curve for a given value of D.
The general values may therefore be represented by a
single curve with varying scales, or with uniform scales and
varying curves.
The surface above referred to may also be cut by planes
parallel to the axes of H and D or D and u, thus cutting out
16
15
14
gj 13
CO
12
11
10
\
\
\
\
\
*v
>.
"^
7000
6000
5000
4000
3000
\
\
\
X
X
X,,
\
^-v
-x^
\
\
>^
X
">
s^
\
N
\
\
^""^
\
x
*""
s
\
\M
00
^v.
^
^
2000
2000 4000 6000 8000 10000
DISPLACEMENT
FIG. 90. Relation between Displacement and Speed. Power constant for
each curve, as shown on the right.
curves, one set showing the relation between H and D for a
fixed value of u, and the other the relation between D and u
for a fixed value of H. The various curves of section are
shown in Figs. 88, 89, and 90, a study of the general charac-
teristics of which may be recommended.
70. APPLICATION OF THE LAW OF COMPARISON TO SHOW
THE GENERAL RELATION BETWEEN SIZE AND
CARRYING CAPACITY FOR A GIVEN SPEED.
In Fig. 91, let AB denote the curve of E.H*.P. for a given
ship of 5000 tons displacement at speeds from 10 to 20
POWERING SHIPS.
357
knots. Now suppose this model taken as the type for a series
of vessels of displacement from 5000 to 20 ooo tons. Then by
the appropriate transformation the E.H.P. for each of these
vessels may be determined from the given curve, as explained
in 68. Again, assuming a constant propulsive coefficient of
JOOOO
8000
6000
4000
10 11 12 13 14 15 16 17
SPEED IN KNOTS
FIG. 91.
13 19 20
say .54, the values of the I.H.P. may be similarly obtained
from a curve CD, the ordinates of which are equal to those
of AB -=- .54. If a constant propulsive coefficient could be
assumed for the 5<DOO-type ship, CD would be its curve of
I H.P. Actually, however, with the same machinery at
varying speeds, the propulsive coefficient will vary so that CD
would not in such case correspond to an experimental curve
derived from any one ship, and we may more properly con-
sider it simply as a curve proportional to E.H.P. Let us
now find the I.H.P. for our series of similar ships, all at 20
358
RESISTANCE AND PROPULSION OF SHIPS.
knots speed. The speed at which the 5ooo-ton ship will
correspond is
D
Taking the ordinate of CD at this speed, let it be denoted
by y. Then the I.H.P. for the similar ship at 20 knots will
be
tf= ( Y
^Sooo/ '
In this way we derive AB, Fig. 92, which shows the
I.H.P. at this speed for the supposed series of ships. The
A/
4 6.8 10 12 14 16 18 20
DISPLACEMENT IN 1000 TON UNITS
FIG. 92.
form of the curve shows plainly that H increases at a much
slower rate than D ; in fact in this particular case the rate is
POWERING SHIPS. 359
nearly as the Square root of the displacement. Next suppose
the weight of the hull and fittings to be .50 of the total dis-
placement for the entire series. This ratio will vary consid-
erably according to the type of structure, and would not,
moreover, remain the same in ships of so great difference in
size. In any case, however, the variation will not be suffi-
cient to affect the general nature of the relations which we
wish to establish, and some such assumption is necessary in
order to introduce the necessary simplification into the rela-
tions involved. The ordinates to the line OD, Fig. 92, will
then equal the displacement, and those between OH and OD
the weight of the hull, etc. Taking the total weight of
machinery and boilers as one-ninth ton per I.H.P. or 9
I.H.P. per ton, and coal at 1.8 pounds per I.H.P. per hour
for all purposes, and assuming enough coal for a seven-days*
run and neglecting the variation in D due to the consumption
of coal and stores, or, otherwise, considering that the mean
displacement for the trip corresponds to the values of the
diagram, it is readily found that the weight of machinery and
coal will be represented by the ordinate between OH and CC.
The remainder lying between CC and the axis of X is
evidently cargo-carrying or earning capacity. It appears that
in the case chosen the 5OOO-ton ship could barely make the
trip without cargo, while as D increases the earning capacity
is seen to rapidly increase, rising to 3000 tons for D = 16 ooo
and to 4200 for D 20 ooo.
Again, if we take it roughly that the operating expenses
will vary with the power, the time being the same, the
diagram shows how rapidly, relative to expenses, the earning
capacity increases as D is increased. Thus for D = 9000
the expenses are represented by 14400 and the earning
3 6o
RESISTANCE AND PROPULSION OF SHIPS.
capacity by about 1000 tons, while by an increase of D to
16000 or by 77 per cent the expenses are increased by
about 45 per cent and the earning capacity by 200 per cent.
Again, in Fig. 93, let us note the variation of carrying
capacity with speed, the length of voyage being constant.
18,000
>
H
16,000
i
14,000
12,000
I 10,000
1
z 8,000
H"
| 6,000
4,000
5
<L 2,000
10
13 14 15 16
SPEED IN KNOTS
FIG. 93:
We will take for illustration the 12 ooo-ton ship. AB is the
curve of I.H.P. The constant ordinate between CD and EF
denotes the constant weight of hull and fittings. The ordi-
nates between EF and GH denote the weight of machinery
and coal, the latter decreasing with the decrease in power, but
increasing with the increase of time. The ordinates between
CD and GH give, then, the total weights on the same suppo-
sitions as before. The remaining ordinates between G H and
X show the variation of carrying capacity with speed, and how
Nearly the latter is bought at the expense of the former.
The particular values for these various relationships in any
POWERING SHIPS. 361
given case will depend on many considerations here omitted.
The general nature of the results, however, is sufficiently
indicated by the diagrams, which are chiefly intended as sug-
gestive applications of the law of comparison to the study of
special problems of this character.
CHAPTER VI.
TRIAL TRIPS.
71. INTRODUCTORY.
THE general purpose of a trial trip is to determine the
speed and power which may be maintained continuously for
a certain distance or time. In addition to this fundamental
purpose it is always desirable to observe such data as bear in
any way on the general problem of resistance and propulsion,
while in particular cases various characteristics may be
observed relating to different points on which information is
desired.
For the direct determination of speed it is sufficient to
obtain simultaneous observations of distance and time. We
shall not here consider in detail the measurement of power,
assuming it to be determined by indicators in the usual
manner.
The details of the various measurements necessary for the
determination of speed may vary somewhat with the length
of the course. We may have (i) a long course, as for exam-
ple from 20 to 100 miles or more, over which but one run, or
at most but one run in each direction, is to be made; or (2) a
short course, as of one or two miles, usually the former, over
which as many runs may be made as desired.
We may also distinguish between the purpose of the trials
as follows: (a) the determination of the speed or power, or
' 362
TRIAL TRIPS. 363.
both, under a single set of conditions, such as full power or
half-power for example; or (b) the determination of the speed
and power under a series of varying conditions so that the
continuous relation between speed and power for widely vary-
ing values of either may be determined.
We may next briefly mention the chief points relating to
the ship which may influence speed, and the conditions to be
fulfilled for maximum speed.
(a) Displacement. This should be as light as possible,
though this consideration is not independent of (b) and (c).
(b) Propeller. The surfaces should be smooth, blades
thin, edges sharp, and it must be well immersed.
(c) Trim. The resistance will not presumably vary sen-
sibly for slight changes of trim, but it may result that a very
considerable change, such for example as might be necessary
to immerse a propeller when the displacement is very light,
would materially increase the resistance for the given dis-
placement. In such case there is also the added loss due to
the obliquity of the thrust.
(d) Bottom. This should be smooth and freshly painted.
(e) Wind and Sea. The wind should be light, and prefer-
ably off the beam or a little ahead, in order to give the benefit
of increased natural draft for the boilers. The sea should, of
course, be smooth.
For the attainment of the fundamental and secondary
purposes of trial trips the following data are requisite or
desirable :
Power.
Distance.
Time.
Revolutions.
364 RESISTANCE AND PROPULSION OF SHIPS.
All conditions affecting the relation of the ship to resist-
ance and propulsion, such as
Mean draft.
Trim at rest.
Condition of bottom if known.
Wave profile alongside of vessel.
Change of trim when under way.
Depth of water.
State and direction of wind and waves.
In addition to these, which are observed at the time of the
trial, the characteristics of the ship and of the propeller are
supposed to be known.
72. DETAILED CONSIDERATION OF THE OBSERVATIONS TO
BE MADE.
Distance. Omitting special reference to power, we first
note that the distance with which we are concerned is that
which the propeller has driven the ship through the water,
and not the distance over the ground or between fixed points
on shore. The influence of tides and currents must, therefore,
be eliminated before the distance actually traversed through
the water can be known.
For the marking of distance itself we have two general
methods buoys or ships at anchor, and range-marks. The
former are more suitable for a long course, where the change
in location caused by swinging to the tide will be of no relative
importance. For a short course such errors would be inad-
missible, and the limits of the course must be marked with all
possible accuracy. In such case range-marks on shore are
made use of. Thus in Fig. 94 AB and CD are two pairs of
TRIAL TRIPS.
365
poles or range-marks determining the two parallel lines AB
and CD. The course is then some line EF at right angles to
these ranges, and at such a distance from the shore that the
depth of water shall be sufficient to avoid sensible retardation
( 19). In addition to the ranges, two requisites are there-
fore necessary in order to definitely fix the course a direction
and a location. The direction is usually known by compass
course, and it is also desirable to mark it by ranges and
buoys. The latter means is also used to indicate its approxi-
TD
FIG. 94.
mate location with reference to the proper depth of water.
At each end of the course there should be plenty of room for
making turns and for gaining headway before entering the run
proper. The considerations developed in 78 show that the
length of this preliminary run should be preferably not less
than about one mile. This seems to be in agreement with
experience, as Mr. Archibald Denny suggests * that the ship
should have a straight run of at least one-half mile, and
better still one mile, before entering the course proper.
* Transactions International Congress of Marine Engineers and Naval
Architects in Chicago, 1893. Paper xxxi.
366 RESISTANCE AND PROPULSION OF SHIPS.
The necessity of a course at right angles to AB and CD
rather than on some oblique line as Ef^ is evident without
special discussion.
Tidal Observations. As will be shown later, the tidal
influence for a short-course trial may be approximately elimi-
nated by appropriate treatment of the other data observed.
With a long-course trial in one direction only such elimina-
tion is not possible, and with a single run in each direction
the data cannot be satisfactorily used without special treat-
ment for the elimination of tidal influence. It is therefore
generally desirable to have made special tidal observations.
For a long course, boats or ships may be anchored at intervals
of five or ten miles along the course, and should make observa-
tions continually as often as once in fifteen minutes throughout
the trial. Such observations should preferably be made at
about the half mean draft of the vessel undergoing trial. The
observation may give simply the component of the tide in
the direction of the course, or perhaps preferably the whole
velocity and its direction. Such observations are usually
made with a so-called patent or taffrail log. This consists
essentially of a small screw propeller, which in the case of
vessels in motion is towed astern, the revolutions being com-
municated through the towing cord or by other means, to a
counter on the ship. In the case of a vessel at anchor near
the line of the course, such a log is buoyed a short distance
from the ship at the appropriate depth, and observations
made as usual. If the component of the tide in the direction
of the course only is desired, the log must be maintained in
this direction. If the entire velocity and the direction are
both to be observed, the log must have a vane attached and
must be free to turn to the tide. As shown by Froude's
TRIAL TRIPS. 367
experiments, such instruments, if made with all the care
appertaining to the production of a piece of truly scientific
apparatus, are quite reliable. As usually furnished, however,
they are liable to a correction, and should be rated in water
known to be still at a speed near that at which they are used.
A slight error in the log is, however, of less importance in
tidal observations than when such instruments are used to
determine directly the velocity of the ship itself. In addition
to the tidal observations, the time of passage of the ship at
each station is also noted.
We may then assume that \ve have thus at our disposal a
mass of data giving at from four to eight points along the
course a series of tidal observations at intervals of about fif-
teen minutes for the whole trial.
For a short course, if tidal observations are made at all,
the number of locations will naturally be much less, according
to circumstances and the degree of accuracy with which the
measure of this disturbance is desired.
Time. For the measure of time careful observation with
accurate watches is sufficient for long-course trials. For
short-course trials the probable error of reading with the usual
form of seconds-hand watch may become sensible. For satis-
factory accuracy stop-watches may be used, or preferably, for
scientific purposes, some form of chronograph by which an
electric signal is recorded the instant contact is made by the
pressure of a key.
Revolutions. For the counting of revolutions on a long-
course trial, the ordinary engine-counter is quite sufficient.
For a short-course trial the same means may be used, though,
as with time, more satisfactory accuracy is attained by the
use of some chronographic arrangement. In Weaver's speed
368 RESISTANCE AND PROPULSION OF SHIPS.
and revolution counter a paper tape is fed at a uniform speed
under a series of electrically controlled pens. One of these
under control of a clock makes a mark every second. Another
is in circuit with a contact maker, and is used to note the
instant of entering and leaving the course. Other pens are
electrically connected with each main shaft and thus register
every revolution. Such an instrument is, of course, only used
in short-course trials.
Special Conditions. Most of these are self-explanatory.
The wave profile may be determined by measuring down
from the rail by batten. The use of photography from a
neighboring vessel may also be suggested. The change of
trim from the condition at rest and when under way may be
determined by the use of a pendulum adjusted to swing in a
longitudinal plane.
73. ELIMINATION OF TIDAL INFLUENCE.
We first suppose the observations made on a short-course
trial. These may be considered as furnishing, for any given
run, the average tidal velocity, and the correction consists
simply in subtracting or adding, as the case may require.
With the long course, where several hours may be required
to traverse its length, the assumption of a sensibly uniform
tide is not admissible, and the more detailed observations
already referred to become necessary.
An approximate method of applying the correction is as
follows :
Let TV TV 7" 3 , etc., denote the lengths in minutes of the
successive time intervals between passing the posts of obser-
vation. Let z/,, ^ 2 , v^ etc., be the tidal velocities per minute
TRIAL TRIPS.
along the course at each successive station at the instant of
passage of the ship, -f- being considered as denoting a velocity
in the same direction as the ship. Then %(v, + 2/J, i(^ +
v,), i( v + *0 etc., are taken as the mean tidal velocities for
the successive intervals. The corresponding tidal influences
will be:
For the first,
second,
T
* 9/_. I _. \ .
" " third, V. + *0;
etc. etc.
The entire tidal correction will be then simply the alge-
braic sum of these partial corrections.
As a method somewhat more accurate in principle, the
following may be suggested :
The tidal characteristics, as is well known, are expressible
in terms of circular functions of time and distance from a
given origin, and the tidal velocity at the ship at any instant
must necessarily vary approximately as a sinusoidal function
of space and time. In Fig. 95 let the ordinates at O, P>
FIG. 95.
g, etc., denote the tidal velocities at the successive points of
observation laid off on a time abscissa. Then OP, PQ, etc.,
370 RESISTANCE AND PROPULSION OF SHIPS.
are the successive time intervals between the points of obser-
vation. Now without attempting any refined analysis of the
tidal components, we may quite closely obtain a continuous
value of the tidal velocity at the ship at each successive
instant by passing through the points A, B, C, etc., a smooth
curve, keeping in mind the characteristics of a sinusoidal
form. If now the area of this curve be found by a planimeter
or by approximate integration, the result will give the total
distance or set due to the tide, + or according to the cir-
cumstances of the case. The total correction found in either
of these ways is then applied to the total length of the course,
thus giving the corrected length, from which in conjunction
with the time the speed is directly found.
We will next consider the elimination of tidal influence for
short-course trials without special observations.
For a measured mile the time required to make a double
run with and against the tide will be so small that without
large error the tidal velocity may be considered constant
throughout. In such case the simple mean of the two runs
will evidently give the true mean speed; thus:
An extension of this, known as the method of " continued
averages," has also been much used. This is as follows:
Suppose four runs made, two in each direction, giving appa-
rent speeds u lf u 91 & 4 . These are treated as shown by the
form on the next page
The third average is then taken as the correct speed, thus
deduced from the speeds observed. This is equivalent to
giving the second and third runs three times the weight of the
TRIAL TRIPS.
371
first and fourth, and the final result may be more readily
found by this formula than by the actual formation of the
successive columns of averages.
Speeds.
ist Averages.
ad Averages.
3d and Final Average.
Ui
Uy
Us
u t
1 + W 2
Ui -f 2W 2 -f 3
i + 3a + 3s -|- 4
2
a -j- W 3
4
2 + 2U 3 -f U^
2
Ms + tit
8
4
2
The correctness of this method depends on certain
assumptions \vhich we may briefly note.
Let the true speed remain constant at u. Let the tidal
velocity be expressed as a function of the time as follows:
v = a -f- bt + cf.
Let the runs be made at equal time intervals which may
be denoted by t. Then for the apparent speeds we shall have
u l = u-\- a ;
w 2 = u a b c ;
, u + a+ 2b -\- ^c\
u 4 = u a ^b gc.
By substituting in the formula for 'continued averages
above, we readily find
8
= u.
It thus appears that if the true speed remains constant,
and if the tidal influence varies as a quadratic function of the
372
RESISTANCE AND PROPULSION OF SHIPS.
time, and if the time intervals are equal, this method will give
the true speed. In the actual case, however, not one of
these conditions will in general be fulfilled. It is therefore
doubtful if the result of four runs thus reduced is any more
accurate than the simple average, or than the average of two
runs made with as small a time interval as possible.
It may also be noted that when the tide is just on the
turn either way the water is frequently affected by eddies
and counter-currents, which though small in velocity are so
confused in distribution as to defy any attempt at elimination.
When, however, the tide is at about half-ebb or half-flow, the
velocity is greater but quite regular. This results naturally
from the sinusoidal character of tidal motion, as a result of
which the change of velocity is least when the velocity is
greatest at about half tide, and the change is greatest when
the velocity is o at full tide or slack water. It follows that
half tide with a clearly defined regular tidal velocity uniformly
distributed over the course is in general preferable to slack or
high water with the accompanying eddies and irregularities.
Further methods of dealing with tidal influences will be
found in the next section.
74. SPEED TRIALS FOR THE PURPOSE OF OBTAINING A CON-
TINUOUS RELATION BETWEEN SPEED, REVOLUTIONS,
AND POWER.
i
We have thus far been concerned with the actual true
speed attained, without reference to the corresponding num-
ber of revolutions or the I.H.P. necessary. For the investi-
gation of the propulsive performance, however, it is quite
desirable to obtain sufficient data to furnish a continuous rela-
tion between these three quantities. For this purpose the
TRIAL TRIPS. 373
short-course trial alone is suitable. The actual conduct of
the trial is in no wise different from that already described,
but it is continued through a decreasing series of speeds with
their appropriate revolutions and powers, to a number of
revolutions as low as can be maintained uniformly by the
engines. The problem of the proper disposition of the data
thus found is more complicated than when speed alone is
desired. We now desire .to determine not only the true
speed, but also the revolutions and I.H.P. which correspond
to it.
Now in practice it is found difficult, even for any one run,
to maintain the revolutions and power constant. With care
the variation may be slight, but at the best the question may
always arise as to the proper interpretation of the data
observed.
Let //,, N tt u l be the power, revolutions, and true speed
for the first run, and //,, N tt u % those for the second, v being
the tidal influence taken as constant for the two runs. Then
the actual observations furnish us with
ff l9 N 19 (, + *);
//, , N t , fa v).
For two runs intended to be at the same speed, the varia-
tion will be so slight that we may properly assume the slip of
the propeller to be the same for both. As a result the true
speed will vary directly with the revolutions, and the true
mean speed will correspond to the mean number of revolu-
tions. That is,
,
- corresponds to
374 RESISTANCE AND PROPULSION OF SHIPS.
Now with power we have in general
H=Bu n ........ (i)
where B and n depend alike on the ship and on the speed.
Where the variation in speed is not large, we may safely take
the variation in B and n as negligible. In the present case,
therefore, we should have
Whence
Jf, = Bu,\
'
A U * + " H * + H *
and
2T
and *^_^ = (_ _L_ L-; . . . . . ( 2 )
The left-hand member of this equation is in the same
general form as (i) above, and it must therefore give the
value of the power which corresponds to the true mean speed.
The relation of this to the observed powers H^ and //, is then
shown by the right-hand side of (2) above. The exponent n
will usually be found between 3 and 4. As an illustration let
I = 3200;
I 3600.
TRIAL TRIPS.
Then by substitution we find
375
LJ H \ TJ~ n \n
^4^-) = 3396 -
The value by simple average would be 3400, and as the
uncertainties in the measurement of power are relatively far
greater than the difference thus resulting, and as these values
of HI and H t represent a variation greater than would be
likely to occur under the conditions assumed, it seems fair to
conclude that in such case we may properly take the mean
REVOLUTIONS
FIG. 96.
power as corresponding to the mean speed. For two runs
thus made we may take, therefore, the mean revolutions,
mean speed, and mean power, as all corresponding.
The next pair of runs being made at a lower power,
another set of means is found, and so for the entire series.
376
RESISTANCE AND PROPULSION OF SHIPS.
Still otherwise, we may effect the general averaging by
graphical process as follows:
Let power be plotted on revolutions as in Fig. 96, each
pair of spots corresponding to a double run intended to be
made at the same power. Then a fair curve passing through
and between the spots may be taken as the best approxi-
mation to the continuous relation between revolutions and
power. We may then plot the mean of the speeds as an
ordinate, on the mean of the revolutions as an abscissa, as in
OB, Fig. 97, thus giving a continuous relation between
speed and revolutions. A straight line OA shows similarly
%
O REVOLUTIONS
FIG. 97.
the speed if the apparent slip were o. Hence the vertical
intercept between OA and OB shows the corresponding loss
in speed at each point. From these diagrams we may then
obtain and plot if desired a curve showing the continuous
relation between power and speed.
TRIAL TRIPS. 377
We may also derive the relation between revolutions and
speed graphically by plotting each separate speed on revolu-
tions as an abscissa, thus obtaining two curves, one for speeds
with and the other for speeds against the tide, with pairs of
points at nearly the same revolutions on each. A mean curve
will then give the desired relation between revolutions and
speed.
A slightly different mode of graphically determining the
relation between revolutions and true speed is that due to
Mr. D. W. Taylor,* which is briefly as follows:
The course is steamed over back and forth, going over
each time the same route off as well as on the course, so that
the time intervals between the middle points of the runs may
vary inversely as the speeds. For each successive run the
revolutions are decreased by as nearly as possible an equal
decrement. This gives a series of runs at certain revolutions
with the tide, and another series with other revolutions
against the tide. Each of these is then plotted, giving curves
of speeds plotted on revolutions, with and against the tide.
A mean curve is then taken as the curve of true speed. The
author then shows by means of an illustrative example that
with a fulfilment of the above conditions well within practi-
cable limits a variable tidal influence will be eliminated with
quite sufficient accuracy for all practical purposes.
The advantage claimed for this method is that a given
range of speed can be satisfactorily covered with a smaller
number of runs than when they are made in pairs as pre-
viously described.
* Journal Am. Soc. of Naval Engineers, vol. iv. p. 587.
3/8 RESISTANCE AND PROPULSION OF SHIPS.
75. RELATION BETWEEN SPEED POWER AND REVOLUTIONS
FOR A LONG-DISTANCE TRIAL.
While the intention may be to carry on such a trial at
uniform power, in any actual case there will naturally arise
considerable variation in the various quantities involved.
The mean speed and the mean revolutions are readily found,
and with any probable amount of variation we may presum-
ably consider them as corresponding the one to the other.
With regard to the power which corresponds to such mean
speed or revolutions, however, there is more room for uncer-
tainty.
The data available will be a series of indicator-cards taken
at intervals throughout the trial, and giving presumably the
mean effective pressures in the cylinders at the instant in
question. The revolutions are also taken at the same time,
and thus each observation gives the data for determining a
value of the power at that instant. The mean of these revo-
lutions, however, will not be the true mean, due to individual
errors of observation, and to the fact that they are merely a
series of detached values of N, while the true mean is
obtained from the continuous counter giving the total number
for the whole run. On this account, therefore, and for finding
the mean power, it seems preferable to use the mean effective
pressures from the cards, and the true mean revdlutions from
the total counter. The method for doing this we may
develop as follows: In any given propeller it is readily seen
that the relation between the mean effective pressure in the
cylinders, or more definitely between the mean effective
pressure reduced to the 1. p. cylinder ( 47), and the thrust
of the propeller or resistance of the ship, must be very nearly
TRIAL TRIPS. 379
constant except for very widely varying conditions. We may
therefore properly assume that the reduced mean effective
pressure will vary very nearly as the resistance. Denoting
these pressures by / 1} /> etc., we may, with all significant
accuracy, assume them expressible in the form /, bN*>
etc. Hence, extending the method outlined in 73 for
mean power, we should here use for the final mean pressure
/ the square of the mean square roots of the various values
/!, /> 2 , etc. This pressure used with the true mean revolu-
tions would then give an approximation to the corresponding
power having all significant accuracy, at least so far as
obtainable from the data at hand.
A still greater refinement, but usually without practical
value, would be the substitution for 2 of an index to be
derived from a special examination of the relation between
speed or revolutions and power.
If the variation in the power is not unusually large, the
square of the mean square root will not sensibly differ from
the simple arithmetical mean, and in such case the latter may
properly be used.
As an illustration of the amount of the variation, let three
values of/ be denoted by 64, 81, and 100. Then the square
of the mean square root is 81, and the arithmetical mean is
81.66, or somewhat less than one per cent in excess.
We therefore conclude that in all usual cases we may, for
the power corresponding to the mean revolutions or speed,
use the simple arithmetical mean of the pressures, as above,
with the mean revolutions. At the same time the substitu-
tion of the square of the mean square root for the arithmeti-
cal mean will undoubtedly give a more correct result, and if
the difference is significant, the latter method should be used.
380 RESISTANCE AND PROPULSION OF SHIPS.
76. LONG-COURSE TRIAL WITH STANDARDIZED SCREW.
We have thus far in all cases assumed the measurement of
distance to be effected by means independent of the ship's
propeller. A method of long-course trial has been used to
some extent, however, by means of which the propeller
itself, having been standardized, is made the instrument for
the measurement of the distance. This is known as the
standardized screw method, and has been used in certain
government trials at the suggestion of the Bureau of Steam
Engineering of the Navy Department.
The ship is first taken on the measured mile and the con-
tinuous relation between speed and revolutions as discussed
in 74 is carefully determined. The ship being in the
same condition as regards draft, trim, condition of bottom,
etc., then puts to sea in smooth water and calm weather, and
runs such a distance or time as may be desired. For the best
accuracy readings of the counter should be taken every ten or
fifteen minutes. In any event the total time and total revolu-
tions are known. Indicator-cards are, of course, taken at
appropriate intervals throughout the run.
Now if the revolutions have been uniform, or if they have
varied somewhat, but not out of a range within which the
apparent slip may be taken as sensibly constant, then, as in
74, the true mean speed and the mean revolutions will
correspond, and the former may be determined directly
from the latter by the aid of the curve of standardization. If
in an extreme case the revolutions are widely variable and the
slip is widely variable with revolutions, this correspondence
will not be accurate. The cause of the error and a method
for its elimination may be seen as follows:
TRIAL TRIPS.
381
Let OX, Fig. 98, be a time abscissa on which are laid off
as ordinates the successive numbers of revolutions per minute
at the instants at which they are taken. A smooth curve AB
drawn through the points thus found gives a continuous rela-
tion between time and revolutions. Then from the curve of
standardization the speed corresponding to each number of
revolutions may be found. These being plotted at the same
points will give a curve CD, showing the continuous relation
between time and speed. The area OCDX being found by
planimeter or by approximate integration, the result will be
FIG. 98.
proportional to the entire distance, and the time being known,
the true mean speed is thus determined. If now the apparent
slip were constant, or OB, Fig. 104, a straight line, the curves
AB and CD would be similar and the integration of AB would
serve the same purpose as that of CD. But the integration
of AB is furnished by the revolution counter, and hence the
mean revolutions thus determined would with constant slip
give the true distance or true speed. Again, if the revolutions
382 RESISTANCE AND PROPULSION OF SHIPS.
are constant, it is evident that both AB and CD become
straight lines parallel to OX, and their constant ratio would
be that between speed and revolutions given by Fig. 97, for
i
the particular value of the latter.
If these conditions are not fulfilled, that is, if revolutions
vary with time, and slip with revolutions, then the curves AB
and CD would no longer be similar, and the use of the former,
or, in other words, of the mean revolutions, would no longer
be theoretically exact.
With the usual amount of variation in these quantities,
however, the error will be negligible 'and the mean revolutions
may be used.
The treatment of the power has already been considered
in 74.
As some of the advantages of this method mention may
be made of the following:
(a) There are no tidal corrections to consider. This arises
from the fact that by means of the standardization the pro-
peller is made the instrument for measuring distance.
(&) The ship under trial is independent of the presence of
other vessels for course-markers and for tidal observations.
(c) The results are not invalidated by departure from a
straight course. Any course desired may be followed so long
as it is not a curve of small enough radius to give rise to a
sensible retardation.
(d) The results attained are constantly known during the
trial, so that the exact status of the ship relative to the
expected performance is known from beginning to end.
(e) Any derangement to the machinery which might
render a trial over a definite course not only inconclusive, but
also devoid of any return of valuable data, will have no effect
TRIAL TRIPS. 383
on the result, since the trial may be considered as ended at
the expiration of any complete ten-minute interval, and
whether the results are considered satisfactory or not, at least
they are known, and valuable data are obtained.
On the other hand the limitation of this mode of trial so
far as relates to similarity of conditions between the standard-
ization trial and regular trial must be carefully noted. The
apparent slip will vary with the condition of the bottom, and
with the draft, trim, and condition of the wind and sea; and
since calm weather and a smooth sea are the only conditions
which can be definitely described, it is necessary that both
trials should be made substantially under such conditions.
77. THE INFLUENCE OF ACCELERATION AND RETARDATION
ON TRIAL-TRIP DATA.
If the revolutions and power or mean effective pressure
are subject to rapid fluctuations, they will no longer have the
same relation to speed as for uniform conditions. If the
pressure rapidly increases, a portion of the resultant increase
of thrust and work done goes toward accelerating the motion
of the ship, and the mean pressure, revolutions, and power
will all be greater than for the existing speed under uniform
conditions. Similarly with a rapidly decreasing pressure the
mean pressure, revolutions, and power will be less than for
the existing speed under uniform conditions.
The variation of the pressure is usually not sufficiently
rapid to make this item of practical importance, though the
possibility of its existence as a disturbing feature may be
borne in mind.
It could only be eliminated by a knowledge of the vary-
384 RESISTANCE AND PROPULSION OF SHIPS.
ing acceleration, an experimental determination of which
would be a matter of no small difficulty. A further problem
involving somewhat similar considerations is that concerned
with the time and distance necessary to effect a change of
speed either of increase or decrease. This is of such general
interest and importance in connection with the subject-
matter of the present chapter that we may properly turn to
its consideration at this point.
78. THE TIME AND DISTANCE REQUIRED TO EFFECT A
CHANGE OF SPEED.
Suppose the mean effective pressure in the cylinders to be
dependent alone on the points of cut-off. With any fixed
condition of these, therefore, and neglecting variations due to
friction, the engine-turning moment will be constant, and
hence its ^equivalent, the moment resisting the transverse
motion of the propeller. Again, assuming that the distribu-
tion of pressure over the surface of the propeller is sensibly
the same for constant turning moment no matter what the
speed of the ship, it follows that the relation between the
actual thrust and the mean effective pressure is geometrical,
and hence if the former remains constant, so will the latter.
While these suppositions may not be quite exact, they must
in any case be very near the truth. Suppose, therefore, the
ship being at rest, that steam of a given pressure is turned on
the engine with the cut-offs in such position that their joint
result would give a mean effective pressure sufficient to give
a thrust which would maintain the ship at a speed of u, knots.
In consequence the ship will gradually gather headway and
after a time attain sensibly the speed u lt during which a dis-
TRIAL TRIPS. 385
tance s will have been traversed. In thus considering thrust
and resistance it is evident that actual thrust and actual or
augmented and not true or towed resistance must be taken.
Let R l denote the resistance at speed u l and in general R
that at speed u. We shall have then a constant thrust equal
to R l acting against a variable hydraulic resistance R. The
difference R l R will be a net force available for the accelera-
tion of the ship. We must also remember that a certain
amount of water is more or less closely associated with the
motion of the ship, and partakes of its acceleration and retar-
dation. We may therefore use i*D instead of D. We have
then
du _g(R l -R)
df~ ~~
In the fundamental formulae of dynamics R is measured
in pounds force and D in pounds mass. As their ratio is
unchanged by a change in the unit, they may both be meas-
ured in tons. We have therefore, as the units in this equa-
tion, feet, seconds, and tons. Now represent R in the gen-
eral form
R = cD*u\ ...... (2)
in which c will be later defined so that R and D may be in
tons and u in feet per second or knots per hour. In this
equation c is not necessarily a constant. This will provide
therefore for any actual law of augmented resistance. We
have then, from (i),
386 RESISTANCE AND PROPULSION OF SHIPS.
ds
Dividing by -;- = //, we have
du __gc(u? - ')
* = ~^u~
Hence from (3) and (4)
du
dt = - ...... (5)
gc u* u*
VD* udu
ds- -- j ...... (6)
gc ?// u
If now to simplify integration we assume c a constant, we
have, reckoning t and s from starting,
From (7) and (8) it appears that when u = u lt both t and
^ become oo . This is simply a result of the form of the
equations, and indicates that under the conditions assumed,
the velocity would go on indefinitely increasing, but would
never actually attain the amount u r This is the natural and
du
necessary result of the decreasing value of -j- as given in (3)
when u approaches u lt and while it may be a suprising result,
it is in exact accord with the physical conditions assumed. In
the actual case, the mean effective pressure is never exactly
constant, and hence the corresponding velocity u^ is con-
stantly variable about a mean value. The important point
practically is not how long or how far it will take to acquire
TRIAL TRIPS.
387
the velocity u lt but rather a velocity sensibly equal to it, as
for example .99^, or .999^. These relations are illustrated
TF; 10,000
8, 7
2
9,000
8,000
7,000
6,000
5,000 g
z
<
4,000
3,000
2,000
1,000
100
500
600
300 300 400
TIME IN SECONDS
FIG. 99. Time and Distance for gaining Speed.
in Fig. 99, where values of (7) and (8) are plotted for succes-
sive values of u. We will now explain the reduction of the
equations to a form suitable for computation.
388 RESISTANCE AND PROPULSION OF SHIPS.
Let Kbe the Admiralty displacement coefficient as defined
in 62, and ^ the ratio between the thrust horse-power
T.H.P. and the I.H.P. Then we have
and T.H.P. = ,
. K.
where the units are the horse-power, ton, and knot.
To reduce to the units, foot-ton per minute, ton, and foot
per second, we have
eD%u * x 33000
and R =
KX 2240 X (1.689)"'
'' X 33000
KX 2240 X (1.689)' X 6o ^
Comparing this with the general form for R in (2), it is
evident that c = .05 le -f- K. This may be taken as the defin-
ing equation for c such that when substituted in (2) it will give
R in tons when D is in tons and u in feet per second. Sub-
stituting this value of c in (7) and (8) we should then have
the values of t in seconds and s in feet, expressed in terms of
D in tons and u in feet per second. While retaining the
second and foot as units for t and s, it will be more conven-
ient to introduce on the right-hand side of the equations the
necessary factors, so that common logarithms may be used
instead of hyperbolic, and speed may be measured in knots
instead of feet per second. Making these substitutions in (7)
and (8), and putting /* = i.i, we find
, ,
t (sec.) -.457 log ; ... (9)
(feet) = .772 log ._.. . . .(10)
TRIAL TRIPS. 389
For illustration take K = 240 and K -r- e = 400, D
5000, w, = 15. Then we have
* = 208.4 log ..... (ii)
225
and * = 5280 log _ ..... (12)
The numerical values of (u) and (12) are plotted in Fig.
99, as above referred to. The curve AB giving the time is
asymptotic to the 15 -knot line, thus showing geometrically the
indefinite approach of the speed to the 15 knots as a limit.
The curve CD shows likewise the relation between time and
distance, approaching more and more nearly a straight line in
form as the 15-knot speed is gradually approximated to.
We will now suppose the vessel going ahead under uniform
conditions at speed u lt and that the engine is suddenly
reversed and driven with full mean effective pressure astern.
In such case we shall have a different geometrical relation
between the longitudinal force due to the screw and the mean
effective pressure, than for motion ahead. In general, how-
ever, we shall have in such case
du_ g(P+R)
dt '
where P = backward pull due to engine, and R = resistance
to velocity of ship forward.
We will now assume, similar to (i),
R = cflu\
and P cTPu
390 RESISTANCE AND PROPULSION OF SHIPS.
where for P, u, is the speed at which the given mean pressure
would drive the ship ahead. We have then for (13)
du g
di =
whence <& = - ^ ^T^TT^ (14)
, _ r-^ udu
g cji* -\-crf'
Integrating in the usual manner, we have
In selecting values of c^ and c^ we may bear in mind the
following considerations: The coefficient , is less than the c
for the go-ahead condition due to the rounded backs of the
blades, and the consequent decreased longitudinal component
of the distributed surface pressures. For a given mean pres-
sure, therefore, the pull would be less than the thrust. As
to the ratio between the two, definite information is lacking;
and in default of such definite data we will take the pull as .8
the thrust. Again, the value of c 1 corresponds to the actual
resistance of the ship to motion ahead when the propeller is
backing. In this condition the latter is sending a stream of
water forward against the stern, thereby producing an excess
of pressure at this point instead of a defect as when working
ahead. The actual resistance to motion ahead is therefore
TRIAL TRIPS. 391
decreased instead of increased, or the augmentation may be
considered as negative. The ratio between c t and the c for
motion ahead at the same speed would be equal therefore to
that between the true resistance minus this decrement, and
the true resistance plus the usual augmentation. Taking the
augmentation and decrement about the same, this ratio would
be not far from .8, and we will therefore take c^ as equal to c iu
Comparing also with the value of c, we have
Taking jj. = i.i, substituting these various values, and
reducing speed to knots and hyperbolic to common loga-
rithms we have
DK f l *\
' = 493 (-785- t-'-); . . . ( l8 )
& K \ I ( u \\~\
s = -959-7- [-30103 - log ^1 + (-) JJ. . (19)
If u = o, we have simply
&K
, = . 38 7_ ; . ....... (20)
D-K
* = .289 = .747*,* ..... (21)
With the same numerical values as before, we have
t=22$ (.785 - tan- ^); ..... (22)
= 6s6o[. 30103- log (i + ()')] ; . (23)
s
and if u = o,
/= 177,
s= 1975.
39 2
RESISTANCE AND PROPULSION OF SHIPS.
The varying values of / and s for any speed u are given in
Fig. 100, which is thus a diagram of the extinction of forward
JSWU
300
^
^
" B
/
'
190
/
/
180
^/
/
jcnn
^
/
/
fi
7
/
150
<
&
/
14U
-ion
1
/
<.
7
s>/
LO s
1000
/
4
/
ri
/
/
z
on U
ui
L_
Z
/
/
yu _
P
bJ
/
/
/
<
fe
/
/
70
Q
crin
/
/
/
60
/
X
/
50
/
/
7
/
/
^
ou
/
/
20
15 14 13 13 11 10
3
987
SPEED
TIME AND DISTANCE FOR STOPPING FROM FULL SPEED
FIG. ioo.
elocity. AB gives the relation between speed and distance,
;d CD that between speed and time.
For simplicity of operation we have assumed that the
TRIAL TRIPS. 393
augmented resistance varies throughout as the square of the
speed. This, of course, is far from being actually the case.
The ratio K -r- e may be expected to vary considerably in the
case of a vessel gaining speed from rest or stopping from full
speed. If the law of variation of K -- e were known that is,
the law of the variation of the actual resistance with speed,
both with engines turning ahead and backing-^-the funda-
mental equations could readily be solved by approximate or
graphical integration. The difference in the result, however,
would be slight, and would not change the general nature of
the results obtained by the substitution of the constant value
of K -i- e. It may be remarked, however, that in the selec-
tion of a value of K a mean rather than maximum value
should be taken.
An interesting conclusion from (21) is that the distance in
which a vessel may be brought to rest from motion ahead is
sensibly independent of the speed, and depends only on the
7
size and the ratio , or if e is taken as practically constant,
on D and K. The reason why there must actually be a ten-
dency toward some such uniformity in the distance run may
readily be seen. In the example taken this distance would
be from five to six lengths. In the case of very full or poorly
formed vessels, or with foul bottom, we might have an average
value of K much less than 240. As an illustration, let K =
1 80 and K - e = 300. Then the distance traversed in such
case would be only three fourths of that under previous con-
ditions, and we should have for the same size of ship
s = 1481.
394 RESISTANCE AND PROPULSION OF SHIPS.
These figures agree with the general results of the experi-
ments reported to the British Association,* by which it was
found that ships could usually be stopped in from 4 to 6
lengths, and that this distance seemed practically independent
of the power, or in other words, of the speed at which the
experiment was made.
We have assumed in all the foregoing that the engine is
instantly reversed, and that the engine instantly gathers its
motion either ahead or back, as the case may be. This is not
actually possible, so that due allowance should be made in
comparing experimental results with those given by the fore-
going approximate theory.
In any case the general nature of these results is correct,
and it is readily seen that they have an important bearing on
the manoeuvring of vessels and on the proper conditions for
trial trips. In measured-mile trials especially the importance
is clearly shown of a long start in order that the ship may
attain sensibly her maximum speed before entering upon the
course.
79. THE GEOMETRICAL ANALYSIS OF TRIAL DATA.
Considering the three items of data, power, speed, and
revolutions, we may plot the former on either of the latter as
abscissa giving curves as in Figs. 101, 102, or we may plot
speed on revolutions giving a curve as in Fig. 103. The
various uses of these curves need no especial explanation.
In using the power-speed curve for purposes of comparison,
61, however, it must be remembered that the given curve
strictly applies only to the ship in the given condition of dis-
* Reports for 1878, p. 421.
TRIAL TRIPS.
395
placement, trim, condition of bottom, and of weather and
water, and that it cannot be applied to the same ship for all
SPEED
FIG. 101.
REVOLUTIONS
FIG. io2.
circumstances without due allowance having been made for
such differences in condition as may exist.
It is frequently of interest to know the index of the speed
with which the power varies at any given speed, and similarly
396
RESISTANCE AND PROPULSION OF SHIPS.
for the variation of resistance with speed, or either with revo-
lutions, or in fact of any quantity which may be expressed in
the general form
y = ax n (i)
For example, we may determine that at 10 knots the
power varies as the cube of the speed and the resistance as
u REVOLUTIONS
FIG. 103.
the square, while at 15 knots the power may vary as the
index 3.7 and the resistance therefore as the index 2.7.
We will now develop methods of deriving the value of
this index.
Let x l be the given value of the abscissa. Then the
usual method has been to assume the index sensibly constant
for a short range on either side of this value, say between the
values x<> and x^ equidistant above and below x^ Then from
the general equation we have
TRIAL TRIPS. 397
whence ^ =
and n = log , + log =
Jo ^^o 2 -
When, however, the curve is at hand showing the relation
between y and x as is usually the case, the following
geometrical method will be found much more rapid and satis-
factory, besides being exact in principle for a continually
varying value of n.
We must first obtain a clear idea of the nature of the
index with which we are here concerned.
Let it be a question of the variation of any function y
with x. Let the value at x^ be y 9 . Then let x be increased
by a small increment such that its new value divided by its
old is (i + e), where e is a quantity very small compared
to i. This will result in a change in the value of y in
a ratio which we may denote by (i +/"). Now obviously
we may put
(I +,)- = !+/).
or = ...... (3)
log (i + e)
This we take as defining the nature of the exponent m
with which we are here concerned. It would therefore follow
that if m = 3 and the speed were increased i per cent it
would result in an increase of the power in the ratio (i.oi) 3 .
So long as the amount of increase is indefinite, the char-
acter of our exponent will lack mathematical distinctness, and
we are therefore naturally led to define m as the ratio of log
(i +/) to log (i -\- e) when e is indefinitely decreased.
398 RESISTANCE AND PROPULSION OF SHIPS.
Denoting the increments by dx and efy, we have then by defi-
nition
(4)
By Maclaurin's theorem of expansion this becomes
dy
m j_ = <fy__^y_
We have also
dy
y _
- ~ - '
We must now show that the exponent m thus defined is
not in general the same as the n of the general equation
y =. ax n .
We have log y log a -\- n log x.
Now remembering that in the cases with which we have
to deal n is not a constant, we differentiate, considering both
n and x as variables. We thus have
Also
whence
dy
~- anx n l -4- ax n loi
dx
2. = ax n ~' L \
x
dy e y
dn
**dx
or
(7)
TRIAL TRIPS.
399
Hence it follows that unless n is constant, m and n will
not be the same. It also follows that in the general case the
exponent found by the expression in (2), being an approxima-
tion to the value of m, will not satisfy the fundamental equa-
tion (i) from which it is found, except for a particular value
of a\ and that a series of values thus found will not in
general satisfy the fundamental equation (i) for any constant
value of a. This still further shows the fact of the difference
ia character between m and n as here defined.
Taking then the value of m as thus defined, we proceed to
show a method for its geometrical determination.
In Fig. 104 let OP be the graphical representation of the
B F
FIG. 104.
law in question. Let A be any given point corresponding to
abscissa OB, and let CA be a tangent to OP at A. Then
dy AB y AB
= and
400 RESISTANCE AND PROPULSION OF SHIPS.
-'-*- ....... < 8 >
or m = the quotient of the abscissa by the subtangent.
In this determination the only uncertainty is that involved
in drawing a tangent to a curve. This may, however, be
done quite accurately as follows: We may evidently assume
without sensible error that the curve in the vicinity of A may
be considered as parabolic of the second degree. We then
take two points E and F equidistant from B and note the
corresponding points 5 and R on the curve. Then by a well-
known property of such curves a line drawn through A
parallel to RS will be tangent to the curve at A.
The exponent n, which is essentially different in character
from m, may be defined simply as the exponent which will
satisfy the given fundamental equation y = ax n . Its value is
then most readily found by taking logarithms, whence
log x
It may be readily seen that the value of n will depend on
the particular unit used for x, but once this unit defined, the
value of n becomes definite. Let x = i, or the unit. Then
we have
a=*y,
or the value of the constant a = the ordinate for x = I.
We may therefore put more generally
y = y^x n .
The exponent m may be seen to relate simply to the
nature of the law at the point in question. We may therefore
term it the index of instantaneous variation. It must neces-
TRIAL TRIPS.
4OI
sarily be the exponent involved when we say that at any
point y varies as a certain power of x. Qn the other hand,
the exponent n depends on the entire law of variation from
the point where x = I to the point in question. It therefore
represents the resultant effect, between these points, of the
10
entire series of values of m. If the law is such that m is con-
stant, m and n become identical as in the common algebraic
curves. The distinction between m and n for such curves as
we are here concerned with must not be forgotten.
The application of the preceding to the variation of power
402
RESISTANCE AND PROPULSION OF SHIPS.
or resistance with speed is obvious. In Fig. 105 are given
examples in which AB is the given fundamental curve, CD is
the locus of the exponent m, and CE is that of the exponent
n, where one mile is the unit. These diagrams are self-
explanatory and will repay careful examination. Evidently
the same investigation may be made with reference to the
power and revolution curve. The speed and revolution curve
gives the apparent slip, which may also, if desired, be plotted
separately as AB, Fig. 106. If OA, Fig. 97, were a straight
SLIP IN PER CENT
o S 8 8
^
,B
A
.
.
-^
5 10 15
SPEED IN KNOTS
FIG. 106.
line the apparent slip would be constant, AB would be hori-
zontal, and the indices relating power to revolutions and
speed would be the same. Practically this is never found,
though frequently the slip will vary but slightly through a
considerable range.
We have in 58 given a method of finding the mean
effective pressure corresponding to the initial friction of the
engine. We will now give another due to Wm. Froude.
In Fig. 107 let AB be a curve of mean pressure, propor-
tional, as we have seen in 47, to the curve of indicated
t TRIAL TRIPS. 403
thrust. Now if we assume that this pressure varies as the
resistance, and the latter for speeds below tt l as speed with
the constant index n, we have only to continue BA by a
curve tangent to it at A and with reference to some at
present unknown axis EX, fulfilling the condition that m = n
at all points. Froude took m = 1.87, and the construction
is readily seen to be an immediate result of the general
method established above for the determination of m. The
construction is as follows : At A draw a tangent to the curve.
SPEED
FIG. 107.
Lay off u^C = u^O -r- m, erect a perpendicular CD, and draw
EX. Then a curve EA run in tangent at E to EX and at A
to AD will approximately fulfil the conditions required for
the continuation of the pressure curve downward, and OE will
therefore be the value of /> , or the value for the initial fric-
tion. Due to the uncertainties of drawing a tangent at the
end of a curve and the uncertainty that the pressure varies
as the index 1.87, or indeed with any other constant index of
the speed, the final result necessarily partakes of the uncer-
404
RESISTANCE AND PROPULSION OF SHIPS.
tainty inherent in all attempts to extend a graphical law
beyond the range covered by the observations.
80. APPLICATION OF LOGARITHMIC CROSS-SECTION PAPER.
In logarithmic cross-section paper, the divisions are similar
to those on a slide-rule, or in other words, the distances from
8 9 10
the origin are proportional to the logarithms of the numbers,
as in Fig. 108. Cross-section paper ruled in this way has so
TRIAL TRIPS. 405
many special applications to various problems in naval archi-
tecture, that a brief description of the more important may
be of interest.
Given an equation y = ax*.
Whence logj = log a + n log x.
Now remembering that on this paper the abscissa and
ordinate are not x and y, but log x and log y, the above
equation is really
ordinate = constant -f- n times abscissa.
This for usual plotting is in the form
y = Ax + B,
and hence represents a straight line.
It follows that any such equation will on this paper be
represented by a straight line inclined to X at an angle whose
tangent is n, and cutting Fat a point which gives a. This
fundamental property has a number of important applications,
of which we may note the following:
(a) If a = i, we have
y = x n .
Hence a straight line on this paper will give a locus of
roots or powers, of index whole or fractional, to determine
which it is simply necessary to draw a line or a series of lines
at an angle tan' 1 ;/ to X.
As commonly made, this paper has but a single logarith-
mic scale from I to 10, subdivided according to the size of
the unit. By drawing on this, however, a number of lines
dependent on the nature of the index, a single sheet may be
made to give the ;/th power or root of any number from o
to oo . For illustration we will take the case of a table of *
3
406
RESISTANCE AND PROPULSION OF SHIPS.
powers. That is, we wish to express graphically the locus of
the equation y x*. Let Fig. 109 denote a square cross-
sectioned with logarithmic scales as described. Suppose that
G
FIG. 109
there were joined to it and to each other on the right and
above, an indefinite series of such squares similarly divided.
Then considering in passing from one square to an adjacent
one to the right or above, that the unit becomes of the next
higher order, it is evident that such a series of squares would,
with the proper variation of the unit, represent all values of
either x or y between o and oo .
Suppose next the original square divided on the horizontal
edge into 2 parts and on the vertical into 3, the points of
division being at A, B, C, E, F y G. Then lines joining these
points as shown in the diagram will be at an inclination to
the horizontal whose tangent is f . Now beginning at O, OE
will give the values of x* for values of x from I to 10. For
greater values of x the line would run into the next adjacent
square, and the location of this line, if continued, may be seen
to be exactly similar to that of BC in the square before us.
TRIAL TRIPS. 407
It follows that, considering the units as of the next nigher
order, the line BC will give values of x* for x between O and
G or 10 and 3i.6-|~ For larger values of x we should run
into the adjacent square above with change of unit for jj/, but
without change for x. We should here traverse a line similar
to GF. Therefore by proper choice of units we may use GF
for values of x* where x lies between 31.6+ and 100. We
should then run into the next square on the right requiring
the unit for x to be of the next higher order, and traverse a
line similar to AD, which takes us finally to the opposite
corner and completes the cycle, the last line giving us values
of x% for x between 100 and 1000. Following this, the same
series of lines would result for numbers of succeeding orders.
A little consideration of the subject will show that the
value of x* for any value of x between I and oo may thus be
read approximately from one or another of these lines; and
if for any value between I and oo , then likewise for any value
between o and I. The location of the decimal point is readily
found by a little attention to the numbers involved. A rule
for its location might be derived, but is of little additional
value in practice. The limiting values of x for any given line
may be marked on it, thus enabling a proper choice to be
readily made.
The principles involved in this case may be readily ex-
tended to any other, and it will be found in general that
if the exponent be represented by , the complete set of
lines may be drawn by dividing one side of the square into
m and the other into n parts, and joining the points of divi-
sion as in Fig. 109. In all there will be (m-\-n i) lines, and
opposite to any point on X there will be n lines correspond-
408 RESISTANCE AND PROPULSION OF SHIPS.
ing to the n different beginnings of the nth root of the mth
power, while opposite to any point on Y will be m lines
corresponding to the m different beginnings of the mth root
of the nth power. Where the complete number of lines
would be quite large, it is usually unnecessary to draw them
all, and the number may be limited to those necessary to
cover the needed range in the values of x.
If, instead of the equation y = x n , we have a constant
term as a multiplier, giving an equation in the more general
m
form y = Bx n , or Bx n , there will be the same number of lines
and at the same inclination, but all shifted vertically through
a distance equal to log B. If, therefore, we start on the axis
of Fat the point B, we may draw in the same series of lines
and in a similar, manner.
It will be noted, of course, that the index m -f- n may be
used either way for the same set of lines. That is, the same
lines will give the f or f power of numbers, or the second or
\ power, etc.
(b) If in the general equation y = ax n the exponent n is
variable, the locus plotted on logarithmic paper will be curved
instead of straight, and the index of instantaneous variation
m, at any given point ( 79), will be the tangent of the incli-
nation of the tangent line at such point. This is readily seen
from 79, (6). Also, n will be the tangent of the inclination
of a line drawn from the point where the locus cuts Y (or
where x i) to the given point. This readily follows from
79, (9). This serves also to clearly illustrate the difference
in character between m and n.
(c) Proportions of the form
TRIAL TRIPS.
are frequently met with. As an equation this is
409
or log j, = log y, + n(\og x 9 log x v ).
Now in Fig. no let A and B denote the points x^ and x^
and C and D be at the heights denoted by y\ and y^ Then,
remembering that the actual distances involved are the
logarithms of the corresponding quantities, we readily see that
BD = AC + nAB or AC + . Therefore, a line from (7
to D will be at an inclination to X whose tangent is n. If,
FIG. no.
therefore, we have any ready means of passing from C, a
point determined by the coordinates x l y 1 , along an oblique
line inclined to the axis of X at an angle whose tangent is n,
it is evident that we shall pass along a continuous series of
points x^y^ so related to x^y^ that the proportion above men-
tioned will be fulfilled. To solve any such proportion, there-
fore, we have simply to start at the given point, x l y lt and
pass along the oblique line until we reach a point whose
abscissa is x t . The corresponding ordinate will be the desired
410 RESISTANCE AND PROPULSION OF SHIPS.
value of y^ Or, vice versa, if we stop at any given value of
the ordinate y the corresponding abscissa will be x^, so
related to the other quantities that the proportion is fulfilled.
To provide the necessary means for moving in the right
direction from any point whatever as x^y^, a series of equi-
distant lines at the proper angle may be ruled. With this aid
a very close approximation to the proper values of x^y^ may
be made. As an instance of the use of this proportion,
suppose that we wish to find the corresponding speeds for two
vessels from the sixth roots of the displacements.
We have - =
Given the u l an^ D l we find the point, considering one
axis as that of speed and the other as that of displacement.
Then by passing along a line inclined at an angle tan' 1 \ to
the axis of D, we shall pass through a continuous series of
points which will give corresponding displacements and
speeds. Stopping at the displacement Z> 2 we read off directly
the desired speed # ; or, vice versa, stopping at a speed z/ 3 we
read off the corresponding displacement Z> 2 .
Having thus found the corresponding speeds, we may by
another application of the same principle find the I.H.P. by
using, for example, the equation
ff. (D.\*
As a further illustration we may take the determination of
Admiralty coefficients from the equation
z?V
~~~-
All of the operations here involved may be carried out on
logarithmic paper and the value of the coefficients rapidly
determined.
STEAMSHIP AND PROPELLER DATA.
IN Tables A and B will be found a collection of data
relating to ship and propeller performance. A few of the
cases given in Table B are not represented in A due to the
difficulty of obtaining satisfactory values of some of the data
required. In Table B it should be noted that the value of
the I.H.P. is that required for one propeller, and not that
required for the ship as a whole in the case of twin or triple
screws.
The data presented in these tables have been gathered from
various sources, as near as possible in all cases to the origi-
nal publication. All ratios and derivative quantities have,
furthermore, been independently computed from the funda-
mental data, and while it can hardly be hoped that among so
many figures all errors have been avoided, it is hoped that
their number is small and their character not significant.
In Table B, column headed Screw, the letters have the
following significance:
C center
S starboard
P , port
M mean
4"
412
STEAMSHIP AND PROPELLER DATA.
ID en n co o en I*** t** M o^ 0*0 o*
en r^ o^ ""> ^ >H D en O O O o en
>O D ""> too O *D t^O O O u-> u->
1-g.S
co co O ^t O i* O mo vr> O co
i/^ ID ID r^ ID co ID r^ o *^ w w
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SI3
1J
enOi^Nr^-^-w O Oi^- r^oo tDQenenOOO^QOOOOr^
Tt o O moo -to HH o O co t^co OcoW'^-O^MCoqrtr^qCJq
C^C4>-< OMMC4I-I
8OOOOQOOOO
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O WOOOCO N O IDO W TfNOCOCO MCOOO 't rt f^ N DrfO
en ^f D T-J- TJ- cno ^ en rfo co ^f ^ ^r w iDiDcni-H rj-cnenrrenM
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TABLE A.
413
w oo c* r^^-o O
O O O *t o r~ eno
vr>O O O O to ir>O
*^O oo o t^
'*'*''3'O>-'
O u-> ir>O O O O tr>O
en M co co o to
S-cg
lS-8
a ^i
oo rococo r-^ r^ en "tf* * to o O "} tr>o i^O O >-i r^M MO
-i o^ O <* N too TJ- en *l-co u->co -t-^-or^ino OO r^o^
O^oo r- t^ o> O^oo oo o^ O^oo r^co r-^ r^oo o r^oo t^oo oo oo
B a
MO r>t-M r^o O Ooo >H 1-1 enr^u^xntnenoo ^-
o >-i TI-N woo T}- ^- o w OOO w N N w OOO
ot^i-iO'^- i ^-O enNcn-^-
2S
NOMOtnenOOt^-^-coOQ'oOooooOooOcnOOooOenO
Oeno^woejqcoqqt-'qoo Ooo oo t^r^q *? "'f ^ O ""* "f 9.
8OOOOOootr> enoo loor^^Qoo O O enenO Otno ^ O
toqqqoooctNooooooooooooenoowooo
Tt to en a ^- en ^v
qqqtnqqtnqqqq tno N N rtco qqqqqMqqqq
dNco'Tt-cn6piiooddQNod6o6 i ^-o'doo
O O OO O t"* tDOO 1-1 "^O O O Tf ~tO IO rf IO O OOO
isplace-
ment.
Tons.
^ O O N o O O tntoenOoo xniootoenwoo O O ^ w tor^Q M
oo O W W M Tj-oo Tf M O O O f N enoo N ^tO oo to OO CO Q O
to Tfoo NOent-iONenNMinMMO OOOOOwenOO
oiH-^-t^o>-ien
414
STEAMSHIP AND PROPELLER DATA.
)5 2 -
C^C^OOOOCOONNCOC>O
(^ N
O Tt-
O r^O f> N
vr> *t to O r~*
ooo^coooco
.c
u-S.2
u .->
tn O m -^- O O 1 O
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coo
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t>-Oo M coc o "- O^" r-O cco M
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rancisco. . . .
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TABLE B.
415
hH <J
co O
or>voco too q o M q o q w vr> q
ooOO"^
O O -frf
ei r^-i-i
II
cnao rr 1- TJ-O O O O O O co too tNWMwwwr^t-ii-i
dwweiNco'Tl-tt OvC oovOooOOOOOO coco co
tCO
u->r>xou-)i-i
too O too O CO
\o t -TO
oo P< C4 N
K
<
H
S-2
2
O r^ co co co Ooo cococo MvCvCOO tttOO to >- O O
M O t t too cocott'-'OOOO>->-'--OOi~iCOcocoo
Ooooooo coo O O O wir^r-^r^r^r^r-^r^i-i 1-1 r^. M <~< o^cc
\n N c< a coq O mvnr^oooOvCOvCcocooo r^o t
tttt>^ O O *+ ^ r>-cd oo co" oo oo* co 06 r^ r>>od t co co M
O O O O ir>mu">u->u->OOOvCvO tttt^"tO u->ir>t
tC> tN tCON tCOt^COCOCO
S S
C w-S
<
bo -
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PQPQ
0QD3 ffl
4i6
STEAMSHIP AND PROPELLER DATA.
JJ 5
^
< u
o> O r~o6 0600 r^
oo coco cQiow M cocoO O
!> ** to tx\O '**> W *^ W x M i> t^ o
M 1-1 cncncnw M I-H >->
O o OOOOC>OO O r^oo co en en
w M u->ir>mu-)u-)u-)tnu-)OOvO ~f-f
OOOOvOOOOOOOOOOoocou-iooooOOO
O O tnvooo O O voxovnvoxntoO O i^oooo O O O
I.iM>f
1/3 "* O W " Ctf . ,
a a u u u
o g ' ' |-
u u u
TABLE B.
417
eo
375 fc
'-
M in en en in
$
Or>enMO*-t M citn^j-M eno co cno r~co
\r> M en pi r- O O ^o" O od en M O vr>oo oo N
Mi-ien MM McncnM MMMM
u-> r^ w en O
^'r^OOOHi
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1
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w
hJ
CQ
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II
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ir>O O O HI m vr>O *^oo i~ r^ ^ en O>O OO r)xn\r>tr>tnenc<
O O O u->ir>o O r^o r^Tj-rrvr>o OOOO cncncnent^r^M
en en eno encnen-3-cnw enenenenuitnintnTfr^-^^-N w en
O enencn
MOOoo Oooao ineno ^-rj-ior^vnO O O u->ir>u->u->co W O
>-i)-ii-ii-iTj- rrco
OO 3C34> oo
UU UQQ QWfefe
4iS
STEAMSHIP AND PROPELLER DATA.
MminCr> Oco N o
wxnm O o M
u> O m O O 9 O O O oo cn<> in O c *^
O CG o "-"c O -< c inc M 10 r>< 10 M
WCOMMCOt^r^OOONO^iOMOO'^'CnQ
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coOu-)n\r)ir>MMcocoNcocOMi-iQococococococoeofor^
u-)n\r)ir>MMcocoNcocOMi-i
l-ii-.NNOO"">NCMOONN
cocococococoOOOOOOOOO'^-'t'^-
invoinvnmioOOr^OiHtnmmmtniovo
8 S SSsl- -
TABLE B.
419
< flu
^ O oo r> O oo 00 vnoo coo O* >-> O N N "^ M coo N rf N m O
i-t wcooo
Ooocooooo
\O O^ * O r>
i^>> w u"> r^ r^ ^H in o w>o O O O oo ^^ HI r*%
corj-cO'l-Nc<r^C7OON>nwi
____., i-iNMC4-iM
Pu w 5
Sl
en cno O
-rr>crH-i cocoir>xnoo cnwoo r>o O^T
COO O C^O W W t-i IH muicocoN u~, ^r
<O<NNNNC<Nl-il-.weNN>-iM
O O *^> 't
O O cocncococncno O O O Q O O
OOcoc r )cocncocoOOOOOOO
en en O N co cooo ooooooaooo O O O O O O >cocococococo
M HH xooooooooooooooooooo O O O O O O M tncococotnco
W
J3 V
1*1
fc
3j
420
STEAMSHIP AND PROPELLER DATA.
Ill
w O to O O w coco ^- O *^ c* f> m t^ t* coo vn ** co o "">
or^cicd cod ** ** w ti od t< t*jp vAooo cocod oo
co^J-Tj-NNMMr^r^r^oOOvONOO
NrtrtOOWNOOO'OOOO Ncoco
O* O *r> too t^-oo oo co coo O cpcocococoQ^O^MvOvO
CONGO O
cio'cc^t'vOHHO>
irjrt-MMiom
t^. o^oo n co coo
d^c5^ccor>vo
r^oo M o O w w
Ooo rxt-t OT|-COCOM\O w t^oo
nr^covOtHOcovOci
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tntnovO ^ w NWW
K
3
^6
S
3 U5
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O c^i co O O O^ O^ r^ r^ co o^ o^ O O *^ O O oo co co
w M mu->iomvococoC4 C*d M H-tmind
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C4 c c<
t tc co
K
g
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8= -5: 59^
TABLE B.
421
< a
-- r
W CM
M oo < r>- O tr> uo co co -- co c- rOO -< & M r>
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INDEX.
PAGE
Acceleration and retardation, influence of, on trial data 383
Admiralty displacement coefficient 340
Admiralty midship-section coefficient 344
Air, indraught of 288
Air-resistance 129
Analysis, geometrical, of trial data 394
Analysis of power required for propulsion 228
Apparent slip 210
Apparent propeller efficiency 226
Area factor, table for 257
Area-ratio 176
Augmentation of resistance due to action of propeller 227
Augmented surface, Rankine's 345
Auxiliaries, power required for 332
Barnaby 196, 294
Beaufoy 34, 42
Bilge-keels, influence of, on resistance 1 27
Blechynden 282, 333
Blenheim, H.M.S no
Bureau of Steam Engineering, Navy Department 380
Calvert 153, 207
Camere 119
Canals, resistance in 109
Carrying capacity, size and speed, relation between 356
Cavitation 291
Change of speed, time and distance required for 384
Coefficient of augmentation 230, 232
Colthurst 53
Comparison, law of 133, 147
Comparison, law of, applied to propeller design 206
Comparison, law of, when ships are not exactly similar 347
Conder 119
423
424 INDEX.
PAGE
Corresponding speeds 138
Cotterill 41. 196, 246
Course for trial trips 365
Data 412, 415
Dead-water 5
Denny 365
Derome 119
Diameter-ratio 182
Disk area denned 1 76
Dubuat 34, 118
Effective horse-power 228
Efficiency denned 165
Efficiency of screw propeller 226, 260
Encyclopaedia Britannica 54
Encyclopaedia Metripolitana 54
Engine friction 324
English's mode of comparison 352
Euler 3, 157
Experimental data connected with equations for propeller design 240
Experimental methods of determining resistance 154
Feathering paddle wheels 200
Foul bottom, influence of, on resistance 53, 131
Friction, engine 324
Froude, R. E 35, 77, 98, 194, 195, 196, 281
Froude, Wm 35, 49> I0 4, 127, 129, 154, 402
Gatewood 54
Greyhound 130, 154
Guide-blade propellers 203
Head-resistance coefficients 34
Helicoidal area denned 1 76
Hichborn 53
Hull efficiency ." 229
Hydraulic propulsion 201
Indicated thrust 236
Indraught of air 288
Initial friction 325, 327
Joessel's experiments 34
Kinematic similitude, law of 133, 147
Lagrange 3
Load friction 330
Location of propellers 286
Logarithmic cross-section paper, application of 404
INDEX. 425
PAG a
Mas, M. de 119
Materials for propeller-blades 304
Maxwell , 19
McDermott 254
Mean pitch 209
Mean pitch, computation of 214
Mean slip defined 209, 21 2
Merkara 104
Middendorf 151
Negative apparent slip 237
New York, U.S.S no
Normand 292
Number of propellers 284
Obliquity of stream and of shaft, influence of, on action of screw propeller. . 218
Paddle-wheel, action of element of 169, 172
Paddle-wheel, propulsive action of 196
Parsons 294
Pitch-angle 177
Pitch defined 176
Pitch, measurement of 316
Pitch, modification of, due to twisting of blade 319
Pitch ratio 1 76
Pollard and Dudebout 54
Powering ships, Chapter V 322
Powering ships, illustrative example 334
Powering ships by law of comparison 337
Powering ships by formulae involving wetted surface 344
Powering ships by use of special constants 346
Propeller design, Chapter IV 240
Propeller design, equations for 255, 278, 279, 280
Propeller design, problems 252
Propeller design, tables 256, 263
Propulsion, Chapter II 162
Propulsive coefficient 228
Propulsive element, action of 164
Projected area defined 176
Racing 290
Rankine 16, 54, 196, 202, 345 .
Reduced wetted surface 48, 50
Resistance, Chapter I i
Resistance, air 1 29
Resistance, eddy 29
426 INDEX.
PAGE
Resistance formulae 151
Resistance, head and tail 30
Resistance in canals 109
Resistance, increase of, due to banks and shoals 109
Resistance, increase of, due to change of trim 1 24
Resistance, increase of, due to irregular movement 107
Resistance, increase of, due to rough water 108
Resistance, increase of, due to slope of currents 1 23
Resistance, oblique, of ship formed bodies 157
Resistance of planes moving normal to themselves 32
Resistance of planes moving obliquely to their normal 37
Resistance, relation of, to density of liquid 104
Resistance, residual 31, 149
Resistance, skin or tangential, of planes 42
Resistance, skin or tangential, of ships 48
Resistance, skin or tangential, values of constants 50
Resistance, stream-line 28
Resistance, surface or skin 29
Resistance, wave-making 29, 93
Risbec 47, 102
Russell, J. Scott 54, 1 1 7, 1 18
Sardegna, bilge-keels on 1 28
Screw propeller, action of element of 177
Screw propeller, definitions 1 74
Screw propeller, geometry of 306
Screw propeller, laying down 310
Screw propeller, propulsive action of 182
Screw turbines 203
Shallow-water resistance 109
Sink, stream-line 18
Size, carrying capacity, and speed, relation between 356
Slip angle defined 1 78
Slip angle, table for connecting with slip ratio 321
Slip defined 168, 209
Source, stream-line 18
Speed, size, and carrying capacity, relation between 356
Speed-trial with standardized screw , 380
Springing of blades 295
Stokes 54
Stream-lines 7
Stream-lines defined 4
Strength of propeller blades 299
INDEX. 427
PAGE
Stopping, distance and time required for 384
Sweet, Elnathan : 118
Taylor, D. W 26, 42, 114, 152, 321, 377
Thornycroft's screw turbine 203
Thrust horse-power 228
Tidal influence, elimination of 368
Tidal observations 366
Tidman -. 42
Trial trips, Chapter VI 362
True propeller efficiency 226
True slip 210
Units of measurement xiii
Wake, constitution of 206
Wake factor 225
Wake, influence of, on equations of propeller action 224
Wake-return factor 232
Waves 54
Waves, combinations of 63
Wave-echoes, formation of 89
Wave-formation due to raotion of ship 75
Waves of translation 69
Waves, shallow-water 67
Waves, tabular data 73~7S
Waves, train of, propagation of 87
Wetted surface 49, 50
Wetted surface, reduced 48, 50
Wetted surface, powering by formulae involving 344
White no, 124
Yarrow 125, 154
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Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo, 2 oo
Kinnicutt, Winslow and Pratt's Purification of Sewage. (In Preparation. :
Laplace's Philosophical Essay on Probabilities. (Truscott and Emory)
i2mo, 2 oo
Mahan's Descriptive Geometry 8vo, i 50
Treatise on Civil Engineering. (1873.) (Wood) 8vo, 5 oo
Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50
Merriman and Brooks's Handbook for Surveyors i6mo, mor. 2 oo
Nugent's Plane Surveying 8vo, 3 50
Ogden's Sewer Construction 8vo, 3 oo
Sewer Design i2mo, 2 oo
Parsons's Disposal of Municipal Refuse 8vo, 2 oo
Patton's Treatise on Civil Engineering 8vo. half leather, 7 50
Reed's Topographical Drawing and Sketching 410, 5 oo
Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 4 oo
Kleiner's Shaft-sinking under Difficult Conditions. (Corning and Peele). . 8vo, 3 oo
Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, i 50
Smith's Manual of Topographical Drawing. (McMillan) 8vo, 2 50
Soper's Air and Ventilation of Subways Large i2mo, 2 50
Tracy's Plane Surveying i6mo, mor. 3 oo
* Trautwine's Civil Engineer's Pocket-book i6mo, mor. 5 oo
Venable's Garbage Crematories in America 8vo, 2 oo
Methods and Dtvices for Bacterial Treatment of Sewage 8vo, 3 oo
Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo
Sheep, 6 50
Law of Contracts 8vo, 3 oo
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8vo, 5 oo
Sheep, 5 50
Warren's Stereotomy Problems in Stone-cutting 8vo, 2 50
* Waterbury's Vest-Pocket Hand-book of Mathematics for Engineers.
2X si inches, mor. i oo
Webb's Problems in the Use and Adjustment of Engineering Instruments.
i6mo, mor. i 25
Wilson's (H. N.) Topographic Surveying 8vo, 3 50
Wilson's (W. L.) Elements of Railroad Track and Construction i2mo, 2 oo
BRIDGES AND ROOFS.
Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 oo
Burr and Falk's Design and Construction of Metallic Bridges 8vo, 5 oo
Influence Lines for Bridge and Roof Computations 8vo, 3 oo
Du Bois's Mechanics of Engineering. Vol. II. . . , Srrall 4:0, 10 oo
Foster's Treatise on Wooden Trestle Bridges 4 to, 5 oo
Fowler's Ordinary Foundations 8vo, 3 50
French and Ives's Stereotomy 8vo, 2 e O
Greene's Arches in Wood, Iron, and Stone 8vo, 2 50
Bridge Trusses 8vo, 2 50
Roof Trusses 8vo, i 25
Grimm's Secondary Stresses in Bridge Trusses 8vo, 2 -o
Heller's Stresses in Structures and the Accompanying Deformations 8vo, 3 oo
Howe's Design of Simple Roof- trusses in Wood and Steel 8vo, 2 oo
Symmetrical Masonry Arches 8vo, 2 50
Treatise on Arches 8vo, 4 oo
7
Johnson Bryan, and Turneaure's Theory and Practice in the Designing of
Modern Framed Structures.. Small 4to, 10 oo
Merrlman and Jacoby's Text-book on Roofs and Bridges:
Part I. Stresses in Simple Trusses 8vo, 2 50
Part II. Graphic Statics 8vo, 2 50
Part III. Bridge Design 8vo, 2 50
Part IV. Higher Structures 8vo, 2 50
Morison's Memphis Bridge Oblong 4to, 10 oo
Sondericker's Graphic Statics, with Applications to Trusses, Beams, and Arches.
8vo, 2 oo
Waddell's De Pontibus, Pocket-book for Bridge Engineers i6mo, mor, 2 oo
** Specifications for Steel Bridges 12010, 50
Waddell and Harrington's Bridge Engineering. (In Preparation.)
Wright's Designing of Draw-spans. Two parts in one volume Svo, 3 50
HYDRAULICS.
Barnes's Ice Formation 8vo, 3 oo
Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from
an Orifice. (Trautwine) 8vo, 2 oo
Bovey's Treatise on Hydraulics 8vo, 5 oo
Church's Diagrams of Mean Velocity of Water in Open Channels.
Oblong 4to, paper, i 50
Hydraulic Motors 8vo, 2 oo
Mechanics of Engineering 8vo, 6 oo
Coffin's Graphical Solution of Hydraulic Problems i6mo, mor. 2 50
Flather's Dynamometers, and the Measurement of Power 12 mo, 3 oo
Folwell's Water-supply Engineering 8vo, 4 oo
Frizell's Water-power 8vo, 5 oo
Fuertes's Water and Public Health i2mo, i 50
Water-filtration Works ... i?.mo, 2 50
Ganguillet and Kutter's General Formula for the Uniform Flow of Water in
Rivers and Other Channels. (Hering and Trautwiue) Svo, 4 oo
Hazen's Clean Water and How to Get It Lar:e ismo. T so
Filtration of Public Water-supplies 8vo, 3 oo
Hazlehurst's Towers and Tanks for Water- works Svo, 2 50
Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal
Conduits Svo , 2 oo
Hoyt and Grover's River Discharge Svo, 2 oo
Hubbard and Kiersted's Water-works Management and Maintenance Svo, 4 co
* Lyndon's Development and Electrical Distribution of Water Power. . . .Svo, 3 oo
Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.)
Svo, 4 oo
Merriman's Treatise on Hydraulics Svo, 5 oo
* Michie's Elements of Analytical Mechanics Svo, 4 oo
* Molitor's Hydraulics of Rivers. Weirs and Sluices Svo, 2 oo
* Richards's Laboratory Notes on Industrial Water Analysis Svo, o 50
Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water-
supply. Second Edition, enlarged Large Svo, 6 oo
* Thomas and Watt's Improvement of Rivers 4to, 6 oo
Turneaure and Russell's Public Water-supplies Svo, 5 oo
Wegmann's Design and Construction of Dams. 5th Ed., enlarged 4to, 6 oo
Water-supply of the City of New York from 1658 to 1895 4to, 10 oo
Whipple's Value of Pure Water Large i2mo, i oo
Williams and Hazen's Hydraulic Tables Svo, i 50
Wilson's Irrigation Engineering Small Svo, 4 oo
Wolff's Windmill ^s a Prime Mover Svo, 300
Wood's Elements of Analytical Mechanics < Svo, 3 oo
Turbines Svo, 2 50
8
MATERIALS OF ENGINEERING.
Baker's Roads and Pavements 8vo, 5 oo
Treatise on Masonry Construction 8vo, 5 oo
Birkmire's Architectural Iron and Steel 8vo, 3 50
Compound Riveted Girders as Applied in Buildings 8vo, 2 oo
Black's United States Public Works Oblong 4to, 5 oo
Bleininger's Manufacture of Hydraulic Cement. (In Preparation.)
* Bovey's Strength of Materials and Theory of Structures 8vo, 7 50
Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 50
Byrne's Highway Construction 8vo, 5 oo
Inspection of the Materials and Workmanship Employed in Construction.
i6mo, 3 oo
Church's Mechanics of Engineering 8vo, 6 oo
Du Bois's Mechanics of Engineering.
Vol. I. Kinematics, Statics, Kinetics Small 4:0, 7 50
Vol. II. The Stresses in Framed Structures, Strength of Materials and
Theory of Flexures Small 4to, 10 oo
Eckel's Cements, Limes, and Plasters 8vo, 6 oo
Stone and Clay Products used in Engineering. (In Preparation.)
Fowler's Ordinary Foundations 8vo, 3 50
Graves's Forest Mensuration 8vo, 4 oo
Green's Principles of American Forestry I2mo, i 50
* Greene's Structural Mechanics 8vo, 2 50
Ho ly and Ladd's Analysis of Mixed Paints, Color Pigments and Varnishes
Large izmo, 2 50
Johnson's C. M.) Chemical Analysis of Special Steels. (In Preparation.)
Johnson's < J . B.j Materials of Construction Large 8vo, 6 oo
Keep's Cast Iron 8vo, 2 50
Kidder's Architects and Builders' Pocket-book i6mo, 5 oo
Lanza's Applied Mechanics 8vo f 7 50
Maire's Modern Pigments and their Vehicles , I2mo, 2 oo
Martens's Handbook on Testing Materials. (Henning) 2 vols 8vo, 7 50
Maurer's Technical Mechanics 8vo, 4 oo
Merrill's Stones for Building and Decoration. 8vo, 5 oo
Merriman's Mechanics of Materials 8vo, 5 oo
* Strength of Materials i2mo, i oo
Metcalf's Steel. A Manual for Steel-users i2mo, 2 oo
Morrison's Highway Engineering 8vo, 2 50
Patton's Practical Treatise on Foundations 8vo, 5 oo
Rice's Concrete Block Manufacture 8vo, 2 o o
Richardson' j Modern Asphalt Pavements 8vo, 3 oo
Richey's Handbook for Superintendents of Construction i6mo, mor. 4 oo
* Ries's Clays: Their Occurrence, Properties, and Uses 8vo, 5 oo
Sabir.'s Industrial and Artistic Technology of Paints ard Varnish 8vo, 3 oo
*Schwarz'sLong1eafPinein Virgin Forest izmo, i 25
Snow's Principal Species of Wood 8vo, 3 50
Spalding's Hydraulic Cement i2mo, 2 oo
Text-book on Roads and Pavements i2mo, 2 oo
Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 oo
Thurston's Materials of Engineering. In Three Parts 8vo, 8 oo
Part I. Non-metallic Materials of Engineering and Metallurgy 8vo, 2 oo
Part IT. Iron and SteeL 8vo, 3 50
Part HI. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8v 2 SO
Tillson's Street Pavements and Paving Materials 8vo, 4 oo
Turneaure and Maurer's Principles of Reinforced Concrete Construction.. .8vo, 3 oo
Waterbury's Cement Laboratory Manual i2mo, i oo
Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on
the Preservation of Timber 8vo, 2 oo
Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and
Steel 8vo, 4 oo
RAILWAY ENGINEERING.
Andrews's Handbook for Street Railway Engineers 3x5 inches, mor. i 25
Berg's Buildings and Structures of American Railroads 4to, 5 oo
Brooks's Handbook of Street Railroad Location i6mo, mor. i 50
Butt's Civil Engineer's Field-book i6mo, mor. 2 50
Crandall's Railway and Other Earthwork Tables 8vo, i 50
Transition Curve i6mo, mor. i 50
* Crockett's Methods for Earthwork Computations 8vo, i 50
Dawson's "Engineering" and Electric Traction Pocket-book i6mo. mor. 5 oo
Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 oo
Fisher's Table of Cubic Yards Cardboard, 25
Godwin's Railroad Engineers' Field-book and Explorers' Guide. . . i6mo, mor. 2 50
Hudson's Tables for Calculating the Cubic Contents of Excavations and Em-
bankments 8vo, i oo
Ives and Hilts's Problems in Surveying, Railroad Surveying and Geodesy
i6mo, mor. i 50
Molitor and Beard's Manual for Resident Engineers i6mo, i oo
Nagle's Field Manual for Railroad Engineers i6mo, mor. 3 oo
Philbrick's Field Manual for Engineers i6mo, mor. 3 oo
Raymond's Railroad Engineering. 3 volumes.
Vol. I. Railroad Field Geometry. (In Preparation.)
VoL II. Elements of Railroad Engineering 8vo, 3 50
Vol. III. Railroad Engineer's Field Book. (In Preparation.)
Searles's Field Engineering i6mo, mor. 3 oo
Railroad Spiral i6mo, mor. i 50
Taylor's Prismoidal Formulae and Earthwork 8vo, i 50
*Trautwine's Field Practice of Laying Out Circular Curves for Railroads.
i2mo. mor. 2 50
* Method of Calculating the Cubic Contents of Excavations and Embank-
ments by the Aid of Diagrams 8vo, 2 oo
Webb's Economics of Railroad Construction Large i2mo, 2 50
Railroad Construction i6mo, mor. 5 oo
Wellington's Economic Theory of the Location of Railways Small 8vo, 5 oo
DRAWING.
Barr's Kinematics of Machinery 8vo, 2 50
* Bartlett's Mechanical Drawing 8vo, 3 oo
* " " " Abridged Ed 8vo, i 50
Coolidge's Manual of Drawing. . . '. 8vo. paper, i oo
Coolidge and Freeman's Elements of General Drafting for Mechanical Engi-
neers Oblong 410, 2 50
Durley's Kinematics of Machines 8vo, 4 oo
Emch's Introduction to Projective Geometry and its Applications 8vo, 2 50
Hill's Text-book on Shades and Shadows, and Perspective 8vo, 2 oo
Jamison's Advanced Mechanical Drawing Svo, 2 oo
Elements of Mechanical Drawing 8vo. 2 50
Jones's Machine Design:
Part I. Kinematics of Machinery Svo, i 50
Part H. Form, Strength, and Proportions of Parts Svo, 3 oo
MacCord's Elements of Descriptive Geometry Svo, 3 oe
Kinematics; or, Practical Mechanism Svo. 5 oo
Mechanical Drawing 4to, 4 oo
Velocity Diagrams Svo, i 50
10
McLeod's Descriptive Geometry.. Large i2mo, i 50
* Mahan's Descriptive Geometry and Stone-cutting 8vo, i 50
Industrial Drawing. (Thompson ) 8vo, 3 50
Meyer's Descriptive Geometry for Students of Engineering 8vo, 2 oo
Reed's Topographical Drawing and Sketching 4to, 5 oo
Reid's Course in Mechanical Drawing 8vo, 2 oo
Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo
Robinson's Principles of Mechanism 8vo, 3 oo
Schwamb and Merrill's Elements of Mechanism 8vo, 3 oo
Smith's (R. S.) Manual of Topographical Drawing. (McMillan) 8vo, 2 50
Smith (A. W.) and Marx's Machine Design 8vo, 3 oo
* Titsworth's Elements of Mechanical Drawing Oblong 8vo : i 25
Warren's Drafting Instruments and Operations i2mo, i 25
Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 50
Elements of Machine Construction and Drawing 8vo, 7 50
Elements of Plane and Solid Free-hand Geometrical Drawing i2mo, i oo
General Problems of Shades and Shadows 8vo, 3 oo
Manual of Elementary Problems in the Linear Perspective of Form and
Shadow i2mo, i oo
Manual of Elementary Projection Drawing i2mo, i 50
Plane Problems in Elementary Geometry i2mo, i 25
Problems, Theorems, and Examples in Descriptive Geometry. 8vo, 2 50
Weisbach's Kinematics and Power of Transmission. (Hermann and
Klein) 8vo, 5 oo
Wilson's (H. M.) Topographic Surveying 8vo, 3 50
Wilson's (V. T.) Free-hand Lettering 8vo, i oo
Free-hand Perspective 8vo, 2 50
Woolf's Elementary Course in Descriptive Geometry Large 8vo, 3 oo
ELECTRICITY AND PHYSICS.
* Abegg's Theory of Electrolytic Dissociation, (von Ende) i2mo, i 25
Andrews's Hand-Book for Street Railway Engineering 3X5 inches, mor. i 25
Anthony and Brackett's Text-book of Physics. (Magie) Large i2mo, 3 oo
Anthony's Theory of Electrical Measurements. (Ball) i2mo, i oo
Benjamin's History of Electricity 8vo, 3 oo
Voltaic Cell 8vo, 3 oo
Betts's Lead Refining and Electrolysis 8vo, 4 oo
Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood). .8vo, 3 oo
* Collins's Manual of Wireless Telegraphy i2mo, i 50
Mor. 2 oo
Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 oo
* Danneel's Electrochemistry. (Merriam) i2mo, i 25
Dawson's "Engineering" and Electric Traction Pocket-book .... i6mo, mor. 5 oo
Dolezalek's Theory of the Lead Accumulator (Storage Battery), (von Ende)
i2mo, 2 50
Duhem's Thermodynamics and Chemistry. (Burgess) 8vo, 4 oo
Flather's Dynamometers, and the Measurement of Power i2mo, 3 oo
Gilbert's De Magnate. (Mottelay) 8vo, 2 50
* Hanchett's Alternating Currents i2mo, i oo
Bering's Ready Reference Tables (Conversion Factors) i6mo, mor. 2 50
* Hotart and Ellis's High-speed Dynamo Electric Machinery 8vo, 6 oo
Holman's Precision of Measurements 8vo. 2 oo
Telescopic Mirror-scale Method, Adjustments, and Tests. .. .Large 8vc , 75
* Karapetoff's Experimental Electrical Engineering 8vo, 6 oo
Kinzbrunner's Testing of Continuous-current Machines 8vo, 2 oo
Landauer's Spectrum Analysis. (Tingle) 8vo, 3 oo
Le Chatelier's High-temperature Measurements. (Boudouard Burgess).. i2mo, 3 oo
Lob's Electrochemistry of Organic Compounds. (Lorenz) 8vo, 3 oo
* London's Development and Electrical Distribution of Water Power . . . ,8vo, 3 oo
11
* Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II 8vo, each, 6 oo
* Michie's Elements of Wave Motion Relating to Sound and Light 8vo, 4 oo
Morgan's Outline of the Theory of Solution and its Results i2mo, i oo
* Physical Chemistry for Electrical Engineers i2mo, i 50
Niaudet's Elementary Treatise on Electric Batteries. (Fishback). . . . I2mo, 2 50
* Norris's Introduction to the Study of Electrical Engineering 8vo, 2 50
* Parshall and Hobart's Electric Machine Design 4to, half mor. 12 50
Reagan's Locomotives: Simple, Compound, and Electric. New Edition.
Large i2mo, 3 50
* Rosenberg's Electrical Engineering. (Haldane Gee Kinzbrunner). .. .8vo, 2 co
Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 2 50
Swapper's Laboratory Guide for Students in Physical Chemistry i2mo, i oo
* Tillman's Elementary Lessons in Heat 8vo, i 50
Tory and Pitcher's Manual of Laboratory Physics Large I2mo, 2 oo
Ulke's Modern Electrolytic Copper Refining 8vo. 3 oo
LAW.
* Brennan's Handbook : A Compendium of Useful Legal Information for
Business Men i6mo, mor. 5 . oo
* Davis's Elements or Law 8vo, 2 50
* Treatise on the Military Law of United States 8vo, 7 oo
* Sheep, 7 50
* Dudley's Military Law and the Procedure of Courts-martial . . . .Large i2mo, 2 50
Manual for Courts-martiaL i6mo, mor. i 50
Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo
Sheep, 6 50
Law of Contracts 8vo, 3 oo
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8vo 5 oo
Sheep, 5 50
MATHEMATICS.
Baker's Elliptic Functions 8vo, i 50
Briggs's Elements of Plane Analytic Geometry. (Bocher) i2mo, i oo
*Buchanan's Plane and Spherical Trigonometry 8vo, i oo
Byerley's Harmonic Functions 8vo, i oo
Chandler's Elements of the Infinitesimal Calculus i2mo, 2 oo
Coffin's Vector Analysis. (In Press.)
Compton's Manual of Logarithmic Computations i2mo, i 50
* Dickson's College Algebra Large i2mo, i 50
* Introduction to the Theory of Algebraic Equations Large i2mo, i 25
Emch's Introduction to Projective Geometry and its Applications 8vo, 2 50
Fiske's Functio -,s of a Complex Variable 8vo, i oo
Halsted's Elementary Synthetic Geometry 8vo, i 50
Elements of Geometry 8vo, i 75
* Rational Geometry i2mo, i 50
Hyde's Grassmann's Space Analysis 8vo, i oo
* Jonnson's (J B.) Three-place Logarithmic Tables: Vest-pocket size, paper, 15
100 copies, 5 oo
Mounted on heavy cardboard, 8X10 inches, 25
10 copies, 2 oo
Johnson's (W. W.) Abridged Editions of Differential and Integral Calculus
Large lamo, i vol.
Curve Tracing in Cartesian Co-ordinates i2mo,
Differential Equations 8vo.
Elementary Treatise on Differential Calculus Large i2mo,
Elementary Treatise on the Integral Calculus Large I2mo,
Theoretical Mechanics I2mo, 3
oo
Theory of Errors and the Method of Least Squares i2mo, i 50
Treatise on Differential Calculus Large i2mo, 3 oo
12
Johnson's Treatise on the Integral Calculus. Large i2mo, 3 oo
Treatise on Ordinary aad Partial Differential Equations. . Large i2mo, 3 50
Karapetoffs Engineering Applications of Higher Mathematics. (In Pre-
paration.)
Laplace's Philosophical Essay on Probabilities. (Truscott and Emory). .i2mo, 2 oo
* Ludlow and Bass's Elements of Trigonometry and Logarithmic and Other
Tables 8vo, 3 oo
Trigonometry and Tables published separately Each, 2 oo
* Ludlow's Logarithmic and Trigonometric Tables 8vo, i oo
Macfarlane's Vector Analysis and Quaternions 8vo, i oo
McMahon's Hyperbolic Functions 8vo, i oo
Manning's Irrational lumbers and their Representation by Sequences and
Series I2mo, i 23
Mathematical Monographs. Edited by Mansfield Merriman and Robert
S. Woodward Octavo, each i oo
No. i. History of Modern Mathematics, by David Eugene Smith.
No. 2. Synthetic Projective Geometry, by George Bruce Halsted.
No. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyper-
bolic Functions, by James McMahon. To 'j. Harmonic Func-
tions, by William E. Byerly. No. 6. Grassmann's Space Analysis,
by Edward W. Hyde. No. 7. Probability and Theory of Errors,
by Robert S. Woodward. No. 8. Vector Analysis and Quaternions,
by Alexander Macfarlane. No. 9. Differential Equations, by
William Woolsey Johnson. No. 10. The Solution of Equations,
by Mansfield Merriman. No. u. Functions of a Complex Variable,
by Thomas S. Fiske.
Maurer's Technical Mechanics 8vo, 4 oo
Merriman's Method of Least Squares 8vo, 2 oo
Solution of Equations 8vo, i oo
Rice and Johnson's Differential and Integral Calculus. 2 vols. in one.
Large i2mo, i 50
Elementary Treatise on the Differential Calculus Large i2mo, 3 oo
Smith's History of Modern Mathematics 8vo, i oo
* Veblen and Lennes's Introduction to the Real Infinitesimal Analysis of One
Variable 8vo, 2 oo
* Waterbury's Vest Pocket Hand-Book of Mathematics for Engineers.
zjXsi inches, mor. i oo
Weld's Determinations 8vo, I co
Wood's Elements of Co-ordinate Geometry 8vo, 2 oo
Woodward's Probability aid Theory of Errors 8vo, i oo
MECHANICAL ENGINEERING.
MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS.
Bacon's Forge Practice I2mo 1 i 50
Baldwin's Steam Heating for Buildings I2mo, 2 50
Bair's Kinematics of Machinery 8vo, 2 50
* Bartlett's Mechanical Drawing 8vo, 3 oo
* " " Abridged Ed 8vo, i 50
Benjamin's Wrinkles and Recipes i2mo, 2 oo
* Burr's Ancient and Modern Engineering and the Isthmian Canal 8vo, 3 50
Carpenter's Experimental Engineering 8vo, 6 oo
Heating and Ventilating Buildings 8vo, 4 oo
Clerk's Gas and Oil Engine. Large i2mo, 4 oo
Compton's First Lessons in Metal Working i2mo, i 50
Connton and De Groodt's Speed Lathe I2mo, i 50
Coolidge's Manual of Drawing 8vo, paper, i oo
Coolidge and Freeman's Elements of General Drafting for Mechanical En-
gineers Oblong 4to, 2 50
13
Cromweil's Treatise on Belts and Pulleys i2mo, i 50
Treatise on Toothed Gearing I2mo, i 50
Durley's Kinematics of Machines 8vo, 4 oo
Flather's Dynamometers and the Measurement of Power, i2mo, 3 oo
Rope Driving izmo, 2 oo
Gill's Gas and Fuel Analysis for Engineers I2mo, i 25
Goss's Locomotive Sparks 8vo, 2 oo
Greene's Pumping Machinery. (In Preparation.)
Hering's Ready Reference Tables (Conversion Factors) i6mo, mor. 2 50
* Hobart and Ellis's High Speed Dynamo Electric Machinery 8vo, 6 oo
Button's Gas Engine 8vo, 5 oo
Jamison's Advanced Mechanical Drawing 8vo, 2 oo
Elements of Mechanical Drawing 8vo, 2 50
Jones's Gas Engine. (In Press.)
Machine Design:
Part I. Kinematics of Machinery 8vo, i 50
Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo
Kent's Mechanical Engineers' Pocket-book i6mo, mor. 5 oo
Kerr's Power and Power Transmission 8vo, 2 oo
Leonard's Machine Shop Tools and Methods 8vo, 4 oo
* Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean).. .8vo, 4 oo
MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oa
Mechanical Drawing 4to, 4 oo
Velocity Diagrams 8vo, i 50-
MacFarland's Standard Reduction Factors for Gases 8vo, i 50
Mahan's Industrial Drawing. (Thompson) 8vo, 3 50
Oberg's Hand Book of Small Tools Large i2mo, 3 oo
* Parshall and Hobart's Electric Machine Design Small 4to, half leather, 12 50
Peele's Compressed Air Plant for Mines 8vo, 3 oo
Poole's Calorific Power of Fuels 8vo, 3 oo
* Porter's Engineering Reminiscences, 1855 to 1882 8vo, 3 oo
Reid's Course in Mechanical Drawing 8vo, 2 oo
Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo
Richard's Compressed Air i2mo, i 50
Robinson's Principles of Mechanism 8vo, 3 oo
Schwamb and Merrill's Elements of Mechanism 8vo, 3 oo
Smith's (O.) Press-working of Metals 8vo, 3 oo
Smith (A. W.) and Marx's Machine Design 8vo, 3 oo
Sorel ' s Carbureting and Combustion in Alcohol Engines . (Woodward and Preston) .
Large 12 mo, 3 oo
Thurston's Animal as a Machine and Prime Motor, and the Laws of Energetics.
i2mo, i oa
Treatise on Friction and Lost Work in Machinery and Mill Work... 8vo, 3 oo
Tillson's Complete Automobile Instructor i6mo, i 50
mor. 2 oo
Titsworth's Elements of Mechanical Drawing Oblong 8vo, i 25
Warren's Elements of Machine Construction and Drawing 8vo, 7 50
* Waterbury's Vest Pocket Hand Book of Mathematics for Engineers.
2j X 5i inches, mor. i oo
Weisbach's Kinematics and the Power of Transmission. (Herrmann
Klein) 8vo, s oo
Machinery of Transmission and Governors. (Herrmann Klein). . .8vo, 5 oo
Wood's Turbines 8vo> 2 50
MATERIALS OF ENGINEERING
* Bovey's Strength of Materials and Theory of Structures 8vo, 7 50
Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 50'
Church's Mechanics of Engineering 8vo, 6 oa
* Greene's Structural Mechanics . . 8vo, 2 50
14
Holley and Ladd's Analysis of Mixed Paints, Color Pigments, and Varnishes.
Large i2mo, 2 50
Johnson's Materials of Construction 8vo, 6 oo
Keep's Cast Iron 8vo, 2 50
Lanza's Applied Mechanics 8vo, 7 50
Maire's Modern Pigments and their Vehicles 12010, 2 oo
Martens 's Handbook on Testing Materials. (Henning) 8vo, 7 50
Maurer's Technical Mechanics 8vo, 4 oo
Mernman's Mechanics of Materials 8vo, 5 oo
* Strength of Materials i2mo, i oo
Metcaif f s Steel. A Manual for Steel-users i2mo, 2 oo
Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 oo
Smith's Materials of Machines I2mo, i oo
Thurston's Materials of Engineering 3 vols., 8vo, 8 oo
Part I. Non-metallic Materials of Engineering and Metallurgy. . .8vo, 2 oo
Part II. Iron and Steel Svo, 3 50
Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8vo, 2 50
Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 oo
Treatise on the Resistance of Materials and an Appendix on the
Preservation of Timber 8vo, a oo
Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and
Steel 8vo, 4 oo
STEAM-ENGINES AND BOILERS.
Berry's Temperature-entropy Diagram. Second Edition, Enlarged. . . .i2mo, 2 oo
Carnot's Reflections on the Motive Power of Heat. (Thurston) i2mo, i 50
Chase's Art of Pattern Making i2mo, 2 50
Creighton's Steam-engine and other Heat-motors 8vo, 500
Dawson's " Engineering" and Electric Traction Pocket-book i6mo, mor. 5 oo
Ford's Boiler Making for Boiler Makers i8mo, i oo
* Gebhardt's Steam Power Plant Engineering 8vo, 6 oo
Goss's Locomotive Performance .... 8vo, 5 oo
Hemenway's Indicator Practice and Steam-engine Economy i2mo, 2 oo
Button's Heat and Heat-engines 8vo, 5 oo
Mechanical Engineering of Power Plants 8vo, 5 oo
Kent's Steam boiler Economy 8vo, 4 oo
Kneass's Practice and Theory of the Injector 8vo, i 50
MacCord's Slide-valves -. Svo, 2 oo
Meyer's Modern Locomotive Construction 4to, 10 oo
Meyer's Steam Turbines Svo, 4 oo
Peabody's Manual of the Steam-engine Indicator i2mo. i 50
Tables of the Properties of Saturated Steam and Other Vapors Svo, i oo
Thermodynamics of the Steam-engine and Other Heat-engines Svo, 5 oo
Valve-gears for Steam-engines Svo, 2 50
Peabody and Miller's Steam-boilers Svo, 4 oo
Pray's Twenty Years with the Indicator Large Svo, 2 50
Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.
(Osterberg) i2mo, i 25
Reagan's Locomotives: Simple, Compound, and Electric. New Edition.
Large 12 mo, 3 50
Sinclair's Locomotive Engine Running and Management i2mo, 2 oo
Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50
Snow's Steam-boiler Practice Svo, 3 oo
Spangler's Notes on Thermodynamics i2mo, i oo
Valve-gears Svo, 2 50
Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 oo
Thomas's Steam-turbines Svo, 4 oo
15
Thurston's Handbook of Engine and Boiler Trials, and the Use of the Indi-
cator and the Prony Brake Cvo, 5 oo
Handy Tables 8vo, i 50
Manual of Steam-boilers, their 7 esigns, Construction, and Operation.. 8vo, 5 oo
Thurston's Manual of the Steam-engine 2 vols., 8vo, 10 oo
Part I. History, Structure, and Theory 8v:>, 6 oo
Fart II. Design, Construction, and Operation 8vo, 6 oo
Steam-boiler Explosions in Theory and in Praci.ice i2mo, i 50
Wehrenfenning's Analysis and Softening of Boiler Feed-water (Patterson) Svo, 4 oo
Weisbach's Heat, Steam, and Steam-engines. (Du Bois) Svo, 5 oo
Whitham's Steam-engine Design Svo, 5 oo
Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. ..Svo, A oo
MECHANICS PURE AND APPLIED.
Church's Mechanics of Engineering Svo, 6 oo
Notes and Examples in Mechanics Svo, 2 oo
Dana's Text-book of Elementary Mechanics for Colleges and Schools. .i2mo, i 50
Du Bois's Elementary Principles of Mechanics:
Vol. I. Kinematics Svo, 3 50
Vol. II. Statics Svo, 4 oo
Mechanics of Engineering. Vol. I Small 4to, 7 50
Vol. II. Small 4to, 10 oo
* Greene's Structural Mechanics Svo, 2 50
James's Kinematics of a Point and the Rational Mechanics of a Particle.
Large i2mo, 2 oo
* Johnson's (W. W.) Theoretical Mechanics isrno, 3 oo
Lanza's Applied Mechanics Svo, 7 50
* Martin's Text Book on Mechanics, Vol. I, Statics i2mo, i 25
* Vol. 2, Kinemdu^s and Kinetics . .i2mo, 1 50
Maurer's Technical Mechanics Svo, 4 oo
* Merriman's Elements of Mechanics 1200, i oo
Mechanics of Materials Svo, 5 oo
* Michie's Elements of Analytical Mechanics Svo, 4 oo
Robinson's Principles of Mechanism Svo, 3 oo
Sanborn's Mechanics Problems Large i2mo, i 50
Schwamb and Merrill's Elements of Mechanism '. . . .Svo, 3 oo
Wood's Elements of Analytical Mechanics Cvo, 3 oo
Principles of Elementary Mechanics i2mo, i 25
MEDICAL.
* Abderhalden's Physiological Chemistry in Thirty Lectures. (Hall and Defren)
Svo, 5 oo
von Behring's Suppression of Tuberculosis. (Bolduan) i2mo, i oo
* Bolduan's Immune Sera i2mo, i 50
Bordet s Contribution to Immunity. (Gay). (In Preparation.)
Davenport's Statistical Methods with Special Reference to Biological Varia-
tions i6mo, mor. i 50
Ehrlich's Collected Studies on Immunity. (Bolduan) Svo, 6 oo
* Fischer's Physiology of Alimentation Large i2mo, cloth, 2 oo
de Fursac's Manual of Psychiatry. (Rosanoff and Collins) Large i2mo, 2 50
Hammarsten's Text-book on Physiological Chemistry. (Mandel). . .' Svo, 4 oo
Jackson's Directions for Laboratory Work in Physiological Chemistry. ..Svo, i 25
Lassar-Cohn's Practical Urinary Analysis. (Lorenz) i2mo, i oo
Mandel's Hand Book for the Bi~-Chemical Laboratory . . i2mo, i 50
* Pauli's Physical Chemistry in the Service of Medicine. (Fischer) i2mo, i 25
Pozzi-Escot's Toxins and Venoms and their Antibodies. (Cohn) I2mo, i oo
Rostoski's Serum Diagnosis. (Bolduan) I2mo, i oo
Ruddiman's Incompatibilities in Prescriptions 8vo. 2 oo
Whys in Pharmacy "mo, i oo
16
Salkowski's Physiological and Pathological Chemistry. (Orndorff) 8vo, 2 SO
* Satterlee's Outlines of Human Embryology I2mo, i 25
Smith's Lecture Notes on Chemistry for Dental Students 8vo, 2 50
Steel's Treatise on the Diseases of the Dog 8vo, 3 50
* Whipple's Typhoid Fever. . Large tamo, 3 oo
Woodhull's Notes on Military Hygiene i6mo, i 50
* Personal Hygiene I2mo, i oo
Worcester and Atkinson's Small Hospitals Establishment and Maintenance,
and S ggestions for Hospital Architecture, with Plans for a Small
Hospital i2mo, i 25
METALLURGY.
Betts's Lead Refining by Electrolysis 8vo, 4 oo
Holland's Encyclopedia of Founding and Dictionary of Foundry Terms Used
in the Practice of Moulding I2mo, 3 oo
Iron Founder i2mo, 2 50
" " Supplement lamo, 2 50
Douglas's Untechnical Addresses on Technical Subjects i2mo, i oo
Goesel's Minerals and Metals: A Reference Book i6mo, mor. 3 oo
* Iles's Lead-smelting i2mo, 2 50
Keep's Cast Iron 8vo, 2 50
LeChatelier's High-temperature Measurements. (Boudouard Burgess) i2mo, 3 oo
Metcalfs Steel. A Manual for Steel-users i2mo, 2 oo
Miller's Cyanide Process i2n?o, i oo
Minet's Production of Aluminium and its Industrial Use. (Waldo) . . .12010, 2 50
Robine and Lenglen's Cyanide Industry. (Le Clerc) 8vo, 4 oo
Ruer's Elements of Metallography. (Mathewson) (In Press.)
Smith's Materials of Machines I2mo, i oo
Tate and Stone's Foundry Practice i2mo, 2 50
Thurston's Materials of Engineering. In Three Parts 8vo, 8 oo
Part I. Non-metallic Materials of Engineering and Metallurgy . . . 8vo, 2 oo
Part II. Iron and SteeL 8vo, 3 50
Part HI. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8vo, 2 50
Ulke's Modern Electrolytic Copper Refining 8vo, 3 oo
West's American Foundry Practice i2mo, 2 50
Moulder's Text Book i2mo, 2 50
Wilson's Chlorination Process i2mo, i 50
Cyanide Processes i2mo, i 50
MINERALOGY.
Barringer's Description of Minerals of Commercial Value Oblong, mor. 2 50
Boyd's Resources of Southwest Virginia 8vo, 3 oo
Boyd's Map of Southwest Virginia Pocket-book form. 2 oo
* Browning's Introduction to the Rarer Elements 8vo, i 50
Brush's Manual of Determinative Mineralogy. (Penfield) . . 8vo, 4 oo
Butter's Pocket Hand-Book of Minerals i6mo, mor. j oo
Chester's Catalogue of Minerals 8vo, paper, i oo
Cloth, i 25
* Crane's Gold and Silver 8vo, 5 oo
Dana's First Appendix to Dana's New " System of Mineralogy. .". .Large 8vo, i oo
Manual of Mineralogy and Petrography i2mo 2 oo
Minerals and How to Study Them i2mo, i 50
System of Mineralogy Large 8vo, half leather, 12 50
Text-book of Mineralogy 8vo, 4 oo
Douglas's Untechnical Addresses on Technical Subjects I2mo, i oo
Eakle's Mineral Tables 8vo, i 25
Stone and Clay Products Used in Engineering. (In Preparation.)
17
Egleston's Catalogue of Minerals and Synonyms 8vo, 2 SO
Goesei's Minerals and Metals: A Reference Book i6mo, mor. 3 oo
Groth's Introduction to Chemical Crystallography (Marshall) 12010, i 23
* Iddmgs's Rock Minerals 8vo, 5 oo
Johannsen's Determination of Rock-forming Minerals in Thin Sections 8vo, 4 oo
* Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe. 12010, 60
Merrill's Non-metallic Minerals: Their Occurrence and Uses 8vo, 4 oo
Stones for Building and Decoration 8vo, 5 oo
* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.
8vo, paper, 50
Tables of Minerals, Including the Use of Minerals and Statistics of
Domestic Production 8vo, i oo
* Pirsson's Rocks and Rock Minerals I2mo, 2 50
* Richards's Synopsis of Mineral Characters I2mo, mor. i 25
* Ries's Clays: Their Occurrence, Properties, and Uses 8vo, 5 oo
* Tillman's Text-book of Important Minerab and Rocks 8vo, 2 oo
MINING.
* Beard's Mine Gases and Explosions Large 12010. 3 oo
Boyd's Map of Southwest Virginia Pocket-boo* form 2 oo
Resources of Southwest Virginia 8vo, 3 oo
* Crane's Gold and Silver 8vo, 5 oo
Douglas's Untechnical Addresses on Technical Subjects i2mo i oo
Eissler's Modern High Explosives 8vo, 4 -o
Goesei's Minerals and Metals : A Reference Book i6mo, mor. 3 oo
Ihlseng's Manual of Mining 8vo, 5 oo
* Iles's Lead-smelting i2mo, 2 50
Miller's Cyanide Process i2mo, i oo
O'Driscoll's Notes on the Treatment of Gold Ores Svo, 2 oo
Peele's Compressed Air Plant for Mines Svo, 3 oo
Riemer ' s Shaft Sinking Under Difficult Conditions . ( Corning and Peele) ... Svo , 300
Robine and Lenglen's Cyanide Industry. (Le Clerc) 8vo, 4 oo
* Weaver's Military Explosives Svo, 3 oo
Wilson's Chlorination Process iimo, i 50
Cyanide Processes xarno, i 50
Hydraulic and Placer Mining. 2rf edition, rewritten i2mo, 2 50
Treatise on Practical and Theoretical Mine Ventilation 12 mo, i 25
SANITARY SCIENCE.
Association of State and National Food and Dairy Departments, Hartford Meeting,
1906 Svo, 3 oo
Jamestown Meeting, 1907 Svo, 3 oo
* Bashore's Outlines of Practical Sanitation I2mo, i 25
Sanitation of a Country House .i2mo, i oo
Sanitation of Recreation Camps and Parks I2mo, i oo
Folwell's Sewerage. (Designing, Construction, and Maintenance) Svo, 3 oo
Water-supply Engineering Svo, 4 oo
Fowler's Sewage Works Analyses i2mo, 2 oo
Fuertes's Water-filtration Works i2mo, 2 50
Water and Public Health i2mo, i 50
Gerhard's Guide to Sanitary Inspections I2mo, i 50
* Modern Baths and Bath Houses 8vo, 3 oo
Sanitation of Public Buildings i2mo, i 50
Hazen's Clean Water and How to Get It Large i2mo, i 50
Filtration of Public Water-supplies Svo, 3 oo
Kinnicut, Winslow and Pratt's Purification of Sewage. (In Press.)
Leach's Inspection and Analysis of Food with Special Reference to State
Control 8vo 7 oo
18
Mason's Examination of Water. (Chemical and Bacteriological) i2mo, i 25
Water-supply. (Considered Principally from a Sanitary Standpoint; . . 8vo, 4 oo
* Merrimztn's Elements of Sanitary Engineering 8vo, oo
Ogden's Sewer Design izmo, oo
Parsons's Disposal of Municipal Refuse 8vo, oo
Prescott and Winslow's Elements of Water Bacteriology, with Special Refer-
ence to Sanitary Water Analysis i2mo, 50
* Price's Handbook on Sanitation i2mo, 50
Richards's Cost of Cleanness. A Twentieth Century Problem i2mo, oo
Cost of Food. A Study in Dietaries i2mo, oo
Cost of Living as Modified by Sanitary Science I2mo, oo
Cost of Shelter. A Study in Economics I2mo, oo
* Richards and Williams's Dietary Computer 8vo, go
Richards and Woodman's Air, Water, and Food from a Sinitary Stand-
point 8vo, a oo
Rideal's Disinfection and the Preservation of Food 8vo, 400
Sewage and Bacterial Purification of Sewage 8vo, 4 oo
Soper's Air and Ventilation of Subways . Large i2mo, a 50
Turneaure and Russell's Public Water-supplies 8vo, 5 oo
Venabl's Garbage Crematories in America 8vo, 2 oo
Method and Devices for Bacterial Treatment of Sewage 8vo, 3 oo
Ward and Whipple's Freshwater Biology i2tno, 2 50
Whipple's Microscopy of Drinking-water 8vo, 3 50
* Typhod Fever Large i2mo, 3 oo
Value of Pure Water Large i2mo, i oo
Winslow's Bacterial Classification ; i2mo, 2 50
Winton's Microscopy of Vegetable Foods 8vo, 7 50
MISCELLANEOUS.
Emmons's Geological Guide-book of the Rocky Mountain Excursion of the
International Congress of Geologists Large 8vo, i 50
Ferrel's Popular Treatise on the Winds 8vo, 4 oo
Fitzgerald's Boston Machinist i8mo, i oo
Gannett's Statistical Abstract of the World 24010, 75
Haines's American Railway Management i2mo, 2 50
* Hanusek's The Microscopy of Technical Products. (Winton) 8vo, 5 oo
Owen's The Dyeing and Cleaning of Textile Fabrics. (Standage). (In Press.)
Ricketts's History of Rensselaer Polytechnic Instituto 1824-1894.
Large i2mo, 3 oo
Rotherham's Emphasized New Testament , Large 8vo, 2 oo
Standage's Decoration of Wood, Glass, Metal, etc i2mo, 2 oo
Thome's Structural and Physiological Botany. (Bennett) i6mo, 225
Westermaier's Compendium of General Botany. (Schneider) 8vo, 2 oo
Winslow's Elements of Applied Microscopy I2mo, i 50
HEBREW AND CHALDEE TEXT-BOOKS.
Green's Elementary Hebrew Grammar I2mo, i 25
Qesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures.
(Tregelles) Smail 4 to, half mor. 5 oo
19
N|
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
AN INITIAL FINE OF 25 CENTS
WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO 5O CENTS ON THE FOURTH
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APR 171933
JftN 2 1935
LD 21-50m-l,'33