'-3
THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
AN ELEMENTARY TREATISE
ON
HYDRODYNAMICS AND SOUND
AN ELEMENTARY TREATISE
ON
HYDKODYNAMICS AND SOUND
BY
A. B. BASSET, M.A., F.RS.
TRINITY COLLEGE, CAMBRIDGE.
SECOND EDITION, REVISED AND ENLARGED.
CAMBRIDGE
DEIGHTON BELL AND CO.
LONDON GEORGE BELL AND SONS
1900
[All Rights reserved.]
Camfarfoge :
FEINTED BY J. AND C. F. CLAY
AT THE UNIVERSITY PBESS.
f//
e^
PREFACE.
HHHE treatise on Hydrodynamics, which I published in 1888,
was intended for the use of those who are acquainted with
the higher branches of mathematics, and its aim was to present to
the reader as comprehensive an account of the whole subject as
was possible. But although a somewhat formidable battery of
mathematical artillery is indispensable to those who desire to
. possess an exhaustive knowledge of any branch of mathematical
" physics, yet there are a variety of interesting and important
^investigations, not only in Hydrodynamics but also in Electricity
^and other physical subjects, which are well within the reach of
Sievery one who possesses a knowledge of the elements of the
Differential and Integral Calculus and the fundamental principles
^of Dynamics. I have accordingly, in the present work, abstained
Kfrom introducing any of the more advanced methods of analysis,
\A
IN such as Spherical Harmonics, Elliptic Functions and the like ;
and, as regards the dynamical portion of the subject, I have
endeavoured to solve the various problems which present them-
selves, by the aid of the Principles of Energy and Momentum, and
have avoided the use of Lagrange's equations. There are a few
problems, such as the helicoidal steady motion and stability of a
solid of revolution moving in an infinite liquid, which cannot be
conveniently treated without having recourse to moving axes ; but
as the theory of moving axes is not an altogether easy branch of
Dynamics, I have as far as possible abstained from introducing
VI PREFACE.
them, and the reader who is unacquainted with the use of moving
axes is recommended to omit those sections in which they are
employed.
The present work is principally designed for those who are
reading for the Mathematical Tripos and for other examinations
in which an elementary knowledge of Hydrodynamics and Sound
is required; but I also trust that it will not only be of service to
those who have neither the time nor the inclination to become
conversant with the intricacies of the higher mathematics, but
that it will also prepare the way for the acquisition of more
elaborate knowledge, on the part of those who have an opportunity
of devoting their attention to the more recondite portions of these
subjects.
The first part, which relates to Hydrodynamics, has been taken
with certain alterations and additions from my larger treatise, and
the analytical treatment has been simplified as much as possible.
I have thought it advisable to devote a chapter to the discussion
of the motion of circular cylinders and spheres, in which the
equations of motion are obtained by the direct method of calcu-
lating the resultant pressure exerted by the liquid upon the solid ;
inasmuch as this method is far more elementary, and does not
necessitate the use of Green's Theorem, nor involve any further
knowledge of Dynamics on the part of the reader than the ordinary
equations of motion of a rigid body. The methods of this chapter
can also be employed to solve the analogous problem of deter-
mining the electrostatic potential of cylindrical and spherical
conductors and accumulators, and the distribution of electricity
upon such surfaces. The theory of the motion of a solid body and
the surrounding liquid, regarded as a single dynamical system, is
explained in Chapter III., and the motion of an elliptic cylinder in
an infinite liquid, and the motion of a circular cylinder in a liquid
bounded by a rigid plane are discussed at length.
The Chapters on Waves and on Rectilinear Vortex Motion
comprise the principal problems which admit of treatment by
elementary methods, and I have also included an investigation
PREFACE. Vll
due to Lord Rayleigh, respecting one of the simpler cases of the
instability of fluid motion.
In the second part, which deals with the Theory of Sound,
I have to acknowledge the great assistance which I have received
from Lord Rayleigh's classical treatise. This part contains the
solution of the simpler problems respecting the vibrations of
strings, membranes, wires and gases. A few sections are also
devoted to the Thermodynamics of perfect gases, principally for
the sake of supplementing Maxwell's treatise on Heat, by giving
a proof of some results which require the use of the Differential
Calculus.
The present edition has been carefully revised throughout,
and a certain amount of new matter has been added. I have
devoted Chapter IX. to the flexion and vibrations of naturally
straight wires and rods ; whilst an entirely new chapter has been
added on the finite deformation of naturally straight and curved
wires, in which I have discussed a variety of questions which
admit of fairly simple mathematical treatment.
FLEDBOROUGH HALL,
HOLYPORT, BERKS.
CONTENTS.
PART I.
HYDRODYNAMICS.
CHAPTER I.
ON THE EQUATIONS OF MOTION OF A PERFECT FLUID.
AKT. PAGE
1. Introduction 1
2. Definition of a fluid 1
3. Kinernatical theorems. Lagrangian and flux methods . . . 2
4. Velocity and acceleration. The Lagrangian method . . . . 2
5. do. The flux method 3
6. The equation of continuity 4
7. The velocity potential 5
8. Molecular rotation . 6
9-10. Lines of flow and stream lines 6
11. Earnshaw's and Stokes' current function 7
12. The bounding surface .......... 8
13. Dynamical Theorem's .......... 9
14. Proof of the Principle of Linear Momentum 10
15. Pressure at every point of a fluid is equal in all directions . . 10
16. The equations of motion -. 11
17-18. Another proof of the equations of motion 12
19. Pressure is a function of the density . 15
20. Equations satisfied by the components of molecular rotation . . 15
21. Stokes' proof that a velocity potential always exists, if it exists at any
particular instant . . . 16
22. Physical distinction between rotational and irrotational motion . . 17
23. Integration of the equations of motion when a velocity potential exists . 18
24. Steady motion. Bernoulli's theorem . 18
25. Impulsive motion > 20
26. Flow and circulation 21
X CONTENTS.
ART. PAGE
27. Cyclic and acyclic irrotational motion. Circulation is independent of
the time 22
28. Velocity potential due to a source 23
29. do. due to a doublet 24
30. do. due to a source in two dimensions .... 24
31. do. due to a doublet in two dimensions . . .25
32. Theory of images . .25
33. Image of a source in a plane 25
34. Image of a doublet in a sphere, whose axis passes through the centre of
the sphere 26
35. Motion of a liquid surrounding a sphere, which is suddenly annihilated 27
36. Torricelli's theorem 29
37. The vena contracta 30
38. Giffard's injector . . . . , f , . . . .31
Examples 32
CHAPTER II.
MOTION OF CYLINDERS AND SPHERES IN AN INFINITE LIQUID.
39. Statement of problems to be solved 37
40. Boundary conditions for a cylinder moving in a liquid ... 38
41. Velocity potential and current function due to the motion of a circular
cylinder in an infinite liquid 40
42. Motion of a circular cylinder under the action of gravity ... 40
43. Motion of a cylinder in a liquid, which is bounded by a concentric
cylindrical envelop 42
44. Current function due to the motion of a cylinder, whose cross section
is a lemniscate of Bernoulli 43
45. Motion of a liquid contained within an equilateral prism ... 44
46. do. do. an elliptic cylinder ... 45
47. Conjugate functions 45
48. Current function due to the motion of an elliptic cylinder ... 46
49. Failure of solution when the elliptic cylinder degenerates into a lamina.
Discontinuous motion 47
50. Motion of a sphere under the action of gravity 49
51. Motion may become unstable owing to the existence of a hollow . 51
52. Definition of viscosity ; and its effect upon the motion of sphere . 52
53. Resistance experienced by a ship in moving through water . . 54
54. Motion of a spherical pendulum, which is surrounded by liquid . 54
55. Motion of a spherical pendulum, when the liquid is contained within a
rigid spherical envelop . 55
Examples 57
CONTENTS. XI
CHAPTER III.
MOTION OF A SINGLE SOLID IN AN INFINITE LIQUID.
ART. PAGE
56. Different methods of solving the problem 61
57. Bertrand's theorem 62
58. Green's theorem 63
59-63. Applications of Green's theorem 64
64. Conditions which the velocity potential must satisfy .... 66
65. Kinetic energy of liquid is a homogeneous quadratic function of the
velocities of the moving solid 67
66. Values of the components of momentum 68
67. Short proof of the expressions for the kinetic energy and momentum 70
68. Motion of a sphere 71
69-71. Motion of an elliptic cylinder under the action of no forces . 72
72. Motion of an elliptic cylinder under the action of gravity . . 77
73. Helicoidal steady motion of a solid of revolution .... 78
74. Conditions of stability 80
75. Application to gunnery 81
76-78. Motion of a circular cylinder parallel to a plane .... 81
Examples ............ 85
CHAPTER IV.
WAVES.
79. Kinematics of wave-motion 87
80. Progressive waves and stationary waves 89
81. Conditions of the problem of wave-motion 90
82-84. Waves in a liquid under the action of gravity .... 91
85-87. Waves at the surface of separation of two liquids ... 93
88-91. Stable and unstable motion 95
92. Long waves in shallow water 99
93. Analytical theory of long waves 100
94. Stationary waves in flowing water ....... 101
95. Theory of group velocity 103
96. Capillarity 103
97. Capillary waves conditions at the free surface 104
98. Capillary waves under the action of gravity 105
99. Discussion of results 105
100. Capillary waves produced by wind . . .... . . 106
Examples 108
Xll CONTENTS.
CHAPTER V.
RECTILINEAR VORTEX MOTION.
AST. PAGE
101. Vortex motion in two dimensions Ill
102. Definition of vorticity 112
103. Velocity due to a single vortex 113
104. Velocity potential due to a vortex 114
105. Conditions which the pressure must satisfy 114
106. Kirchhoff's elliptic vortex 115
107. Discussion on the stability of a vortex 117
108. Motion of two vortices of equal vorticities 118
109. Motion of two vortices of equal and opposite vorticities . . .118
110. Motion of a vortex in a square corner 119
111. Motion of a vortex inside a circular cylinder 120
112. Rankine's free spiral vortex 121
113. Fundamental properties of vortex motion 122
114. Proof that the vorticity is an absolute constant .... 124
Examples 125
PAET II.
THEORY OF SOUND.
CHAPTER VI.
INTRODUCTION.
115. Noises and musical notes . . 131
116. Connection between the characteristics of a note and the geometrical
constants of a wave 132
117. Velocity of propagation of sound in gases and liquids .... 132
118. Intensity 132
119. Pitch 132
120. Compound notes and pure tones . 133
121. Timbre 134
122. Beats 134
CHAPTER VII.
VIBRATIONS OF STRINGS AND MEMBRANES.
123. Transverse and longitudinal vibrations 136
124. Equation of motion for transverse vibrations 137
125. Solution for a string whose ends are fixed 138
126. Initial conditions . 139
CONTENTS. Xlll
ART. PAGE
127. Motion produced by a given displacement 140
128. Motion produced by an impulse applied at a point .... 141
129. Motion produced by a periodic force ....... 142
130. Free vibrations gradually die away on account of friction . . . 143
131. Forced vibrations 144
132. Normal coordinates. Kinetic and potential energy .... 144
133. Longitudinal vibrations 145
134. Transverse vibrations of membranes 146
135. Nodal lines of a square membrane 147
136. Circular membrane 148
Examples 149
CHAPTER VIII.
FLEXION OF WIRES.
137. Equations of equilibrium of a wire 151
138. Value of the flexural couple 152
139. Conditions to be satisfied at the ends 153
140. The elastica . 154
141. Kirchhoff's kinetic analogue 156
142143. Stability under thrust 156
144. Greatest height consistent with stability 159
145. Equations of motion of a wire 161
146. Equation of motion for the flexural vibrations of a naturally straight
wire 161
147. Conditions at a free end 162
148. Equation of motion and conditions at a free end, when the rotatory
inertia is neglected 162
149. Period of an infinite wire 163
150. Flexural vibrations of a wire of given length 163
151. Period equations 163
152. Extensional vibrations 165
Examples 166
CHAPTER IX.
THEORY OF CURVED WIRES.
153. General equations of motion 170
154. Value of the flexural couple . 171
155. Its components about two arbitrary axes are proportional to the
changes of curvature 173
156. Value of the torsional couple 175
157. Potential energy of a deformed wire 176
158. Torsional couple is constant when the wire is naturally straight . 176
159. Integration of the equations of equilibrium of a naturally straight wire . 177
XIV CONTENTS.
ART. PAGE
160. Discussion of the three first integrals . . . . . 178
161. A helix is a possible figure of equilibrium 178
162. Discussion of the terminal stresses 179
163. Stability of a naturally straight wire under thrust and twist . . 180
164 16?.' Equilibrium and stability of a naturally straight wire which is
deformed into a circle 181
166. Vibrations of a circular ring 183
CHAPTER X.
EQUATIONS OF MOTION OF A PERFECT GAS.
167. Fundamental equations of the small vibrations of a gas . . . 186
168. Displacement in a plane wave is perpendicular to the wave front . 187
169. Newton's value of the velocity of sound 188
170. Thermodynamics of gases 189
171. The second law of Thermodynamics . . . ' . . . . . 191
172. Specific heats of a gas 192
173. Specific heats of air . . 192
174. Equations of the adiabatic and isothermal lines of a perfect gas . 193
175. Elasticity of a perfect gas 194
176. Velocity of sound in air 195
177. Intensity of sound 195
CHAPTER XI.
PLANE AND SPHERICAL WAVES.
178. Motion in a closed vessel 197
179 181. Motion in a cylindrical pipe 197
182. Reflection and refraction 199
183. Change of phase, when reflection is total . . . . . . 202
184. Spherical waves 203
185. Symmetrical waves in a spherical envelop 204
186. Waves in a conical pipe 205
187. Sources of sound 205
188. Diametral vibrations 205
189. Motion of a spherical pendulum surrounded by air . . ... 206
190. Scattering of a sound wave by a small rigid sphere .... 208
Examples 210
PAKT I.
HYDRODYNAMICS.
CHAPTER I.
ON THE EQUATIONS OF MOTION OF A PERFECT FLUID.
1. THE object of the science of Hydrodynamics is to in-
vestigate the motion of fluids. All fluids with which we are
acquainted may be divided into two classes, viz. incompressible
fluids or liquids, and compressible fluids or gases. It must
however be recollected that all liquids experience a slight com-
pression, when submitted to a sufficiently large pressure, and
therefore in strictness a liquid cannot be regarded as an incom-
pressible fluid ; but inasmuch as the compression produced by
such pressures as ordinarily occur is very small, liquids may be
usually treated as incompressible fluids without sensible error.
The physical interest arising from the study of the motion of
gases, is due to the fact that air is the vehicle by means of
which sound is transmitted. We shall therefore devote the first
part of this volume to the discussion of incompressible fluids
or liquids, reserving the discussion of gases for the second part,
which deals with the Theory of Sound.
We must now define a fluid.
2. A fluid may be defined to be an aggregation of molecules,
which yield to the slightest effort made to separate them from
each other, if it be continued long enough.
A perfect fluid, is one which is incapable of sustaining any
tangential stress or action in the nature of a shear; and it
will be shown in 15 that the consequence of this property
is, that the pressure at every point of a perfect fluid is equal
in all directions, whether the fluid be at rest or in motion. A
perfect fluid is however an entirely ideal substance, since all fluids
B. H. 1
2 EQUATIONS OF MOTION OF A PERFECT FLUID.
with which we are acquainted are capable of offering resistance
to tangential stress. This property, which is known as viscosity,
gives rise to an action in the nature of friction, by which kinetic
energy is gradually converted into heat.
In the case of gases, water and many other liquids, the effect
of viscosity is so small that such fluids may be approximately
regarded as perfect fluids. The neglect of viscosity very much
simplifies the mathematical treatment of the subject, and in the
present treatise, we shall confine our attention to perfect fluids.
Before entering upon the dynamical portion of the subject,
it will be convenient to investigate certain kinematical proposi-
tions, which are true for all fluids.
Kinematical Theorems.
3. The motion of a. fluid may be investigated by two different
methods, the first of which is called the Lagrangian method, and
the second the Eulerian or flux method, although both are due
to Euler.
In the Lagrangian method, we fix our attention upon an
element of fluid, and follow its motion throughout its history.
The variables in this case are the initial coordinates a, b, c of the
particular element upon which we fix our attention, and the time.
This method has been successfully employed in the solution of
very few problems.
In the Eulerian or flux method, we fix our attention upon a
particular point of the space occupied by the fluid, and observe
what is going on there. The variables in this case are the
coordinates x, y, z of the particular point of space upon which
we fix our attention, and the time.
Velocity and Acceleration.
4. In forming expressions for the velocity and acceleration
of a fluid, it is necessary to carefully distinguish between the
Lagrangian and the flux method.
I. The Lagrangian Method.
Let u, v, w be the component velocities parallel to fixed axes,
of an element of fluid whose coordinates are x, y, z and x + 8x,
y 4- &/, z + 8z at times t and t + 8t respectively, then
u = dxjdt = x, v=y, w = z ( 1 ),
VELOCITY AND ACCELERATION. 3
where in forming x, y, z we must suppose ac, y, z to be expressed
in terms of the initial coordinates a, b, c and the time.
The expressions for the component accelerations are
f*=u=x, f v = y> f z = z .................. (2),
where u, v, w are supposed to be expressed in terms of a, b, c
and t,
II. The Flux Method.
5. Let 8Q be the quantity of fluid which in time St flows
across any small area A, which passes through a fixed point P
in the fluid ; let p be the density of the fluid, q its resultant
velocity, and e the angle which the direction of q makes with
the normal to A drawn towards the direction in which the
fluid flows. Then
BQ = pqA&t cose,
, 1 dQ
therefore q = . -7- .
pA cos e at
Now A cos e is the projection of A upon a plane passing
through P perpendicular to the direction of motion of the fluid ;
hence SQ is independent of the direction of the area, and is the
same for all areas whose projections upon the above-mentioned
plane are equal. Hence the velocity is equal to the rate per unit
of area divided by the density, at which fluid flows across a plane
perpendicular to its direction of motion.
The velocity is therefore a function of the position of P and
the time.
In the present treatise the flux method will almost exclu-
sively be employed. We may therefore put u = F (ac, y, z, t) ;
whence if u + Su be the velocity parallel to x at time t + St of
the element of fluid which at time t was situated at the point
t, y + vSt, z + wSt, t + St)-F(x, y, z, t).
Therefore the acceleration,
Bu du du du du
f x = hm^- = J.+U-T- + V-J- + W-7-.
J ct dt dx dy dz
Hence if dfdt denotes the operator
d/dt + udjdx + vdjdy + wdjdz,
the component accelerations will be given by the equations
f _ou , _rto f _^w /o\
J*=fo> Jy-^t' Jz ~ dt"
12
4 EQUATIONS OF MOTION OF A PERFECT FLUID.
The Equation of Continuity.
6. If an imaginary fixed closed surface be described in a fluid,
the difference between the amounts of fluid which flow in and
flow out during a small interval of time 8t, must be equal to
the increase in the amount of fluid during the same interval,
which the surface contains.
The analytical expression for this fact is called the equation
of continuity.
Let Q be any point (x, y, z), and consider an elementary
parallelepiped SxSy&z.
The amount of fluid which flows in across the face GB in
time St is
puSySzSt.
The amount which flows out across the opposite face AD is
puSySzSt + -j- (pu) SxSySzSt,
Qw/
whence the gain of fluid due to the fluxes across the faces GB,
AD is
-T- (pu) 8xSySz8t.
Treating the other faces in a precisely similar manner, it
follows that the total gain is
EQUATION OF CONTINUITY. 5
The amount of fluid within the element at time t is pSxSySz,
and therefore the amount at time t + St is
p + -n &t) SxSySz.
at /
The gain is therefore
dp
Equating this to (4) we obtain the equation
dp d (pu) d (pv) d (pw) _
~77 "I 7 I 7 ~\ 7 \J ( ).
at ax ay az
This equation is called the equation of continuity.
In the case of a liquid, p is constant, and (5) takes the simple
form
du dv dw
y + -y 'f -5- U (O>
dx ay az
We shall hereafter require the equation of continuity of a
liquid referred to polar coordinates. This may be obtained in
a similar manner by considering a polar element of volume
r 2 sin O&r&d&Q), and it can be shown that if u, v, w be the velocities
in the directions in which r, 0, &> increase, the required equation is
d(vsin0) dw _ /( _,
+ r \j/ + r ^~ = Q ( 7 >
- - ^-
dr dd da
If or, 6, z be cylindrical coordinates, the equation is
d(tsm) dv dw_
j h -Tn + & -y- =
diff dd dz
The Velocity Potential.
7. In a large and important class of problems, the quantity ,.t.n
udx + vdy + wdz is a perfect differential of a function of x, y, z (
which we shall call
,
dd> dd> dd>
whence u = -r -, v = ^r~, w= , (9).
dx ay dz
Substituting these values of u, v, w in (6) we obtain
5"^ "I" T ~^ T """" ^ (-l-v)j
or as it is usually written
6 EQUATIONS OF MOTION OF A PERFECT FLUID.
This equation is called Laplace's equation, from the name of
its discoverer ; it is a very important equation, which continually
occurs in a variety of branches of physics. The operator V 2 is
called Laplace's operator.
We can now obtain the transformation of Laplace's equation
when polar coordinates are employed. For in this case
udr + vrdQ + wr sin Odw = d,
d(f) 1 d(f> 1 d
whence u = -~ , t = --j2, w = = a -r- ............ (11).
dr r ad r sin dw
Substituting in (7) we obtain
1 d ( . d$
d ( 2 d>\
dr\ T dr)
__
-"- (l
The equation of continuity and the theory of the velocity
potential may therefore be employed to effect transformations,
which it would be very laborious to work out by the usual
methods for the change of the independent variables.
8. The existence of a velocity potential involves the conditions
that each of the three quantities
dw dv du dw dv du
dy dz ' dz dx' dx dy
should be zero ; when such is not the case we shall denote these
quantities by 2, 2??, 2 The quantities , rj, for reasons which
will be explained hereafter, are called components of molecular
rotation, they evidently satisfy the equation
d% .dy d
~J -- r ~l -- r 1~ "
dx dy dz
When a velocity potential exists, the motion is called irrota-
tional ; and when a velocity potential does not exist, the motion
is called rotational or vortex motion.
Lines of Flow and Stream Lines.
9. DEF. A line of flow is a line whose direction coincides
with the direction of the resultant velocity of the fluid.
The differential equations of a line of flow are
dx _ dy _ dz
u v w'
LINES OF FLOW AND STREAM LINES. 7
Hence if Xi( x > y> z > ^) =ct i' X?( x > U> z > = a 2 be any two
independent integrals, the equations % = const., ^ = const., are
the equations of two families of surfaces whose intersections
determine the lines of flow.
DEF. A stream line or a line of motion, is a line whose
direction coincides with the direction of the actual paths of the
elements of fluid.
The equations of a stream line are determined by the simul-
taneous differential equations,
x = u, y = v, z = w,
where ae, y, z must be regarded as unknown functions of t. The
integration of these equations will determine x, y, z in terms of
the initial coordinates and the time.
10. When a velocity potential exists, the equation
udx + vdy + wdz =
is the equation of a family of surfaces, at every point of which the
velocity potential has a definite constant value, and which may be
called surfaces of equi-velocity potential.
If P be any point on the surface, > = const., and dn be an
element of the normal at P which meets the neighbouring surface
(/> + S> at Q, the velocity at P along PQ, will be equal to d/dn ;
hence d must be positive, and therefore a fluid always flows
from places of lower to places of higher velocity potential.
The lines of flow evidently cut the surfaces of equi-velocity
potential at right angles.
11. The solution of hydrodynamical problems is much sim-
plified by the use of the velocity potential (whenever one exists),
since it enables us to express the velocities in terms of a single
function . But when a velocity potential does not exist, this
cannot in general be done, unless the motion either takes place
in two dimensions, or is symmetrical with respect to an axis.
In the case of a liquid, if the motion takes place in planes
parallel to the plane of xy, the equation of the lines of flow is
udy vdx = (13).
The equation of continuity is
du dv _
dx dy
8 EQUATIONS OF MOTION OF A PERFECT FLUID.
which shows that the left-hand side of (13) is a perfect differ-
ential cfyr, whence
The function -|r is called Earnshaw's current function.
When the motion takes place in planes passing through the
axis of 2, the equation of the lines of flow may be written
TO- (wd-sr udz) = ..................... (15),
where ty, z are cylindrical coordinates.
By (8) the equation of continuity is
d(vu) dw _
~~~j -- > OT ~T~ = "j
ttCT (LZ
which shows that the left-hand side of (15) is a perfect differential
dty, whence
1 . (27 ).
dt \pj pdy pdy p dy <
d / \ _ f du 77 dv dw
dt \pJ p dz p dz p dz
These equations may also be written in the form
9 /\ du 77 du t du ,
r7-=--j- + -j-+-- r -,&c. &c.
dt \p/ p dx p dy p dz
21. It was stated in 7, that in many important problems,
the motion is such that a velocity potential exists. The con-
dition that such should be the case is, that , 77, should each
vanish. We shall now prove that, when the fluid is under the
action of a conservative system of forces, a velocity potential will
always exist whenever it exists at any .particular instant.
Let us choose the particular instant at which a velocity
potential exists, as the origin of the time ; then by hypothesis
|, 77, vanish when t = ; also the coefficients of these quantities
in (27), will not become infinite at any point of the interior of the
fluid; it will therefore be possible to determine a quantity L,
which shall be a superior limit to the numerical values of these
coefficients. Hence f, 77, cannot increase faster than if they
satisfied the equation
But if + 77 + = lp, we obtain by adding the above equations
dt
whence fl = Ae? Lt .
Now H = when t = 0, therefore A = ; and since O is the
sum of three quantities each of which is essentially positive, it
follows that , 77, must always remain zero, if they are so at any
particular instant. The above proof is due to Sir G. Stokes 1 .
1 "On the friction of fluids in motion," Section II, Trans. (Jamb, Phil. Soc.
vol, VHI.
MOLECULAR ROTATION. 17
22. There is, as was first shown by Sir G. Stokes, an important
physical distinction in the character of the motion which takes
place, according as a velocity potential does or does not exist.
Conceive an indefinitely small spherical element of a fluid
in motion to become suddenly solidified, and the fluid about it
to be suddenly destroyed. By the instantaneous solidification,
velocities will be suddenly generated or destroyed in the different
portions of the element, and a set of mutual impulsive forces will
be called into action.
Let x, y, z be the coordinates of the centre of inertia G of the
element at the instant of solidification, x + x', y + y', z + z those
of any other point P in it ; let u, v, w be the velocities of G along
the three axes just before solidification, u', v', w' the velocities of P
relative to G] also let u, v, w be the velocities of G, u 1} v ly w^ the
relative velocities of P, and f, 77, the angular velocities just
after solidification. Since all the impulsive forces are internal,
we have
u u, v = v, w = w.
We have also by the Principle of Conservation of Angular
Momentum,
2m \y' (w l - w') - z' (v^ - v')} = 0, &c.
m denoting an element of the mass of the element considered.
But MJ = rjz' y', and u' is ultimately equal to
du , du t du ,
and similar expressions hold good for the other quantities. Sub-
stituting in the above equation, and observing that
r/ = 0, and 2m#' 2 = 2m?/' 2 = 2m/ 2 ,
fdw (
dz;
, <. T
we have c= *
* 2 \
We see then that an indefinitely small spherical element of
the fluid, if suddenly solidified and detached from the rest of the
fluid, will begin to move with a motion of translation alone, or
a motion of translation combined with one of rotation, according as
udx + vdy + wdz is, or is not, an exact differential, and in the latter
case the angular velocities will be determined by the equations
,. _ dw _ cfo _ du _ dw 9 .. _ dp _ du
dy dz' dz dx' dx dy'
B. H.
18 EQUATIONS OF MOTION OF A PERFECT FLUID.
On account of the physical meaning of the quantities f, 77, ,
they are called the components of molecular rotation, and motion
which is such that they do not vanish is called rotational or vortex
motion; when they vanish, the motion is called irrotational.
In the foregoing investigation, it has been assumed that the
pressure is a function of the density, and also that the fluid is
under the action of a conservative system of forces ; it therefore
follows that vortex motion cannot be produced and, if once set up,
cannot be destroyed by such a system of forces. It can however
be shown that the theorem is not true if the pressure is not a
function of the density. If therefore by reason of any chemical
action, the pressure should cease to be a function of the density
during any interval of time however short, vortex motion might
be produced, or if in existence might be destroyed.
23. The equations of motion can be integrated whenever
a force and a velocity potential exist; for putting
= _p:- F,
J P
and multiplying (22) by dx, dy, dz respectively and adding, we
obtain
,~ du , . dv , dw ,
Now in the present case
du du du dv dw
^i = -n+u-j- + v- r +w- r -
ot at dx ax dx
where is the resultant velocity. Integrating, we obtain
^ + }>q* = F(t) (28),
where F is an arbitrary function.
24. When the motion of a liquid is steady, du/dt, dv/dt and
dw/dt are each zero, and in this case the general equations of
motion can always be integrated. It will however be necessary
to distinguish between irrotational and rotational motion.
STEADY MOTION. 19
In the former case d/dt = and F (t) is constant ; whence a
first integral is
p/p+V+1>q*=C ..................... (29).
This result, from the name of its discoverer, is called Bernoulli's
Theorem.
When the motion is rotational, let
and the general equations of motion may be written
du _ dR}
dt das
dv
dw ,. dR
.(30).
(i) Let the motion be steady and in two dimensions; then
w = % = r) = 0, and none of the quantities are functions of z or t ;
whence substituting the values of u and v from (14) in terms of
i/r we obtain
^? = _ 2 ^ r ^? = _2^
da! dx ' dy dy '
which shows that 2=F' (-\/r), where jPis an arbitrary function; and
therefore
.................. (31).
P
From these results it follows that the sum of the first three
terms is constant along a stream line, but varies as we pass from
one stream line to another; also that the molecular rotation is
constant along a stream line.
(ii) In the same way it can be shown that when the motion
is symmetrical with respect to an axis,
.................. (32),
where CD is the molecular rotation, and i/r is Stokes' current function.
(iii) In the general case of the steady motion of a liquid,
multiply (30) by u, v, w and add and we obtain
dR dR dR _ /00 .
.................. (33) -
22
20 EQUATIONS OF MOTION OF A PERFECT FLUID.
Multiply by , rj, % and add, and we obtain
These equations show that R is constant along a stream line
and a vortex line ; whence it is possible to draw a family of
surfaces each of which is covered with a network of stream lines
and vortex lines. Let \ = const, be such a surface ; then since X
contains stream lines and vortex lines, it follows that
u\ x + v\ y + w\ z = 0,
where \ x = d\/dx, &c. These equations show that if we write
R= F(\) equations (33) and (34) will be satisfied; whence a
first integral of the equations of motion is
f
where \ is a surface which contains both stream lines and vortex
lines.
Impulsive Motion.
25. The equations which determine the change of motion
when a fluid is acted upon by impulsive forces, may be deduced in
manner similar to that employed in 16.
Let u, v, w and u', v', w' be the velocities of the fluid just
before and just after the impulse ; p the impulsive pressure.
Since impulsive forces are equal to the change of momentum
which they produce, it follows by considering the motion of a small
parallelepiped 8x8y8z, that
p (u' u) 8x8y8z = p8y8z ( p + --f- 8x \ 8y8z,
whence the equations of impulsive motion are
f>(u '- u} = -f x
P( ' ~dy
.(36).
FLOW AND CIRCULATION. 21
Multiplying by das, dy, dz and adding we obtain
- dp/p = (u r - u) dx + (v f - v) dy + (w 1 - w) dz (37).
In the case of a liquid p is constant, whence differentiating
with respect to x, y, z, adding and taking account of the equation
of continuity, we obtain
V 2 p = (38).
If the liquid were originally at rest, it is clear that the motion
produced by the impulse must be irrotational, whence if $ be its
velocity potential
P = -P* (39),
which is a very important result.
Flow and Circulation.
26. The line integral f(udx + vdy+wdz), taken along any
curve joining a fixed point A with a variable point P, is called the
flow from A to P.
If the points A and P coincide, so that the curve along which
the integration takes place is a closed curve, this line integral is
called the circulation round the closed curve.
If the motion of a liquid is irrotational, and (f> A , P A , and is independent of the path from A
to P; also the circulation round any closed curve is zero, provided
(f) be a single-valued function. Cases however occur in which > is
a many-valued function ; and when this is the case, the value of
the circulation will depend upon the position of the closed curve
round which the integration is taken, being zero for some curves,
whilst for others it has a finite value.
For example, when the motion is in two dimensions, < satisfies
the equation
ttljS-o
da?^ dy*
and it can be verified by trial, that a particular solution of this
equation is
= m tan" 1 y/as.
This value of $ therefore gives a possible kind of irrotational
motion. Let 6 be the least value of the angle tan" 1 y/x; then since
22 EQUATIONS OF MOTION OF A PERFECT FLUID.
the equation 6 = tan" 1 y\x is satisfied by 9 4- 2w7r, where n is any
positive or negative integer, it follows that the most general value
of > is
at any other point P is a function of
the distance alone of P from the origin ; and Laplace's equation
becomes
^ + ?^ =0 .
dr 2 r dr
Therefore m
and j =-5'
dr r 2
The origin is therefore a singular point, from or to which the
stream lines diverge or converge, according as m is positive or
negative. In the former case the singular point is called a source,
in the latter case a sink.
The flux across any closed surface surrounding the origin is,
where dfl is the solid angle subtended by dS at the origin, and e
is the angle which the direction of motion makes with the normal
to 8 drawn outwards.
The constant m is called the strength of the source.
24 EQUATIONS OF MOTION OF A PERFECT FLUID.
29. A doublet is formed by the coalescence of an equal source
and sink. To find its velocity potential, let there
P be a source and sink at S and H respectively, and
let be the middle point of SH, then
H S
m
m
+ HP
_mSHcosSOP
OP 2
Now let SH diminish and m increase inde-
finitely, but so that the product m . SH remains
finite and equal to /*, then
/* cos SOP
flZ
if the axis of z coincides with OS.
Hence the velocity potential due to a doublet, is equal to the
magnetic potential of a small magnet whose axis coincides with
the axis of the doublet, and whose negative pole corresponds to
the source end of the doublet.
30. When the motion is in two dimensions, and is sym-
metrical with respect to the axis of z, Laplace's equation becomes
Therefore
^ T* I _ ""r r\
dr* r dr
= m log r,
d __ m
dr~r'
where r is the distance of any point from the axis. This value of
= m log SP - m log HP
SH u, cos SOP
= m 75 cos SOP = -
OP r
Theory of Images.
32. Let H lt H 2 be any two hydrodynamical systems situated
in an infinite liquid. Since the lines of flow either form closed
curves or have their extremities in the singular points or bound-
aries of the liquid, it will be possible to draw a surface S, which
is not cut by any of the lines of flow, and over which there is
therefore no flux, such that the two systems H l} H 2 are completely
shut off from one another.
The surface S may be either a closed surface such as an
ellipsoid, or an infinite surface such as a paraboloid.
If therefore we remove one of the systems (say H 2 ) and
substitute for it such a surface as S, everything will remain
unaltered on the side of S on which H 1 is situated ; hence the
velocity of the liquid due to the combined effect of HI and H 2 will
be the same as the velocity due to the system H l in a liquid
which is bounded by the surface S.
The system H 2 is called the image of H^ with respect to the
surface S, and is such that if _fiT 2 were introduced and S removed,
there would be no flux across S.
The method of images was invented by Lord Kelvin, and has
been developed by Helmholtz, Maxwell and other writers; it affords
a powerful method of solving many important physical problems.
33. We shall now give some examples.
Let S, S' be two sources whose strengths
are m. Through A the middle point of SS'
draw a plane at right angles to SS'. The
normal component of the velocity of the liquid
at any point P on this plane is
m
SI*
m
cos PSA + cos P8'A = 0.
o r*
26 EQUATIONS OF MOTION OF A PERFECT FLUID.
Hence there is no flux across AP. If therefore Q be any
point on the right-hand side of AP, the velocity potential due
to a source at S, in a liquid which is bounded by the fixed plane
AP, is
mm
(j)= ~sQ"srQ'
Hence the image of a source 8 with respect to a plane is an
equal source, situated at a point S' on the other side of the plane,
whose distance from it is equal to that of S.
34. The image in a sphere, of a doublet whose axis passes
through the centre of the sphere, can also be found by elementary
methods.
Let 8 be the doublet, the centre of the sphere, a its radius,
and let OS =/.
The velocity potential of a doublet situated at the origin and
whose axis coincides with OS, has already been shown to be
racos#
9= ^ 5
whence if R, be the radial and transversal velocities
-n _d _2mcos#
"> == ~T n >
dr r 3
_ 1 d(f> _ m sin 6
r dd r 3
Hence if we have a doublet at S, the component velocity
along OP is
cos OSP cos OPS - ^ sin OSP sin OPS
cos OSP cos OPS + cos (OPS - OSP)} (41).
Let us take a point H inside the sphere such that OH = a 2 //;
IMAGE OF A DOUBLET IN A SPHERE. 27
then it is known from geometry that the triangles OPH and OSP
are similar, and therefore the preceding expression may be written
- - (cos OPH cos OHP + cos SPH}.
But the normal velocity due to a doublet of strength m' placed
at H is by (41)
{cos OPH cos OHP + cos SPH},
and therefore the normal velocity will be zero if
m/
for all positions of P. But by a well-known theorem,
_
SP~ HP'
and therefore the condition that the normal velocity should vanish,
is that
m' = ma 3 // 3 .
Whence the image of a doublet of strength m in a liquid
bounded by a sphere is another doublet placed at the inverse
point H, whose strength is ma 3 // 3 .
The theory of sources, sinks and doublets furnishes a powerful
method of solving certain problems relating to the motion of
solid bodies in a liquid 1 .
We shall conclude this chapter by working out some examples.
35. A mass of liquid whose external surface is a sphere of
radius a, and which is subject to a constant pressure II, surrounds
a solid sphere of radius b. The solid sphere is annihilated, it is
required to determine the motion of the liquid.
It is evident that the only possible motion which can take
1 If a magnetic system be suddenly introduced into the neighbourhood of a
conducting spherical shell, it can be shown that the effect of the induced currents
at points outside the shell, is initially equivalent to a magnetic system inside the
shell, which is the hydrodynamical image of the external system ; and that the law
of decay of the currents, is obtained by supposing the radius of the shell to diminish
according to the law ae~ a , where a- is the specific resistance of the shell.
Analogous results hold good in the case of a plane current sheet ; hence all results
concerning hydrodynamical images in spheres and planes are capable of an electro-
magnetic interpretation. See C. Niven, Phil. Trans. 1881.
28 EQUATIONS OF MOTION OF A PERFECT FLUID.
place, is one in which each element of liquid moves towards the
centre, whence the free surfaces will remain spherical. Let R', R
be their external and internal radii at any subsequent time, r
the distance of any point of the liquid from the centre. The
equation of continuity is
whence r 2 v = F(t).
The equation for the pressure is
1 dp _ dv dv
p dr dt dr
F'(t) dv*
r 2 2 dr
. T7II /\
whence = A + - } - Aw 2 .
p r
When r = R', p = II ; and when r = R, p = ; whence if V, V be
the velocities of the internal and external surfaces
Since the volume of the liquid is constant,
also
whence
(R 3 + c 3 )
Putting z = R*V 2 , multiplying by 2R 2 and integrating, we
obtain
)* ^1 '
which determines the velocity of the inner surface.
If the liquid had extended to infinity, we must put c = oo , and
we obtain
TORRICELLl's THEOREM. 29
whence if t be the time of filling up the cavity
= V2nJo VF^Tl 3 '
Putting b 3 x = R 3 , this becomes
The preceding example may be solved at once by the Principle
of Energy.
The kinetic energy of the liquid is
S
'
' Ur -- - il
( R (R 3 + c 3 )^
The work done by the external pressure is
T
whence |H (b* - R 3 ) = V*R*p
4-n-n r*dr = fH-Tr (a 3 -
(R 3
36. The determination of the motion of a liquid in a vessel of
any given shape is one of great difficulty, and the solution has
been effected in only a comparatively few number of cases. If,
however, liquid is allowed to flow out of a vessel, the inclinations
of whose sides to the vertical are small, an approximate solution
may be obtained by neglecting the horizontal velocity of the
liquid. This method of dealing with the problem is called the
hypothesis of parallel sections.
Let us suppose that the vessel is kept full, and the liquid is
allowed to escape by a small orifice at P.
Let h be the distance of P below the
free surface, and z that of any element
of liquid. Since the motion is steady,
the equation for the pressure will be
30 EQUATIONS OF MOTION OF A PERFECT FLUID.
Now if the orifice be small in comparison with the area of the
top of the vessel, the velocity at the free surface will be so small
that it may be neglected ; hence if II be the atmospheric pressure,
when z = 0, p = Tl,v = and therefore C = II/p. At the orifice
p = II, z = h, whence the velocity of efflux is
and is therefore the same as that acquired by a body falling from
rest, through a height equal to the depth of the orifice below the
upper surface of the liquid. This result is called Torricelli's
Theorem.
The Vena Contracta.
37. When a jet of liquid escapes from a small hole in the
bottom of a cistern, it is found that the area of the jet is less than
the area of the hole ; so that if a- be the area of the hole and a' that
of the jet, the ratio }.
If A be the sectional area of the jet at the vena contracta, the
quantity of fluid which escapes per unit of time is
The momentum per unit of time is
Apq* = 2A(p- H).
The energy per unit of time is
In Giffard's Injector, a jet of steam issuing by a pipe from the
upper part of the boiler, is directed at an equal pipe leading back
into the lower part of the boiler, the jet being kept constantly just
surrounded with water. Now if we assume that the velocity of the
steam jet is equal to the velocity at which the water flows into
the pipe leading to the lower part of the boiler, which must be
very nearly true ; it follows from the preceding equations that
velocity of steam jet _ /p
velocity of water jet V " ' .
quantity of steam jet _ /, w! the angular velocities of the element in latitude
and longitude respectively.
4. Fluid is moving in a fine tube of variable section K, prove
that the equation of continuity is
where v is the velocity at the point s.
5. If F (x, y, z, f) is the equation of a moving surface, the
velocity of the surface normal to itself is
- ~ d f where R 2 = (dFfdx) 2 + (dF/dy) 2 + (dF/dz)*.
-ft Ctt
6. If x, y and z are given functions of a, b, c and t, where a,
b and c are constants for any particular element of fluid, and if
u, v and w are the values of x, y, z when a, 6, c are eliminated,
prove analytically that
d 2 x _ du du du du
dt 2 dt dx dy dz'
7. If the lines of flow of a fluid lie on the surfaces of coaxial
cones having the same vertex, prove that the equation of con-
tinuity is
r dt + T dr ( Up ) + 2pU + COS6C e dd> ^ = '
8. Show that
x 2 /(akt 2 ) 2 + Jet 2 {(y/b) 2 + 0/c) 2 } = 1
is a possible form of the bounding surface at time t of a liquid.
9. A fine tube whose section k is a function of its length s, in
the form of a closed plane curve of area A filled with ice, is moved
in any manner. When the component angular velocity of the
tube about a normal to its plane is O, the ice melts without
change of volume. Prove that the velocity of the liquid relatively
B. H. 3
34 EQUATIONS OF MOTION OF A PERFECT FLUID.
to the tube at a point where the section is K, at any subsequent
time when co is the angular velocity is
where 1/c = jk~^ds t the integral being taken once round the tube.
10. A centre of force attracting inversely as the square of the
distance, is at the centre of a spherical cavity within an infinite
mass of liquid, the pressure on which at an infinite distance is -BT,
and is such that the work done by this pressure on a unit of area
through a unit of length, is one half the work done by the attrac-
tive force on a unit of volume of the liquid from infinity to the
initial boundary of the cavity ; prove that the time of filling up
the cavity will be
'3\*1
a being the initial radius of the cavity, and p the density of the
liquid.
11. A solid sphere of radius a is surrounded by a mass of
liquid whose volume is 4?rc 3 /3, and its centre is a centre of attrac-
tive force varying directly as the square of the distance. If the
solid sphere be suddenly annihilated, show that the velocity of the
inner surface when its radius is x, is given by
&a? {(a* + c 3 )' - x\ = ( ~ + | ytic 3 ) (a 3 - a?) (c 3 + orf,
V op u /
where p is the density, II the external pressure and //, the absolute,
force.
12. Prove that if or be the impulsive pressure, , ' the
velocity potentials immediately before and after an impulse acts,
V the potential of the impulses,
w + pV+ p ($' ) = const.
13. If the motion of a homogeneous liquid be given by a
single valued velocity potential, prove that the angular momentum
of any spherical portion of the liquid about its centre is always
zero.
14. Homogeneous liquid is moving so that
, w= 0,
EXAMPLES. 35
and a long cylindrical portion whose section is small, and whose
axis is parallel to the axis of z, is solidified and the rest of the
liquid destroyed. Prove that the initial angular velocity of the
cylinder is
Bft-Aa-ZFy
A+B
where A, B, F are the moments and products of inertia of the
section of the cylinder about the axes.
15. Fluid is contained within a sphere of small radius ; prove
that the momentum of the mass in the direction of the axis of x is
greater than it would be if the whole were moving with the
velocity at the centre by
Ma?
-R-
Op
where p x = dp/da; &c.
16. The motion of a liquid is in two dimensions, and there is
a constant source at one point A in the liquid and an equal sink
at another point B ; find the form of the stream lines, and prove
that the velocity at a point P varies as (AP . -BP)" 1 , the plane of
the motion being unlimited.
If the liquid is bounded by the planes a; = 0, a; = a, y = 0, y = a,
and if the source is at the point (0, a) and the sink at (a, 0), find
an expression for the velocity potential.
17. The boundary of a liquid consists of an infinite plane
having a hemispherical boss, whose radius is a and centre 0. A
doublet of unit strength is situated at a point 8, whose axis
coincides with OS, where OS is perpendicular to the plane. P is
any point on the plane, OP = y, OS=f. Prove that the velocity
of the liquid at P is
18. Prove that
=f(t) {(r 2 + a 2 - 2as)- + (r 2 + a 2 + 2as)~* - r- 1 } + ^ (t)
is the velocity potential of a liquid, and interpret it. Find the
surfaces of equal pressure if gravity in the negative direction of
the axis of z be the only force acting.
^ 9
O -
36 EQUATIONS OF MOTION OF A PERFECT FLUID.
19. Liquid enters a right circular cylindrical vessel by a
supply pipe at the centre 0, and escapes by a pipe at a point A
in the circumference ; show that the velocity at any point P is
proportional to PBfPA . PO, where B is the other end of the
diameter AO. The vessel is supposed so shallow that the motion
is in two dimensions.
CHAPTER II.
MOTION OF CYLINDERS AND SPHERES IN AN
INFINITE LIQUID.
39. THE present chapter will be devoted to the consideration
of certain problems of two-dimensional motion, and we shall also
discuss the motion of a sphere in an infinite liquid.
If a right circular cylinder is moving in a liquid, the pressure
of the liquid at any point of the cylinder passes through its axis,
and therefore the resultant pressure of the liquid on the cylinder
reduces to a single force, which can be calculated as soon as the
pressure has been determined. Now the pressure at any point of
the liquid is found by means of the equation
and therefore p can be determined as soon as the velocity potential
is known. Hence the first step towards the solution of problems
of this character is to find the velocity potential.
If the cylinder is not circular, the resultant pressure of the
liquid upon its surface will usually be reducible to a single force
and a couple, and the problem becomes more complicated. The
motion of cylinders, which are not circular, can be most con-
veniently treated by means of the dynamical methods explained
in the next chapter. In the present chapter we shall show how
to find the motion of an infinite liquid, in which cylinders of
certain given forms are moving, and we shall also work out the
solution of certain special problems relating to the motion of
circular cylinders and spheres.
38 MOTION OF CYLINDERS AND SPHERES.
40. If the liquid be at rest, and a cylinder of any given form
be set in motion in any manner, the subsequent motion of the
liquid will be irrotational and acyclic, and is therefore completely
determined by means of a velocity potential. It is however more
convenient to employ Earnshaw's current function i/r. This
function, when the motion is irrotational, satisfies the equation
_
da? df
at all points of the liquid.
The integral 1 of this equation is
TJr = f(a>+iy) + F(a:-iy) .................. (2),
also u = ~, v = -- ....................... (3).
dy dx
We must now consider the boundary conditions to be satisfied
by ty.
If the liquid is at rest at infinity (which will usually be the
case), d-^rjdx and d-fr/dy must vanish at infinity. If any portions
of the boundary consist of fixed surfaces, the normal component
of the velocity must vanish at such fixed boundaries, and there-
fore the fixed boundaries must coincide with a stream line. This
requires that ty = const, at all points of fixed boundaries.
When the cylindrical boundary is in motion, the component
velocity of the liquid along the normal, must be equal to the
component velocity of the cylinder in the same direction.
(i) Let the cylinder be moving with velocity U parallel to the
axis of a, and let 6 be the angle which the normal to the cylinder
makes with this axis ; then at the surface
u cos + v sin 6 = Ucos 9.
Now cos 6 = dyjds ; sin 6 = dxjds ; therefore by (3)
d =v -dy
ds ds'
Integrating along the boundary, we obtain
where A is a constant.
1 The easiest way of showing that (2) is a solution of (1), is to differentiate the
right-hand side of (2) twice with respect to x, and twice with respect to y and add.
Since the result is zero, this shows that (2) satisfies (1) ; also since (2) contains two
arbitrary functions, it is the most general solution that can be obtained.
BOUNDARY CONDITIONS. 39
(ii) If the cylinder be moving with velocity V parallel to the
axis of y, it can be shown in the same manner that the surface
condition is
^ = -Vx + B ......................... ,.(5).
(iii) Let the cylinder be rotating with angular velocity w ;
then at the surface
u cos + v sin 9 = wy cos 6 + wx sin 6,
d-dr dr
or -~ = u>r -7- ,
as as
Therefore ^r = -^wr z + C ........................ (6),
where r = (a? + y 2 )*.
When there are any number of moving cylinders in the liquid,
conditions (4), (5) and (6) must be satisfied at the surfaces of each
of the moving cylinders.
In addition to the surface conditions, -ty* must satisfy the
following conditions at every point of space occupied by the
liquid ; viz. ty must be a function which is a solution of Laplace's
equation (1), and which together with its first derivatives must be
finite and continuous at every point of the liquid.
If we take any solution of (1), and substitute its value in (4),
(5) or (6), we shall in many cases be able to determine the current
function due to the motion of a cylinder, whose cross section is
some curve, in one of the three prescribed manners.
In most of the applications which follow -\Jr will be of the
form
- t y) .................. (7),
From these equations we see that
0+^=2t/(a; + y) ..................... (9),
and therefore when ty is known, can be found by equating the
real and imaginary parts of (9).
40 MOTION OF CYLINDERS AND SPHERES.
Motion of a Circular Cylinder.
41. Let
Transforming to polar coordinates, and using De Moivre's
theorem, we obtain
Tjr=- Fa 2 #/r 2 ........................ (10).
When r = a, ty = Vx ; equation (10) consequently deter-
mines the current function, when a circular cylinder of radius a
is moving parallel to the axis of y, in an infinite liquid with
velocity V.
By (9) the velocity potential is
< = - Vtfyfr* ........................ (11).
42. Let us now suppose that the cylinder is of finite length
unity, and that the liquid is bounded by two vertical parallel
planes, which are perpendicular to the axis of the cylinder.
In order to find the motion when the cylinder is descending
vertically under the action of gravity, let /3 be the distance of the
axis of the cylinder at time t from some fixed point in its line of
motion which we shall choose as the origin, and let (x, y) be the
coordinates of any point of the liquid referred to the fixed origin,
the axis of y being measured vertically downwards; also let (r, 6)
be polar coordinates of the same point referred to the axis of the
cylinder as origin. By (11)
Fa 2 . Fa 2 -3
and therefore since dftjdt = V,
tfV . , a 2 F 2 2a 2 F 2 .
d> = --- sin0+ ---- s
n* ff* rv&
and therefore at the surface, where r = a,
Also fl*"" I j ) + I - ;> J
\drj \r dvj
therefore when r = a,
q* = F 2 .
CIRCULAR CYLINDER. 41
Whence
pjp = a Fsin - F 2 cos 20 - F 2 + g (/3 + a sin 0) + C. . .(12).
The horizontal component of the pressure is evidently zero ;
the vertical component is
Y=-a ( painOdG.
J o
Substituting the value of p from (12) and integrating, we
obtain
F = - Trpa?(V+g).
Hence if cr be the density of the cylinder, the equation of
motion is
Tro-a 2 V = F + no-go?,
or (*r+p)F = (cr-p)flr (13).
Integrating this equation, we obtain
Tr (o- - p) at
F=t> + v /'* ,
( + />)
where u is the initial velocity measured vertically downwards.
We therefore see that the cylinder will move in a vertical
straight line, with a constant acceleration which is equal to
g(
Assuming F=r m , the equation reduces to ra 2 - w 2 = 0, whence
m = n, and therefore the required solution is
= (Ar n + Br~ n ) e in ..................... (15).
In this solution n may have any value whatever, and the real
and imaginary parts of the above expression will be independent
solutions of (14).
Let us now suppose that the radius of the outer cylinder, which
is supposed to be fixed, is c; and let the inner one be started
with velocity U.
Since the velocity of the liquid at the surface of the outer
cylinder must be wholly tangential, the boundary condition is
i=0, when r = c ., ................. (16).
dr
At the surface of the inner cylinder, which is moving with
velocity U, the component velocities of the cylinder and liquid
along the radius must be equal ; whence the boundary condition
at the inner cylinder is
-j?=Ucos&, when r = a ............... (17),
dr
being measured from the direction of U.
If in (15) we put n = 1, the function
= (Ar + Blr)cos0 ..................... (18)
1 This transformation can be most easily effected, by forming the equation
of continuity in polar coordinates.
LEMNISCATE OF BERNOULLI. 43
is a solution of Laplace's equation ; if therefore we can determine
A and B so as to satisfy (16) and (17), the problem will be
solved.
Substituting from (18) in (16) we obtain
Ac*-B = 0.
Substituting in (17) we obtain
Aa 2 -B= Ua\
Solving and substituting in (18) we obtain
Ua* ( , c 2 \
9 = - Y , r + - cos 0.
c 2 a 2 V r)
If we put c= oo , we fall back on our previous result of a
cylinder moving in an infinite liquid.
We can now determine the impulsive force which must be
applied to the inner cylinder, in order to start it with velocity U.
By 25, equation (39), it follows that if a liquid which is
at rest be set in motion by means of an impulse, and be the
velocity potential of the initial motion, the impulsive pressure
at any point of the liquid is equal to p.
Hence if M be the mass of the cylinder, F the impulse, the
equation of motion is
= F+pal (bcosddO
Jo '
UTrpa 2 (c 2 + a 2 )
= * J b ,
r 2 -
whence since M =
The Lemniscate of Bernoulli.
44. The lemniscate of Bernoulli is a bicircular quartic curve
whose equation in Cartesian coordinates is (# 2 + y 2 ) 2 = 2c 2 (a? y z ),
or in polar coordinates r 2 = 2c 2 cos 20. In order to find the current
function when a cylinder, whose cross section is this curve, is
moving parallel to x in an infinite liquid, let us put u = x + ty,
v = x ly, and assume
44 MOTION OF CYLINDERS AND SPHERES.
Now w 2 -c 2 = r 2 (cos20 + isin20)-c 2 ;
whence at the surface where r 2 = 2c 2 cos 20, the right-hand side
becomes
2C 2 cos 2 26 + ic* sin 40 - c 2 = c 2 (cos 20 + t sin 20) 2 ;
u r (cos 4- fc sin 0)
whence - - -r = -7^ ^- -- . ' = r (cos - t sm 0)/c.
(w 2 c 2 )* c (cos 20 + i sm 20)
Therefore at the surface
tyx = Ur sin = E/y.
The value of -fy x given by (19), is therefore the current function
due to the motion parallel to x with velocity U, of a cylinder
whose cross section is a lemniscate of Bernoulli.
If we put ^-i
it can be shown in a similar manner, that ty y is the current
function, when a cylinder of this form is moving parallel to y with
velocity V ; and that i/r 3 is the current function, when the cylinder
is rotating with angular velocity a> about its axis.
If the cross section be the cardioid r = 2c (1 + cos 0), the vajues
of ty x and 1/r,, can be obtained by writing (w* c*) 2 , (v* c*) 2 for
(u 2 c 2 )*, (y 2 c 2 )* in the preceding formulae ; but the value of ^ 3
can be so simply obtained. See Quart. Jour. vol. xx. p. 246.
An Equilateral Triangle.
45. The preceding methods may also be employed, to find the
motion of a liquid contained within certain cylindrical cavities,
which are rotating about an axis.
Let ty = %A {(x + iy) 3 + (x- iy} s ]
= A(a?- 3#?/ 2 ) = Ar* cos 30.
Substituting in (6), the boundary condition becomes
A(a?-3xy 2 ) + $a)(a? + f) = C .............. (20).
If we choose the constants so that the straight line x=a, may
form part of the boundary, we find
. G) 2&>a 2
ELLIPTIC CYLINDER. 45
Hence (20) splits up into the factors
(x a) ; x 4- y V3 + 2a ; x y \/3 + 2a.
The boundary therefore consists of three straight lines forming
an equilateral triangle, whose centre of inertia is the origin.
Hence -fy is the current function due to liquid contained in an
equilateral prism, which is rotating with angular velocity &> about
an axis through the centre of inertia of its cross section. The
values of i/r and <, when cleared of imaginaries, are
^ = ^-r3 cos 30 rf> = -^- r 3 sin 30.
6a 6a
An Elliptic Cylindrical Cavity.
46. Let i/r = \A {(x + lyf + (x - iy) 2 }
= A(x*-f).
Substituting in (6) we find
ca
2C
the equation of the boundary becomes
*..
a* + b*~ '
and a>(^-60
2 (a 2 + & 2 ) v
i|r is therefore the current function due to the motion of liquid
contained in an elliptic cylinder, which is rotating about its axis.
Elliptic Cylinder.
47. The problem of finding the motion of an elliptic cylinder
in an infinite liquid cannot be solved by such simple methods as
the foregoing ; in order to effect the solution we require to employ
the method of Conjugate Functions.
Def. If % and 77 are functions of x and y, such that
+ 7=/(* + y) ................... (21),
then % and ij are called conjugate functions of x and y.
If we differentiate (21) first with respect to x, and afterwards
46 MOTION OF CYLINDERS AND SPHERES.
with respect to y, eliminate the arbitrary function, and then
equate the real and imaginary parts, we shall obtain the equations
dx dy' dy dx"
Comparing these equations with (8), we see that < and ty are
conjugate functions of # and y.
From equations (22) we also see that
dx dx dy dy
V 2 |=0, V 2 77 = ..................... (24).
Equation (23) shows that the curves = const., 77 = const., form
an orthogonal system ; and equations (24) show that and 77 each
satisfy Laplace's equation.
If and ^r are conjugate functions of x and y, and and
i) are also conjugate functions of x and y, then and ty are
conjugate functions of and 77.
For + n|r = F (x + ty),
and f + irj =f(x + iy),
whence eliminating x + iy, we have
From this proposition combined with (24), it follows that if
the equation V 2 -\Jr = be transformed by taking and 77 as
independent variables
(25)
48. We can now find the current function due to the motion
of an elliptic cylinder.
Let x + iy = c cos ( iij)
= c cos cosh 77 + ^0 sin sinh 77,
then x = c cos cosh 77}
fc . r (26),
y = c sin if sinn 77]
whence the curves 77 = const., f = const., represent a family of
confocal ellipses and hyperbolas, the distance between the foci
being 2c.
If a and 6 be the semi-axes of the cross section of the elliptic
cylinder 77 = /3, then,
a = c cosh /8, b = c sinh /3.
ELLIPTIC CYLINDER. 47
If /3 is exceedingly large, sinh /3 and cosh /3 both approximate
to the value ^ce^ ; and therefore as the ellipse increases in size, it
approximates to a circle whose radius is |ce^.
It can be verified by trial, that (25) can be satisfied by a series
of terms of the form e~ nr > (A n cos n% + B n sin nf) ; and if n be a
positive quantity not less than unity, this is the proper form of A|T
outside an elliptic cylinder, since it continually diminishes as 77
increases.
When the cylinder is moving parallel to its major axis with
velocity U, let us assume
\Jr a . = J.e~ 1) sin .
Substituting in (4) we obtain
A~P sin f = Uc sinh /3 sin + G,
where 77 = /3 is the equation of the cross section of the cylinder.
Since this equation is to be satisfied at every point of the
boundary, we must have (7 = 0, A = Uce 1 * sinh /S; whence
fa= Uce-v+P sinh ft sin f (27).
When the cylinder is moving parallel to its minor axis with
velocity V, it may be shown in the same manner that
-fy y =- Fce-^ cosh cos (28).
Lastly let us suppose that the cylinder is rotating with angular
velocity to about its axis. Then
a? + y 2 = c* (cos 2 f cosh 2 77 + sin 2 sinh 2 77)
= c 2 (cosh 277 + cos 2).
Let us therefore assume
^ 3 = Be~*> cos 2f
Substituting in (6) we obtain
Be-* cos 2 + &>c 2 (cosh 2/3 + cos 2f) = G,
whence B = - wtfe, G = J we 2 cosh 2/3,
and therefore ^s = - i *>c 2 e- 2 "-*> cos 2f (29).
49. If we suppose that /3 = 0, the ellipse degenerates into a
straight line joining the foci, and (28) becomes
ty y = _ Fee-" cos (30).
It might therefore be supposed that (30) gives the value of the
current function, due to a lamina of breadth 2c, which moves with
48 MOTION OF CYLINDERS AND SPHERES.
velocity V, perpendicularly to itself. This however is not the case,
inasmuch as the velocity at the edges of the lamina becomes
infinite, and therefore the solution fails. To prove this, we have
dty _ city dx dty dy
dtj dx dij dy dij
. , ..dty . dty
= c smh i} cos -jr- + c cosh 77 sin g -~ ,
cLco ^y
and -=- = c cosh rt sin -~ + c sinh 77 cos -~ .
dg dx dy
whence squaring and adding, we obtain
c 2 (sinh 2 T? cos 2 + cosh 2 77 sin 2 f ) q* = ( ^Y + (^ Y= F 2 c 2 e-^ . . . (31).
The coordinates of an edge are x = c, y = ; and therefore in
the neighbourhood of an edge 77 and f are very small quantities ;
and therefore by (31) the velocity in the neighbourhood of an
edge is
which becomes infinite at the edge itself, where t] and ff are zero.
It therefore follows that the pressure in the neighbourhood of an
edge is negative, which is physically impossible.
Since the pressure is positive at a sufficient distance from the
edge, there will be a surface of zero pressure dividing the regions
of positive and negative pressures ; and it might be thought that
the interpretation of the formulae would be, that a hollow space
exists in the liquid surrounding the edges, which is bounded by a
surface of zero pressure. But the condition that a free surface
should be a surface of zero (or constant) pressure, although a
necessary one, is not sufficient ; it is further necessary, that such
a surface should be a surface of no flux, which satisfies the kine-
matical condition of a bounding surface 12, equation (17); and it
will be found on investigating the question, that no surface exists,
which is a surface of zero (or constant) pressure, and at the same
time satisfies the conditions of a bounding surface. The solution
altogether fails in the case of a lamina.
When the velocity of the solid is constant and equal to V,
the easiest way of dealing with a problem of this character is to
reverse the motion by supposing the solid to be at rest, and that
the liquid flows past it, the velocity at infinity being equal to - V.
MOTION OF A SPHERE. 49
The correct solution in the case of a lamina has been given by
Kirchhoff 1 , and he has shown that behind the lamina there is a
region of dead water, i.e. water at rest, which is separated from
the remainder of the liquid by two surfaces of discontinuity, which
commence at the two edges of the lamina, and proceed to infinity
in the direction in which the stream is flowing. Since the liquid
on one side of this surface of discontinuity is at rest, its pressure is
constant ; and therefore since the motion is steady, the pressure,
and therefore the velocity of the moving liquid, must be constant
at every point of the surface of discontinuity. It may be added
that a surface of discontinuity is an imaginary surface described
in the liquid, such that the tangential component of the velocity
suddenly changes as we pass from one side of the surface to the
other.
Motion of a Sphere.
50. The determination of the velocity potential, when a solid
body of any given shape is moving in an infinite liquid, is one of
great difficulty, and the only problem of the kind which has been
completely worked out is that of an ellipsoid, which of course
includes a sphere as a particular case.
We shall however find it simpler in the case of a sphere to
solve the problem directly, which we shall proceed to do.
Let the sphere be moving along a straight line with velocity
V, and let (r, 0, , hence by 7, equation (12), the equation
of continuity is
_0 ......... (32).
dr 2 r dr r 2 dv 2 r 2 d6
The boundary condition, which expresses the fact that the
normal component of the velocity of the liquid at the surface of
the sphere is equal to the normal component of the velocity of
the sphere itself, is
(33).
Equation (33) suggests that must be of the form F(r) cos 9 ;
we shall therefore try whether we can determine F so as to satisfy
1 See also, Michell, Phil. Trans. 1890, p. 389 ; Love, Proc. Gamb. Phil. Soc.
vol. vn. p. 175.
B. H. 4
50 MOTION OF CYLINDERS AND SPHERES.
(32). Substituting this value of , we find that (82) will be
satisfied, provided
*rw_sr_ ........... (34)
dr 2 r dr r*
To solve (34), assume F = r m ; whence on substitution we
obtain
(m-l)(w + 2)=0;
which requires that m = 1 or 2.
A particular solution of (32) is therefore
Since the liquid is supposed to be at rest at infinity, d(f)/dr =
when r = oo , and therefore .4 = 0. To find B, substitute in (33)
and put r = a, and we find
A = -%Va?,
Va 3 cos
whence > = ----- ^ ..................... (35).
This is the expression for the velocity potential due to the
motion of a sphere in an infinite liquid.
In order to determine the motion, when a sphere is descending
vertically under the action of gravity, let 7 be the distance of its
centre at time t from some fixed point in its line of motion, which
we shall choose as the origin ; let the axis of z be measured
vertically downwards and let x, y, z be the coordinates of any
point of the liquid referred to the fixed origin.
-D /OK\ JL Va?(z-^)
*= - -.l'
and therefore since 7= V,
d_ _ Fa 3 cos0 F 2 a 3 3F 2 a 3 cos 2
~di~ ^Zr* '~2r*~ 2r 3
and therefore at the surface where r = a,
also 2 =
and therefore
p/p = G + g ( 7 + a cos 0) + % Va cos + $ F 2 (9 cos 2 - 5) . . . (36).
MOTION OF A SPHERE. 51
If Z be the force due to the pressure of the liquid, which
opposes the motion,
f*
= 2-Tra 2 p cos 6 sin Odd
Jo
= |7rpa 3 (^V+g) (37)
by (36). If therefore a be the density of the sphere, the equation
of motion is
f Tro-a 3 F = - f Trpa* (%V + g) + frato-g
or (cr + ^-p) F=(o- p) g (38).
Hence the sphere descends with vertical acceleration
In order to pass to the case in which the sphere is projected
with a given velocity and no forces are in action, we must put
g = 0, and we see that V= const. = its initial value ; hence the
sphere continues to move with its velocity of projection, and
the effect of the liquid is to produce an apparent increase in the
inertia of the sphere, which is equal to half the mass of the liquid
displaced.
It also follows that if the sphere be projected in any manner
under the action of gravity, it will describe a parabola with vertical
acceleration g (a- p)/(a- + ^p).
51. Let us now suppose that the sphere is moving with
constant velocity V under the action of no forces. The equation
determining the pressure is
p z
Since d/dz and q vanish at infinity, it follows that C = TI/p,
where II is the pressure at infinity, whence
p U d$_
p /> dz * q '
and therefore at the surface,
P P
The right-hand side of this equation will be a minimum when
6 = |TT, in which case it becomes II /p f F 2 . Hence if
42
52 MOTION OF CYLINDERS AND SPHERES.
the pressure will become negative within a certain region in the
neighbourhood of the equator, and the solution fails. When V
exceeds the critical value (8II/op)* it is probable that a region of
dead water exists behind the sphere, which is separated from the
rest of the liquid by a vortex sheet.
52. In discussing the motion of a cylinder, we found that
if the solid were projected in a liquid and no forces were in action,
the solid would continue to move in a straight line with its original
velocity of projection; and we called attention to the fact that
this result was contrary to experience ; and that one reason of
this discrepancy between theory and observation arose from the
fact that all liquids are more or less viscous, the result of which
is that kinetic energy is gradually converted into heat. The
motion of viscous fluids is beyond the scope of an elementary
work such as the present, but a few remarks on the subject will
not be out of place.
Let us suppose that fluid is moving in strata parallel to the
plane xy, with a variable velocity U, which is parallel to the
axis of x. Let U be the velocity of the stratum AB, U + 8U
of the stratum CD, and let Bz be the distance between AB
and CD.
If the fluid were frictionless, the action between the fluid
on either side of the plane AB would be a hydrostatic pressure p,
whose direction is perpendicular to this plane, and consequently
no tangential action or shearing stress could exist ; if however
the fluid is viscous, the action between the fluid on either side of
the plane AB, usually consists of an oblique pressure (or tension),
and may therefore be resolved into a normal component perpen-
dicular to the plane, and a tangential component in the plane.
The usual theory of viscosity supposes, that if F be the tan-
gential stress on AB per unit of area, F+8F the corresponding
stress on CD, then the latter stress is proportional to the relative
velocity of the two strata divided by the distance between the
strata, so that
u^su^-u
if + d* x - ^ ,
oz
whence proceeding to the limit
dU
Fee-;-.
dz
VISCOSITY. 53
We may therefore put F '= /A-T (39),
CL%
where /i is a constant. The constant /i is called the viscosity ; it
is a numerical quantity whose value is different for different fluids,
and also depends upon the temperature.
The viscosity is a quantity which corresponds to the rigidity
in the Theory of Elasticity. If a shearing stress F be applied
parallel to the axis of x, and in a plane parallel to the plane xy, to
an elastic solid, it is known from the Theory of Elasticity, that
F = ndafdz,
where a is the displacement parallel to x. Whence the ratio of
the shearing stress F, to the shearing strain da/dz produced by it,
is equal to a constant n, which is called the rigidity. Now in the
hydrodynamical theory of viscous fluids, dll/dz is equal to the
rate at which shearing strain is produced by the shearing stress F;
hence (39) asserts that the ratio of the shearing stress to the rate
at which shearing strain is produced, is equal to a constant /*,
which is called the viscosity.
If the shearing stress F is applied in the plane z = c, and if
U = uz/c, (39) becomes
F = fjiu/c (40),
where u is the velocity of the fluid in the plane z = c. Hence
if u="L, and c = 1, then F=/JL. We may therefore define the
viscosity as follows 1 .
The viscosity is equal to the tangential force per unit of area, on
either of two parallel planes at the unit of distance apart, one of
which is fixed, whilst the other moves with the unit of velocity, the
space between being filled with the viscous fluid.
Equation (40) shows that the dimensions of p are [ ML~ l T~ *].
If we put v = fju/p, where p is the density, the quantity v is
called the kinematic coefficient of viscosity. The dimensions of v
are [L*T~ 1 ].
The equations of motion of a viscous fluid are known, and the
motion of a sphere which is descending under the action of gravity
in a slightly viscous liquid, such as water, has been worked out by
myself; and I have shown that if the sphere be initially projected
downwards with velocity V, its velocity at any subsequent time will
1 Maxwell's Heat, p. 298, fourth edition.
54 MOTION OF CYLINDERS AND SPHERES,
be approximately given by the equation
where /- about a fixed axis. Prove that the
velocity potential with reference to the principal axes of the
cylinder is wxy (a? 6 2 )/(a 2 + 6 2 ), and that the surfaces of equal
pressure when the angular velocity is constant, are the hyperbolic
cylinders
3a 2 + 6 2 36 2 + a 3
5. If =f(x, y), '\lr = F (x, y) are the velocity potential and
current function of a liquid, and if we write
and from these expressions find and ->/r; prove that the new
values of and ty will be the velocity potential and current
function of some other motion of a liquid.
Hence prove that if = cc* y 2 , -^ = 2xy, the transformation
gives the motion of a liquid in the space bounded by two confocal
and coaxial parabolic cylinders.
58 MOTION OF CYLINDERS AND SPHERES.
6. In example 4 prove that the paths of the particles relative
to the cylinder are similar ellipses, and that the paths in space are
similar to the pericycloid
x = (a + 6) cos 6 + (a 6) cos ( - ~\ 6,
\a-bj
.
a-bj
7. Water is enclosed in a vessel bounded by the axis of y
and the hyperbola 2 (a? 3y 2 ) + x + my = 0, and the vessel is set
rotating about the axis of z. Prove that
$ = 2 (3x*y - f) + xy - m (a? - f\
fy == 2 (x 3 - 3xy 2 ) + i (^ - 2/ 2 ) +
8. The space between two confocal coaxial elliptic cylinders
is filled with liquid which is at rest. Prove that if the outer
cylinder be moved with velocity U parallel to the major axis, and
the inner with relative velocity V in the same direction, the
velocity potential of the initial motion will be
, TT , * IT cosn (- 17) ,
= uc cosh 77 cos tVc- , ; -~ smh a cos t,
cosh (p a)
where rj = (3, i] = a are the equations of the outer and inner
cylinders respectively, and 2c the distance between their foci.
9. If in the last example the outer cylinder were to rotate
with angular velocity H, and the inner with angular velocity o>,
prove that initially
cosh 2 (?;- a) . ,. cosh 2 (/3- 77) . ..
d> = lie 2 ^-T-o7Z - \ sm 2 S - f wc i. n)a - 4. sm 2
smh 2 (/3 - a) smh 2 (ft - a)
10. The transverse section of a uniform prismatic vessel is of
the form bounded by the two intersecting hyperbolas represented
by the equations
V2 (x 2 - vy 2 ) + x 2 + y* = a 2 , \/2 (y 2 - x 2 ) + x* + y* = b 2 .
If the vessel be filled with water and made to rotate with
angular velocity co about its axis, prove that the initial component
velocities at any point (x, y) of the water will be
ft)
respectively.
EXAMPLES. 59
11. In the midst of an infinite mass of liquid at rest is a
sphere of radius a, which is suddenly strained into a spheroid of
small ellipticity. Find the kinetic energy due to the motion of
the liquid contained between the given surface, and an imaginary
concentric spherical surface of radius c; and show that if this
imaginary surface were a real bounding surface which could not be
deformed, the kinetic energy in this case would be to that in the
former case in the ratio
C 5 (3a 5 +2c 6 ) : 2(c 5 -a 5 ) 2 .
12. The space between two coaxial cylinders is filled with
liquid, and the outer is surrounded by liquid extending to infinity,
the whole being bounded by planes perpendicular to the axis. If
the inner cylinder be suddenly moved with given velocity, prove
that the velocity of the outer cylinder to that of the inner, will be
in the ratio
26 2 c 2 p : p (a 2 6 2 - a 2 c 2 + 6 4 + 6 2 c 2 ) + a- (a 2 - Z> 2 ) (6 2 - c 2 ),
where a and 6 are the external and internal radii of the outer
cylinder, a- its density, c the radius of the inner cylinder and p the
density of the liquid.
13. A solid cylinder of radius a immersed in an infinite liquid,
is attached to an axis about which it can turn, whose distance
from the axis of the cylinder is c, and oscillates under the action
of gravity. Prove that the length of the simple equivalent
pendulum is
a and p being the densities of the cylinder and liquid.
14. Liquid of density p is contained between two confocal
elliptic cylinders and two planes perpendicular to their axes. The
lengths of the semi-axes of the inner and outer cylinders are
c cosh a, c sinh a, c cosh /3, c sinh ft respectively. Prove that if the
outer cylinder be made to rotate about its axis with angular
velocity fl, the inner cylinder will begin to rotate with angular
velocity
lip cosech 2 (/3 a)
p coth 2 (/3 - a) + \ and -ty- be any two functions, which throughout the
interior of a closed surface 8 are single valued, and which together
with their first and second derivatives, are finite and continuous at
every point within S ; then
j!
ay ay az
...... (1)
^^dxdydz ...... (2),
where the triple integrals extend throughout the volume of S, and the
surface integrals over the surface of 8, and dn denotes an element
of the normal to 8 drawn outwards.
Integrating the left-hand side by parts, we obtain
7 T ^
dx dx
where the brackets denote that the double integral is to be taken
between proper limits. Now since the surface is a closed surface,
any line parallel to x, which enters the surface a given number of
times, must issue from it the same number of times; also the
^-direction cosine of the normal at the point of entrance will be
of contrary sign to the same direction cosine at the corresponding
point of exit ; hence the surface integral
64 MOTION OF A SINGLE SOLID.
Treating each of the other terms in a similar manner, we find
that the left-hand side of (3)
The second equation (2) is obtained by interchanging > and i/r.
59. We may deduce several important corollaries.
(i) Let -v/r = 1, and let < be the velocity potential of a liquid;
then V 2 (f> = 0, and we obtain
d8 ............... (4).
The right-hand side is the analytical expression for the fact
that the total flux across the closed surface is zero; in other words
as much liquid enters the surface as issues from it.
(ii) Let and i/r be both velocity potentials, then
(iii) Let = ty, where is the velocity potential of a liquid;
then
>Y fdy} 7 7 7 ff , dd> 7L , ,-,
+(j 1 +( j ) \dxdydz** ^-cSf...(6).
dee/ \dy) \dz) j JJ Y dn
If we multiply both sides of (6) by |p, the left-hand side is
equal to the kinetic energy of the liquid ; and the equation shows
that the kinetic energy of a liquid whose motion is acyclic and
irrotational, which is contained within a closed surface, depends
solely upon the motion of the surface.
60. Let us now suppose that liquid contained within such a
surface is originally at rest, and let the liquid be set in motion by
means of an impulsive pressure p applied to every point of the
surface. The motion produced must be necessarily irrotational,
and acyclic; also if < be its velocity potential, it follows from
25 (39) that p = p/dn = 0, and therefore
whence d(j>/dx, d}dy, and d/dz are each zero, and therefore the
liquid is reduced to rest.
62. In proving Green's Theorem, we have supposed that the
region through which we integrate, is contained within a single
closed surface, but if the region were bounded externally and
internally by two or more closed surfaces, the theorem would still
be true, provided we take the surface integral with the positive
sign over the external boundary, and with the negative sign over
each of the internal boundaries.
63. Let us suppose that the liquid is bounded internally by
one or more closed surfaces S 1} S 2 &c., and externally by a very
large fixed sphere whose centre is the origin. If T be the kinetic
energy of the liquid,
where the square brackets indicate that the integral is to be taken
over each of the internal boundaries.
If the liquid be at rest at infinity, the value of at S cannot
contain any term of lower order than m/r, where m is a constant,
whence
dfdn = d/dr = m/r* ;
also if dl be the solid angle subtended by dS at the origin,
therefore
which vanishes when r = oo . Hence the kinetic energy of an
infinite liquid bounded internally by closed surfaces is
(7 >-
where the surface integral is to be taken over each of the internal
boundaries.
B. H. 5
66 MOTION OF A SINGLE SOLID.
The preceding expression for the kinetic energy shows that,
if the motion is acyclic and the internal boundaries of the liquid
be suddenly reduced to rest, the whole liquid will be reduced to
rest.
>
64. When a single solid is moving in an infinite liquid, the
velocity potential must satisfy the following conditions ;
(i) must be a single valued function, which at all points of
the liquid satisfies the equation V 2 < = 0.
(ii) < and its first derivatives must be finite and continuous
at all points of the liquid, and must vanish at infinity, if any
portion of the liquid extends to infinity.
(iii) At all points of the liquid which are in contact with a
moving solid, d/dn must be equal to the normal velocity of the
solid, where dn is an element of the normal to the solid drawn
outwards ; if any portion of the liquid is in contact with fixed
boundaries, d(j>{dn must be zero at every point of these fixed
boundaries.
The most general possible motion of a solid may be resolved
into three component velocities parallel to three rectangular axes
(which may either be fixed or in motion), together with three
angular velocities about these axes.
Let us therefore refer the motion to three rectangular axes
Ox, Oy, Oz fixed in the solid, and let fa be the velocity potential
when the solid is moving with unit velocity parallel to Ox, and let
^ be the velocity potential when the solid is rotating with unit
angular velocity about Ox. Let fa, fa, %a> %s be similar quantities
with respect to Oy and Oz. Also let u, v, w be the linear velocities
of the solid parallel to, and ca lf o> 2 , 6> 3 be its angular velocities
about the axes.
We can now show that the velocity potential of the whole
motion will be
= Ufa + Vfa + Wfa + !%! + G> 2 X 2 + 6)3^3 (8).
For if \, fi, v be the direction cosines of the normal at any
point x, y, z on the surface of the solid, we must have at the
surface
dfa _ dfa_ dfa _
~1 ~~~ ">> 7 M/ 7 V,
an dn dn
dy l dy 2 dy z
-p = vy u.z, -y^ = \z vx, -p- = /mx \y.
dn 9 dn dn
KINETIC ENERGY. 67
Hence - = (u yw 3 + 2&> 2 ) A, + (v
= normal velocity of the solid.
65. If we substitute the value of < from (8) in (7), it follows
that T is a homogeneous quadratic function of the six velocities
u, v, w, &>!, ft> 2 , ^ an d therefore contains twenty-one terms. If
we choose as our axes Ox, Oy, Oz, the principal axes at the centre
of inertia of the solid, the kinetic energy of the latter will be
equal to
where M is the mass of the solid, and A 1} B lt 0^ are its principal
moments of inertia. Hence the kinetic energy T of the system,
being the sum of the kinetic energies of the solid and liquid, is
determined by the equation,
2T= Pu? + Qv* + Rw* + ZP'vw + 2Q'wu + ZR'uv
+ Aw? + Ba> 2 2 + (7ft> 3 2 + 2J/&> 2 &) :! + Z&tojO! + 2<7'ft)!< 2
+ 2ft)! (Lu + Mv + Nw)
+ 2ft) 2 (L'u + M'v + N'w)
+ Zw s (L"u+M"v + N"w) ................................. (9).
The coefficients of the velocities in the preceding expression
for the kinetic energy, are called coefficients of inertia. The
quantities P, Q, R are called the effective inertias of the solid
parallel to the principal axes, and the quantities A,B, C are called
the effective moments of inertia about the principal axes. If the
liquid extend to infinity, and there is only one moving solid, the
coefficients of inertia depend solely upon the form and density
of the solid and the density of the liquid, and not upon the
coordinates which determine the position of the solid in space.
The values of these coefficients are
...(10)
by (5) ; with similar expressions for the other coefficients.
When the form of the solid resembles that of an ellipsoid,
which is symmetrical with respect to three perpendicular planes
through its centre of inertia, and the motion is referred to the
52
68 MOTION OF A SINGLE SOLID.
principal axes of the solid at that point, the kinetic energy must
remain unchanged when the direction of any one of the component
velocities is reversed ; hence the kinetic energy cannot contain
any of the products of the velocities, and must therefore be of the
form ;
2T = Pu* + Qv 2 + Rw* + Ato* + B(oJ+Ca>,? (11).
If in addition, the solid is one of revolution about the axis of z,
the kinetic energy will not be altered if u is changed into v, and
2 2 ) + C&> 3 2 (12).
Although every solid of revolution must be symmetrical with
respect to all planes through its axis, it is not necessarily sym-
metrical with respect to a plane perpendicular to its axis. The
solid formed by the revolution of a cardioid about its axis is an
example of such a solid. In this case the kinetic energy will be
unaltered when the signs of u, v or a> 3 are changed, and also when
u is changed into v, and w l into w 2 ; hence in this case
2T = P (w 2 + v 2 ) + Rw 3 + A (w^ + ft> 2 2 ) + <7o> 3 2 + 2Nw fa + &> 2 ). . .(13).
If the solid moves with its axis in one plane, (say zx), v and eoj
must be zero, and the last term may be got rid of, by moving the
origin to a point on the axis of z, whose distance from the origin is
N/R. This point is called the Centre of Reaction.
66. We must now find expressions for the component linear
and angular momenta.
Since we are confining our attention to acyclic irrotational
motion, it follows from 59 that the motion of the liquid at any
instant depends solely upon the motion of the surface of the solid;
hence the motion which actually exists at any particular epoch,
could be produced instantaneously from rest, by the application of
suitable impulsive forces to the solid ; and since impulsive forces
are measured by the momenta which they produce, it follows that
the resultant impulse which must be applied to the solid, must be
equal to the resultant momentum of the solid and liquid.
Let , ij, be the components parallel to the principal axes of
the solid, of the impulsive force which must be applied to the
solid, in order to produce the actual motion which exists at time t;
and let X, /*, v be the components about these axes, of the
impulsive couple. Then , 77, % are the components of the linear
COMPONENT MOMENTA. 69
momentum, and X, /*, v of the angular momentum of the solid and
liquid.
Let p denote the impulsive pressure of the liquid, and let us
consider the effect produced upon the solid by the application of
the impulse whose components are f, 77, f, X, \i, v.
By the ordinary equations of impulsive motion
where I, m, n are the direction cosines of the normal at any point
of 8.
But if > be the velocity potential of the motion instantaneously
generated by the impulse, which is equal to the velocity potential
which actually exists at time t, it follows from 25, that p = pcf),
whence
since at the surface of the solid
I = d 'H) > ^-i !*> v be the component impulses, which must be
applied to the solid, in order to generate from rest the motion
which actually exists at time t, it follows from the first proposition
of 57, that
But since T is a homogeneous quadratic function of the
velocities,
orr dT dT dT dT dT dT
2J. =u-j- +v j- + w j- + o>i j h o> 2 -j h ft> 3 j .
du dv dw dw l dco 2 dw A
Comparing these two equations, we obtain (14).
We are now in a position to solve a variety of problems
connected with the motion of a single solid in an infinite liquid.
MOTION OF A SPHERE. 71
Motion of a Sphere.
68. Let us suppose that the centre of the sphere describes
a plane curve, and let u, v be its component velocities parallel
to the axes of a; and y. Since every diameter of a sphere is
a principal axis, the axes of x and y may be supposed to be fixed
in direction, whence
and since on account of symmetry P Q, we have
where P = M-
= M+ Trpct? I cos 2 9 sin ddO
Jo
where M' is the mass of the liquid displaced. Whence
T
and therefore
Let us now suppose that the sphere is descending under the
action of gravity, and that the axis of y is drawn vertically
downwards; we shall also suppose that the sphere is initially
projected with a velocity, whose horizontal and vertical components
are U and V.
Since the momentum parallel to x is constant throughout the
motion,
= const. = (M + %M') U,
whence u = U.
To determine the force acting on the system, draw two
horizontal planes above and below the sphere and at a considerable
distance from it. Then the force on the whole system due to
gravity is
% + tt(Pi -P2 + 9pz} dA - Mg' + h,
when p 1} p 2 are the hydrostatic pressures on the upper and lower
planes, z the distance between them, and h that portion of the
pressure due to the motion of the liquid. The integral vanishes ;
also h vanishes when the planes are at an infinite distance from
72 MOTION OF A SINGLE SOLID.
the sphere ; whence the force is equal to (M M') g. The equa-
tion giving the vertical motion is therefore
or
Whence if a- be the density of the sphere,
dv _ cr p
dt~ ff + ^p ff>
and the sphere will describe a parabola with vertical acceleration
fj (a p)l( its angular velocity
about its axis. On account of symmetry none of the products can
appear, and therefore
T=%(Pi? + Qv* + A< a *) .................. (15),
where P, Q, A are constant quantities.
Let us now suppose that no forces are in action, and that the
solid and liquid are initially at rest ; and let the cylinder be set in
motion by means of an impulsive force F, whose line of action
passes through its axis, and an impulsive couple which produces
an initial angular velocity ft.
Let us refer the motion to two fixed rectangular axes x and y,
the former of which coincides with the direction of F, and let 6 be
the angle which the major axis of the cross section makes with the
axis of x at time t.
Resolving the momenta along the axes of x and y, we obtain
f cos i} sin 6 = F
sin 6 + 17 cos 6 0,
whence since
= Pw, 77 = Qt>,
we obtain
Pu = Fcos0, Qv = -Fsin0 ............... (16).
MOTION OF AN ELLIPTIC CYLINDER. 73
Since the kinetic energy remains constant throughout the
motion, it follows that if we substitute the values of u, v from (16)
in (15), and put (3 for the initial value of 6, we shall obtain
or Afc=A&+ F* - (sin 2 0- sin 2 y3) ...... (17).
We shall presently show that Q > P ; it therefore follows
that if
Q-P
/
V
APQ 3
6 will never vanish, and the cylinder will make a complete
revolution.
The integration of equation (17) requires elliptic functions, but
without introducing these quantities, we can easily ascertain the
character of the motion of the centre of inertia of the cylinder.
Let (x, y) be the coordinates of the centre of inertia referred to
the fixed axes of x and y ; then
x = u cos v sin 0, y=usm0 + v cos (18).
Substituting the values of u, v from (16) we obtain
y = F(p-Q)sin0cos0.
These equations show that the centre of inertia of the cross
section of the cylinder, moves along a straight line parallel to the
direction of F with a uniform velocity F/Q, superimposed upon
which is a variable periodic velocity, and that at the same time
it vibrates perpendicularly to this line. This kind of motion
frequently occurs in hydrodynamics, and a body moving in such a
manner is called a Quadrantal Pendulum.
If
which is the limiting case between oscillation and rotation, the
74 MOTION OP A SINGLE SOLID.
equations of motion admit of complete integration. Putting
A \P Q)'
(17) becomes
whence
Therefore ^ = ^
6=/sin<9
It = log tan
dy IA
y=
dx F
I A .
IA
.
= pj cosec ~ ~p sln "
F I A
x = pj log tan %6 + -y cos 0.
Putting IA/F = c, and eliminating 6 we obtain the equation of
the path, viz.
F
x = pi- l g
y
The curves described by the centre of inertia of the cylinder
in the three cases have been traced by Prof. Greenhill, and are
shown in Figures 1, 2, 3 of the accompanying diagram.
70. We shall now show that for an elliptic cylinder Q > P.
When the cylinder is moving with unit velocity parallel to x,
we have shown in 48 that
and therefore
Now at the surface
sinh & sin ,
/3 cos .
= c sinh /3/cos
= c 2 sinh 2 /8/ * cos* j-dg
Jo
P=M-pf folds
=M(I + ^
whence
where P.
MOTION OF AN ELLIPTIC CYLINDER.
75
jf
PH
70 MOTION OF A SINGLE SOLID.
Similar results can be proved to be true in the case of an
ellipsoid ; from which it is inferred that when any solid body is
moving in an infinite liquid, the effective inertias corresponding
to the greatest, mean, and least principal axes, are in descending
order of magnitude.
71. If the cylinder be projected parallel to a principal axis
without rotation, it will continue to move in a straight line with
uniform velocity ; but if the direction of projection is not a
principal axis, it will begin to rotate, and its angular velocity at
any subsequent time will be determined by putting = in (17).
We shall now show that if the cylinder be projected parallel to
a principal axis, its motion will be stable or unstable according as
the direction of projection coincides with the minor or major
axis.
Let us first suppose the cylinder projected parallel to its major
axis, and that a slight disturbance is communicated to it. The
equation determining the angular velocity is obtained by putting
n = /3=0 in (17); whence
and therefore differentiating, and remembering that in the begin-
ning of the disturbed motion 6 is a small quantity, we obtain
Since Q> P, the coefficient of 6 is negative, which shows that
the motion is unstable.
If the cylinder is projected parallel to its minor axis we must
put /3 = ^TT ; also if ^ = \ir 6, % will be a small quantity in the
beginning of the disturbed motion; Avhence (17) becomes
whence A% + F* - X =
Since the coefficient of % is positive, the motion is stable.
It can also be shown that if an ellipsoid be projected parallel
to a principal axis, without rotation, the motion will be unstable
unless the direction of projection coincides with the least axis.
MOTION OF AN ELLIPTIC CYLINDER. 77
We shall however presently show, that if an ovary ellipsoid be
projected parallel to its axis, the motion will be stable, provided a
sufficiently large angular velocity be communicated to the solid
about its axis.
72. Let us now investigate the motion of an elliptic cylinder,
which descends from rest under the action of gravity.
Let the axis of y be horizontal, and the axis of x be drawn
vertically downwards.
The equations of momentum are
sin 6 + 7} cos & = 0,
|- ( cos - 77 sin 0) = (M- M') g,
du
from the last of which we obtain
f cos 6 - 77 sin 6 = (M-M')gt
Solving these equations and recollecting that = Pu, 77 = Qv,
we obtain
Pu = (M- M'} gt cos 8, Qv = -(M- M') gt sin 0.
Substituting these values of u and v in (18), we obtain
/cos 2 6 sin 2 Q\ , , , , ,,. \
* = \~~p~ + ~Q J ( ~ } n
< 19 >-
sm e cos e
The equation of energy gives
(M - MJ gH* + AP=2(M- M') gx.
If we differentiate this equation with respect to t, we can
eliminate x by means of (19) ; but the resulting equation would be
difficult to deal with. We see however from the first of (19), that
x is always positive, and therefore the cylinder moves downwards
with a variable velocity, which depends upon the inclination of its
major axis to the vertical, as well as upon the time. We also see
from the second equation, that the horizontal velocity vanishes,
whenever the major axis becomes horizontal or vertical ; but
if the motion should be of such a character that always lies
between and TT, the horizontal velocity will never vanish.
78
MOTION OF A SINGLE SOLID.
Helicoidal Steady Motion of a Solid of Revolution.
73. In the figure let 00 be the axis of the solid of revolution,
its centre of inertia, and let the solid be rotating with angular
velocity ft about its axis.
A
Let the solid be set in motion by means of an impulsive force
F along OZ, and an impulsive couple G about OZ, and let a be
the angle which OC initially makes with OZ.
Let i/r be the angle which the plane ZOO makes at time t
with a plane ZOX, which is parallel to some fixed plane, and let
the former plane cut the equatorial plane in OA ; also let ZOC = 6.
It will be convenient to refer the motion to three moving axes,
OA, OB, 00, where OB is the equatorial axis which is perpendicular
to OA.
Resolving the linear momentum of the system along OZ, OX
and a line Y perpendicular to the plane ZOX, we obtain
cos 6 + % sin 6) cos ^ 77 sin i/r = 0,
cos 6 + sin 6) sin i/r + 17 cos -|r = 0,
whence
(20).
Since the components of momentum parallel to the axes of X
and Y (which are fixed in direction, but not in position because
is in motion) are zero throughout the motion, the angular
HELICOIDAL STEADY MOTION. 79
momentum about OZ is constant 1 , whence
(21).
The equation of energy gives
PU* + Rw* + A (a)! 2 + 2 ) = const ............. (22),
also = Pu, = Rw,
and therefore by (20) and (21) this becomes
., ,/sin 2 ^ cos 2 0\ {Q + <7H (cos a - cos 0)}* .-
F-(n-+^n-) + L A ,/, - 2L + Affl = const.
V P R J A sm 2
= its initial value ...... (23).
This equation determines the inclination of the axis.
So far our equations have been perfectly general, we shall now
introduce the conditions of steady motion. These are
= a, ijr = fjL, 0=0 = ............... (24).
Now a) l = TJr sin a = p sin a, whence (21) becomes
^/isin 2 a = # ........................ (25).
Differentiating (23) with respect to t and using (24) and
(25) we obtain
A p? cos a -Clp + (-l F*cosa = ......... (26).
This is a quadratic equation for determining p, when O and F
are given. Now //, must necessarily be a real quantity, and there-
fore the condition that steady motion may be possible is that
C 2 W>4,F*-A cos 2 a - ............... (27),
\-tt -i I
and since Rw = = F cos a,
the condition becomes
(28).
1 It was shown by Hayward (Trans. Camb. Phil. Soc. Vol. x.) that when the
origin as well as the axes are in motion, the Principle of Angular Momentum is
expressed by three equations of the form
dv'
--vf+wf- x'0 2 + //0i = N.
Since the direction of the axes of X, Y and Z are fixed, 0j = 2 =0; also since the
momenta ' and 77' parallel to X and Y are zero, the equation reduces to dv'jdt = N ;
which gives ' = const., when N=0. This result can of course be proved by
elementary methods.
80 MOTION OF A SINGLE SOLID.
If the solid of revolution is oblate (such as a planetary
ellipsoid) R > P, and therefore (28) is always satisfied ; but if the
solid is prolate (such as an ovary ellipsoid) P > R, and therefore
steady motion will not be possible unless ft exceeds a certain
value.
In order to find the path described by the centre of inertia
of the solid in steady motion, we have, since ^ fit,
/ */l : 1\ ,
x = (u cosa + w sma) cosy = Jf I -^- =- ) sma cosa cospt,
\Jt " i
i\ 1 \ .
y = (u cosa + w sin a) sin-Jr = F ( -^ ^ sina cosa sin/u,
\xl ft
r, /sin 2 a cos 2 a\
.gr = ;cosa u sma =-^(^^5 I n~ >
V " Jt 1
which shows that the centre of inertia describes a helix.
74. In order to find whether the steady motion is stable or
unstable, differentiate (23) with respect to t, and we obtain
A'0+f(0) = Q (29),
where
(7ft
cosa
The condition for steady motion is, that /(a) = 0, which leads
to (26), whence writing = a + x where ^ is small, (29) becomes
and the condition of stability requires that f (a) should be
positive. Now
/' (a) = Ap? (1 + 2 cos 2 a) - SOft/i cosa + ~ - F n - ~ - cos2a,
whence eliminating ft by (26) this becomes
- ~ (1-3 cos
The condition that the right-hand side should be positive
is that
* (^ - iY sin 2 a (9 cos 2 a - 1) > 0,
which requires that a should lie between cos" 1 ^ and 0, or between
TT cos" 1 and TT.
CONDITIONS OF STABILITY. 81
As a particular example let the solid be projected point
foremost; then = and = 0, and therefore since 6 is a small
quantity in the beginning of the disturbed motion
/(*) =
If therefore R > P the motion is always stable, whether
there is or is not rotation, and consequently the forward motion
of a planetary ellipsoid is always stable ; but if P > R, it follows
that since F=Rw, the motion will be unstable unless
The motion of an ovary ellipsoid is therefore unstable, unless
the ratio of its angular velocity to its forward velocity exceeds
a certain value.
75. These results have an application in gunnery.
When an elongated body, such as a bullet, is fired from
a gun with a high velocity, the effect^ of the air upon its motion
cannot be neglected ; and if the air is treated as an incompressible
fluid, the previous investigation shows that the bullet will tend
to present its flat side to the air, and also to deviate from its
approximately parabolic path, unless it be endowed with a rapid
rotation about its axis. Hence the bores of all guns destined
for long ranges are rifled, by means of which a rapid rotation
is communicated to the bullet before it leaves the barrel. The
effect of the rifling tends to keep the bullet moving point
foremost, and to ensure its travelling along an approximately
parabolic path in a vertical plane. Moreover when a bullet is
moving with a high velocity, the effect of friction cannot be
neglected ; and it is obvious that when the bullet is moving
with its flat side foremost, the effect of frictional resistance will
be much greater than when it is moving point foremost, and
therefore the bullet will not travel so far in the former as in the
latter case. The hydrodynamical theory therefore explains the
necessity of rifling guns.
Motion of a Cylinder parallel to a Plane.
76. We have thus far supposed that the liquid extends to
infinity in all directions ; we shall now suppose that the liquid
B. H. 6
82 MOTION OF A SINGLE SOLID.
is bounded by a fixed plane, and shall enquire what effect the
plane boundary produces on the motion of a circular cylinder.
Let the axis of y be drawn perpendicularly to the plane, and
let the origin be in the plane, and let (x, y) be the coordinates
of the centre of the cylinder, (u, v) its velocities parallel to and
perpendicular to the plane.
The kinetic energy of the solid and liquid must be a homo-
geneous quadratic function of u and v, but since the kinetic
energy is necessarily unchanged when the sign of u is reversed,
the product uv cannot appear. We may therefore write
T = % (Ru 2 + R'v*) ..................... (30).
The coefficients R, R' depend upon the distance of the
cylinder from the plane, and are therefore functions of y but
not of x ; and as a matter of fact their values are equal. It will
not however be necessary to assume the equality of R and R',
since our object will be attained provided we can show that
R and R' diminish as y increases.
In order to produce from rest the motion which actually
exists at time t, we must apply to the cylinder impulsive forces
whose components are X, Y; and we must also apply at every
point of the plane boundary an impulsive pressure, which is just
sufficient to prevent the liquid in contact with the plane from
having any velocity perpendicular to the plane. The work done
by the impulsive pressure is zero, whilst the work done by the
impulses X, Y is
%(Xu+Yv) ........................ (31),
which must be equal to T. Now (30) may be written in the form
)
whence comparing (31) and (32) we see that
*-*(+
X= C ^ = Ru, Y = ~ = R'v (33),
du dv
and therefore y=ir +*) ( 34 )-
\K H /
The last equation gives the kinetic energy in terms of the
impulsive forces X, Y applied to the cylinder.
Let us now suppose that the cylinder instead of being at
a distance y from the plane, is at a distance y 1} where y-^ >y ; and
EFFECT OF A PLANE BOUNDARY. 83
let R l} RI be the values of R, R' at y 1 . Then if the cylinder were
set in motion by the same impulses, the work done would be
(33) -
Now the effect of the plane boundary is to produce a con-
straint, and the effect of this constraint evidently diminishes
as the distance of the cylinder from the plane increases, and
therefore by Bertrand's theorem, 1\ > T. Hence
4-4,1 + F'
JT/ ~ ~rv )
R 1 H J
is positive for all values of X and Y, which requires that
R > R! , R' > RI ',
hence R, R' diminish as y increases, and consequently their
differential coefficients with respect to y are negative.
77. We can now determine the motion of the cylinder.
The momentum parallel to x is equal to dTjdu and is con-
stant ; whence
Ru = const. = X (36).
Since the kinetic energy is constant, we have
Ru 2 + R'v* = const. = 2T (37).
Differentiating (37) with respect to t, and eliminating dufdt
by (36), we obtain
J TV J T)\
GLft CLl\
dy dy) '
From this equation we can ascertain the effect of the plane
boundary ; for if the cylinder is projected perpendicularly to the
plane, u = 0, and
dR
2R' dy '
Now dR'/dy is negative, and therefore v is positive; whence it
follows that whether the cylinder be moving from or towards the
plane, the force exerted by the liquid upon the cylinder will
always be a repulsion from the plane, which is equal to
M& dR'
2R' dy '
Hence if the cylinder be in contact with the plane, and a small
62
84 MOTION OF A SINGLE SOLID.
velocity perpendicular to the plane be communicated to it, the
cylinder will begin to move away from the plane with gradually
increasing velocity. This velocity cannot however increase indefi-
nitely, for that would require the energy to become infinite, which
is impossible, since the energy remains constant and equal to its
initial value. If RJ denote the value of R' when the cylinder is
in contact with the plane, v the initial velocity, and v the velocity
when the cylinder is at an infinite distance from the plane, the
equation of energy gives
7? ' r 2 _ 7? ',2
.n/ u ^1,^ u .
The value of R^' = M + M', since the motion is the same as if
the plane boundary did not exist, whence the ratio of the terminal
to the initial velocity is
v -= /
V V -<
M+M''
Let us now suppose that the cylinder is projected parallel to
the plane with initial velocity ii . By (38) the initial acceleration
v perpendicular to the plane is
. _ M 2 dR
V ~2Rfy''
and since dRjdy is negative, the cylinder will be attracted towards
the plane, and will ultimately strike it.
78. All the results of the last two sections are true in the case
of a sphere, and can be proved in the same manner. Moreover the
motion will be unaltered, if we remove the plane boundary, and
suppose that on the other side, an infinite liquid exists in which
another equal cylinder or sphere is moving with velocities u, v.
The second cylinder or sphere is therefore the image of the first.
Our results are therefore applicable to the case of two equal
cylinders or spheres, which are moving with equal and opposite
velocities along the line joining their centres; or to the case
in which the cylinders or spheres are projected perpendicularly
to the line joining their centres, with velocities which are equal
and in the same direction.
These results have however a wider application, for according
to the views of Faraday and Maxwell, the action which is observed
to take place between electrified bodies is not due to any direct
action which electrified bodies exert upon one another, but to
EXAMPLES. 85
something which takes place in the dielectric medium surrounding
these bodies ; and although the preceding hydrodynamical results
do not of course furnish any explanation of what takes place in
dielectric media, they establish the fact that two bodies which are
incapable of exerting any direct influence upon one another, are
capable of producing an apparent attraction or repulsion upon one
another, when they are in motion in a medium which may be
treated as possessing the properties of an incompressible fluid.
EXAMPLES.
1. A light cylindrical shell whose cross section is an ellipse is
rilled with water, and placed at rest on a smooth horizontal plane
in its position of unstable equilibrium. If it is slightly disturbed,
prove that it will pass through its position of stable equilibrium
with angular velocity w, given by the equation
2. An elliptic cylindrical shell, the mass of which may be
neglected, is filled with water, and placed on a horizontal plane
very nearly in the position of unstable equilibrium with its axis
horizontal, and is then let go. When it passes through the position
of stable equilibrium, find the angular velocity of the cylinder,
(i) when the horizontal plane is perfectly smooth, (ii) when it is
perfectly rough ; and prove that in these two cases, the squares of
the angular velocities are in the ratio
(a 2 - 6 2 ) 2 + 46 2 (a 2 + 6 2 ) : (a 2 - & 2 ) 2 ,
2a and 26 being the axes of the cross section of the cylinder.
3. A pendulum with an elliptic cylindrical cavity filled with
liquid, the generating lines of the cylinder being parallel to the
axis of suspension, performs finite oscillations under the action of
gravity. If I be the length of the equivalent pendulum, and I' the
length when the liquid is solidified, prove that
~
where M is the mass of the pendulum, m that of the liquid, h the
86 MOTION OF A SINGLE SOLID.
distance of the centre of gravity of the whole mass from the axis
of suspension, and a, b the semi-axes of the elliptic cavity.
4. Find the ratio of the kinetic energy of the infinite liquid
surrounding an oblate spheroid, moving with given velocity in
its equatorial plane, to the kinetic energy of the spheroid; and
denoting this ratio by P, prove that if the spheroid swing as the
bob of a pendulum under gravity, the distance between the axis of
the suspension and the axis of the spheroid being c, the length of
the simple equivalent pendulum is
(1 + P) c + 2a 2 /5c
I -pi a
where a is the equatorial radius, a and p the densities of the
spheroid and liquid respectively.
5. A pendulum has a cavity excavated within it, and this
cavity is filled with liquid. Prove that if any part of the liquid
be solidified, the time of oscillation will be increased.
6. A closed vessel filled with liquid of density p, is moved in
any manner about a fixed point 0. If at any time the liquid
were removed, and a pressure proportional to the velocity potential
were applied at every point of the surface, the resultant couple
due to the pressure would be of magnitude G, and its direction
in a line OQ. Show that the kinetic energy of the liquid was
proportional to ^pcoG cos 6, where w is the angular velocity of the
surface, and 6 the angle between its direction and OQ.
7. Liquid is contained in a simply-connected surface S ; if w
is the impulsive pressure at any point of the liquid due to any
arbitrary deformation of S, subject to the condition that the
enclosed volume is not changed, and c/ the impulsive pressure
for a different deformation, show that
8. If a sphere be immersed in a liquid, prove that the kinetic
energy of the liquid due to a given deformation of its surface, will
be greater when the sphere is fixed than when it is free.
CHAPTER IV.
WAVES.
79. BEFORE discussing the dynamical theory of waves, we
shall commence by explaining what a wave is.
Let us suppose that the equation of the free surface of a liquid
at time t, is
y = a sin (mx nt) (1),
where the axis of x is horizontal, the axis of y is measured vertically
upwards, and a, ra and n are constants.
The initial form of the free surface, i.e. its form when t 0, is
y a sin mx, which is the curve of sines. The maximum values
of y occur when mx = (2i+ ^) TT, where i is zero or any positive
or negative integer ; and this maximum value is equal to a. The
minimum values of y occur when mx = (2i + f) TT, where i is
zero or any positive or negative integer ; and this minimum
value is equal to a. As x increases from to ^7r/m, y increases
from to a, and as x increases from ^7r/m to -rr/m, y decreases
from a to 0. As x increases from tr/m to f7r/w, y is negative ;
and when x has the latter value, y has attained its greatest
negative value, which is equal to a ; as x increases from ITT/TO
to 2-TT/m, y numerically decreases from a to 0.
The values of y comprised between
x = 2t7T/m and x = 2 (i + 1) irjm,
evidently go through exactly the same cycle of changes.
When the motion of a liquid is such, that its free surface is
represented by an equation such as (1), the motion is called wave
motion.
WAVES.
The quantity a, which is equal to the maximum value of y,
is called the amplitude ; and the distance 2?r/?n, between two
consecutive maxima values of y, is called the wave length.
In order to ascertain the form of the free surface at time t, let
us transfer the origin to a point = nt/m ; then if x' be the abscissa
at time t referred to the new origin, of the point whose abscissa
referred to the old origin is x, we have x = x' + and
y = a sin (mx + m% nt) = a sin mx'.
The form of the free surface at time t, is therefore obtained by
making the point which initially coincided with the origin, travel
along the axis of x, with velocity n/m.
The velocity of this point is called the velocity of propagation
of the wave.
If V be the velocity of propagation, and X the wave length, we
thus obtain the equations
m-2-Tr/X, V = n/m (2),
and therefore (1) may be written
O-FO (3).
If n, and therefore F, were negative, equations (1) and (3)
would represent a wave travelling in the opposite direction.
The position of the free surface at time t, is exactly the same
as at time t + 2i7r/w, or t + 2iX/F, since n - 2-TrF/X ; the quantity
X/Fis called the periodic time, or shortly the period, and is equal
to the time which the crest of one wave occupies in travelling
from its position at time t to the position occupied by the next
crest at the same epoch. If r denote the period, we evidently
have
* = Vr (4),
and (1) may be written in the form
/ *\
(5),
which is a form sometimes convenient in Physical Optics.
From (4) we see that for waves travelling with the same
velocity, the period increases with the wave length.
The reciprocal of the period, which is the number of vibrations
executed per unit of time, is called the frequency. If therefore we
KINEMATICS OF WAVE MOTION. 89
have a medium which propagates waves of all lengths with the
same velocity, equation (4) shows that the number of vibrations
executed in a second increases as the wave length diminishes.
This remark is of importance in the Theory of Sound.
Let us now suppose that two waves are represented by the
equations
y = a sin (x - Vt),
\
y' = a sin (x Vt e).
X
The amplitudes, wave lengths and velocities of propagation of
the two waves are equal, but the second wave is in advance of the
first; for if in the first equation we put t + e/V for t, the two
equations become identical. It therefore follows that the distance
at which the second wave is in advance of the first is equal to e.
The quantity e is called the phase of the wave.
80. Waves which are represented by equations such as (1),
are called progressive waves ; their wave lengths are equal to
2?r/m, and their velocities of propagation to n/m. If such waves
axe travelling along the surface of water under the action of
gravity, they may be conceived to have been produced by com-
municating to the free surface an initial displacement y = a sin mx,
together with an initial velocity an cos mx. We therefore see
that the wave length depends solely on the initial displacement,
but that the velocity of propagation depends upon the initial
velocity as well as upon the initial displacement.
If we combine the two waves, which are obtained by writing
n for n in (1) and add the results, we shall obtain
y = a sin (mx nt) + a sin (mx + nt)
= 2a sin mx cos nt (6).
Such a wave is called a stationary wave. It is produced by
means of an initial displacement alone, and gives the form of the
free surface at time t, when the latter is displaced into the form of
the curve y = 2a sin mx, and is then left to itself.
In equations (1) or (6), m, which is equal to 2?r/\, is always
supposed to be given, and the problem we have to solve, consists
in finding the value of n, which determines the velocity of
propagation.
90 WAVES.
81. In most problems relating to small oscillations, the motion
is supposed to be sufficiently slow for the quadratic terms which
occur in the equations of motion to be neglected. Under these
circumstances, the equations become linear and usually admit of a
solution in which the time enters in the form of the factor e inf .
Throughout the present chapter this hypothesis will be made ;
but it may be remarked that a solution of the complete equations
of motion has been obtained by Gerstner, which leads to a species
of trochoidal wave involving molecular rotation. The theory of
waves involving molecular rotation is not of any great interest in
the dynamics of a frictionless liquid ; but it is of the highest im-
portance in the dynamics of actual liquids, which are viscous,
inasmuch as the motion of a viscous liquid always involves mole-
cular rotation.
We shall now proceed to consider the irrotational motion of
liquid waves in two dimensions, under the action of gravity.
The solution of the problem involves the determination of a
velocity potential , which satisfies the following three conditions :
(i) (f> must satisfy Laplace's equation, and together with its
first derivatives, must be finite and continuous at every point of
the liquid.
(ii) must satisfy the given boundary conditions at the fixed
boundaries of the liquid.
(iii) must be determined, so that the free surface of the
liquid is a surface of constant pressure.
To find the condition to be satisfied at the free surface, let the
origin be taken in the undisturbed surface, and let the axis of a
be measured in the direction of propagation of the waves, and let
the axis of z be measured vertically upwards.
The pressure at any point of the liquid is determined by the
equation
p/P + gz + + W = C (7).
The equation of the surfaces of constant pressure is p = const.,
and since the free surface is included in this family of surfaces,
and must also satisfy the kinetnatical condition of a bounding
surface, it follows from 12 (17) that
dp dp dp dp
-TT+u-^ + v-^+w^ = (8).
dt dx dy dz
WAVES IN A LIQUID OF GIVEN DEPTH. 91
Substituting the value of p from (7) in (8), and neglecting
squares and products of the velocity, we obtain
+ '" ........................ <
This is the condition to be satisfied at a free surface, where
2=0.
Waves in a Liquid of given Depth.
82. We shall now find the velocity of propagation of two-
dimensional waves travelling in an ocean of depth h.
The equation of continuity is
2+2-
The boundary condition at the bottom of the liquid is
^ = 0, when z = -h .................. (11).
To satisfy (10) assume
> = F (z) cos (mx - nt) .................. (12).
Substituting in (10) we obtain
the solution of which is
F = P cosh mz + Q sinh mz ;
whence = (P osh mz + Q sinh mz) cos (mx nt).
Substituting in (11) and (9) we obtain
P sinh mh = Q cosh mh,
Pn 2 = Qmg ;
whence eliminating P and Q, and taking account of (2), we obtain
which determines the velocity of propagation.
If the lengths of the waves are large in comparison with the
depth of the liquid, h/\ is small, and the preceding result becomes
V 2 = gh .............................. (14),
which determines the velocity of propagation of long waves in
shallow water.
92 WAVES.
If the depth of the liquid is large in comparison with the wave
length, h/X is large, and tanh 27rh/\ 1 approximately, whence
F'=<7X/27r ........................ (15),
which determines the velocity of propagation of deep sea waves.
The last result may be obtained directly, for the value of F
may be written in the form
F= Ae mz + Be- mz ,
and since >, and therefore F, cannot be infinite when z = oo ,
B = 0, and (9) at once gives the required result.
83. Equation (15), which determines the velocity of propaga-
tion of deep-sea waves, may be written in the form n 2 mg',
whence the form of the free surface at time t is
y= a cos (ma; m*g*t) .................. (16),
corresponding to the initial form y = a cos mx. If the initial form
were y=F (x), Fourier's theorem enables us to express F (x) in
the form of a definite integral involving cos ma;; hence by (16)
the form of the free surface at any subsequent time can be written
down in terms of these quantities ; but the difficulty of evaluating
the definite integral is usually so great, that the results are rarely
of much practical utility.
As an example of the use of definite integrals, let us suppose
that the initial form of the free surface is
y = a exp (+ xjc),
where the upper and lower signs are to be taken on the positive
and negative sides of the origin respectively. When x is positive
cos xujc . du
2 f
exp(-a?/c)=-
7TJO
cos mxdm
~l+mV '
whence the equation of the free surface at time t is
_ 2ac f cos (mx mlg*t) dm
~~
84. Returning to the general case, we see that is of the
form
= A cosh m (z + h) cos {mx nt).
93
If 77 be the elevation of the free surface above the undisturbed
surface, we must have
7) = d/dz when z = (17),
whence substituting the value of > in (17), and suitably choosing
the origin we obtain
ij = Amn~ l sinh mh sin (mx nt).
Let (x, z) be the coordinates of an element of liquid when
undisturbed, (f, f ) its horizontal and vertical displacements, also
let x = x + z = z + ; then
= d(f>/dx' = Am cosh m (/ + h) sin (mx' nt)
= d/dz' = Am sinh m (z' + h) cos (mx' - nt).
Since the displacement is small we may put x = x', z = z' as a
first approximation, and we obtain
= a cosh m (z -f h) cos (mx nt)
a sinh m (z + h) sin (mx nt),
where Amjn = a ; whence the elements of liquid describe the
ellipse
2 /cosh 2 m (z + h) + 2 /sinh 2 m ( z + h) = a 2 .
When the depth of the liquid is very great we may put h = oc ,
and the hyperbolic functions must be replaced by exponential
ones ; we shall thus obtain
= Ae mz cos(mx nt)
r n = Amn~ l sin (mx nt),
and the elements of liquid will describe the circles
2 + 2 = (Am/n) 2 e mz .
Waves at the Surface of Separation of Two Liquids.
85. Let us first suppose that two liquids of different densities
(such as water and mercury) are resting upon one another, and
are in repose except for the disturbance produced by the wave
motion ; also let the liquids be confined between two planes
parallel to their surface of separation. Let p, p be the densities
of the lower and upper liquids respectively, h, h' their depths,
and let the origin be taken in the surface of separation when in
repose.
94 WAVES.
In the lower liquid let
< = A cosh m (z + h) cos (tnx ni),
and in the upper let
>' = A' cosh m (z h') cos (mx nt)
also let 77 = a sin (mx nt)
be the equation of the surface of separation. At this surface, the
condition that the two liquids should remain in contact requires
that
drj/dt = dfyjdz = dfy'jdz, when z = 0.
Whence - na = mA sinh mh = mA' sinh mh'.
If Sp, Sp' be the increments of the pressure due to the wave
motion just below and just above the surface of separation, then
Sp + gprj + pd/dt = 0,
and Sp' + gp'r) + p'dfi/dt = 0,
and since Sp = Sp', we obtain
g(p-p')f) = - pd(f>/dt + p'dfi/dt
= n ( Ap cosh mh + A'p cosh mh') sin (mx nt)
= (p coth mh + p coth mh') n^y/m,
whence
' ,. (is),
-
m (p coth ra/i + p coth ra/i )
where m = 27r/\.
86. When X is small compared with h and A', then rw/i, mh'
are large, and coth mh and coth m^' may be replaced by unity : we
thus obtain
If p > p, U" is negative and therefore n is imaginary ; hence if
the upper liquid is denser than the lower, the motion cannot be
represented by a periodic term in t, and is therefore unstable.
If the density of the upper liquid is small compared with that
of the lower, we have approximately
U^gnr 1 (l-Zp'/p).
If the liquid is water in contact with air, p'/p = '00122, hence
if the air is treated as an incompressible fluid
U- = ' 99756 xm- 1 .
STABILITY AND INSTABILITY. 95
87. Secondly, let us suppose that the upper liquid is moving
with velocity V, and the lower with velocity V\ then we may put
(j) = Vx + A cosh m (z + h) cos (mx nt)
ft = V'x + A' cosh m (z h') cos (mx nt).
Let the equation of the surface of separation be
F = i) a sin (mx nt) = 0.
Then in both liquids F must be a bounding surface, and
therefore by 12 equation (17), when z = 0,
dF d$dF d$dF =Q
dt doc dx dz dt]
dF d^dF d'dF_
dt dx dx dz drj
Whence an m Va + mA sinh mh = 0,
an m Va mA' sinh mh' = 0.
Hence if U = n/m be the velocity of propagation,
A sinh mh = a ( V U)
If Sp, 8p' be the increments of pressure at the surface of
separation due to the wave motion
Sp/p + gr} + d(j>/dt + ^ { V Am cosh mh cos (mx nt)}- = ^V-,
Sp'/p + gy + dfijdt + \ { V - A'm cosh mh' cos (mx - nt)} 2 = % V-.
Therefore since Bp = &p',
ag (p p') = A mp (V U) cosh mh Amp ( V U) cosh mh'
or g (p - p') = mp ( V- U) 2 coth mh + mp' ( V - 7") 2 coth mh' . . . (19),
which determines U.
Stability and Instability.
88. We shall now consider a question which has excited
a good deal of attention of late years, viz. the stability or
instability of fluid motion.
If a disturbance be communicated to the two liquids which
are considered in 85 87, the surface of separation may be
conceived to be initially of the form 17 = a sin mx or a cos mx,
where m is a given real quantity, whose value depends upon the
nature of the disturbance. An equation of this kind does not of
96 WAVES.
course represent the most general possible kind of disturbance,
but inasmuch as by Fourier's theorem, any arbitrary function
can be expressed in the form of a series of sines or cosines, or by
a definite integral involving such quantities, an equation of this
form is sufficient for our purpose.
We have pointed out that the object of the wave motion
problem is to determine n ; if therefore n should be found to
be a real quantity, the subsequent motion will be periodic, and
therefore stable ; but if n should turn out to be an imaginary or a
complex quantity, the final solution will involve real exponential
quantities, and therefore the motion will tend to increase with
the time and will be unstable.
To understand this more clearly, it must be recollected that
we have neglected the quadratic terms in the equations of motion.
The validity of this hypothesis depends firstly on the condition
that the disturbance is a small quantity, from which it follows
that the initial displacements and velocities must also be small
quantities ; secondly, on the condition that these quantities
remain small during the subsequent motion. If the solution
thus obtained consists of periodic terms whose amplitudes are
small quantities of the same order as the disturbance, the
second condition is fulfilled, and the system oscillates about its
undisturbed configuration ; but if the solution should contain an
exponential term of the form e**, where k is positive, the system
will tend to depart from its undisturbed configuration, and the
solution will only represent the state of things in the beginning of
the disturbed motion; and the subsequent history of the motion
cannot be ascertained without finding the complete solution of the
equations of motion, in which the quadratic terms are taken into
account. In the former case the motion is stable, and in the
latter unstable.
To express this analytically, let us employ complex quantities,
and assume that the initial form of the free surface is the real
part of
77 = (A - iB) e inue ,
where A, B and m are real; and let n be of the form a -I- ift.
Since the form of the free surface at any subsequent time is
this becomes 77 = (4 - iB) e<
STABILITY AND INSTABILITY. 97
the real part of which is
i) = ?* {A cos (mx at) + B sin (mx at)} ...... (20).
If therefore /3 is positive, the amplitude will tend to increase
with the time, and the motion will be unstable. In such a
case the two liquids will, after a short time, become mixed
together, and will usually remain permanently mixed, if they are
capable of mixing; but if they are incapable of remaining
permanently mixed, the lighter liquid will gradually work its
way upwards, and a stable condition will ultimately be arrived at.
89. If one liquid is resting upon another, equilibrium is
possible when the heavier liquid is at the top, but in this case
the equilibrium is unstable ; for since p > p, it follows from (18)
that n- is negative and therefore n is of the form + t/3. Hence in
the beginning of the disturbed motion, the free surface is of the
form
77 = Ae*?* cos (mx e).
If the upper liquid is moving with velocity V, and the lower
with velocity V, the values of U or n/m are determined by the
quadratic (19); and the condition of stability requires that the
two roots of this quadratic should be real.
Putting k, k' for m coth mh and m coth mh r , (19) becomes
The condition that the roots of this quadratic in U should
be real, is
g (kp + k'p') (p - p'} - kk'pp' ( V - VJ > 0.
It therefore follows that if p > p, that is if the lower liquid
is denser than the upper liquid, the motion may be stable. But if
p > p] or if no forces are in action, so that g=0, the motion will
be unstable.
90. If no forces are in action, and both liquids are of unlimited
extent so that h = h' = oo , the equation for determining U becomes
p(V- C7) 2 + p'(F'-0 2 =0,
the roots of which are
B. H.
98 WAVES.
Hence U, and therefore n, is a complex quantity, and we may
therefore put
U = a. i ft = n/ra,
where a and /3 are determined from (21). If therefore the initial
form of the free surface is
77 = ae tma; ,
its form at any subsequent time may be written
where a'+ &' = a.
If there is no initial displacement, ^ = when t = 0, in which
case a'=b' = ^a. To express this result in real quantities, let
a = A iB, and (22) becomes
V) = {A cos m (x at} + B sin m (x at}} cosh mfit,
corresponding to an initial displacement
?7 = A cos mx + B sin mx.
91. When the initial velocity is zero there are three cases
worthy of notice.
(i) Let p = p, V = V, so that the densities of the two
liquids are equal, and their undisturbed velocities are equal and
opposite ; then from (21), a = 0, /3 = V, whence
77 = (A cos mx 4- B sin mx) cosh mVt.
(ii) Let p = p, V = 0, then a = F, /3 = |F, whence
77 = [A cos m (x \ Vt) + B sin m(x ^ Vt)} cosh \rnVt,
hence the waves travel in the direction of the stream and with
half its velocity.
(iii) Let p = p', V = V. In this case the roots are equal,
but the general solution may be obtained from (21) by putting
V = V(l +7), where 7 ultimately vanishes. We thus obtain
and therefore since 7 is small, (22) may be written
77 = e im <*- F <> {a + ^mVyt [a (l-i)- 2a 7 ]}.
Putting c = ^mVy {a (1~ t) 2a'}, this becomes
LONG WAVES. 99
Let a = A iB, c = C iD, then the real part is
77 = (A + Ct) cos m (x - Vt) + (B + Dt) sin m (a; - Vt),
corresponding to an initial displacement 77 = A cos mx + B sin mx.
If the initial velocity 7} is zero, (7 = mBV, _D = mJ.F, and
17 = (A + mBVt) cos m (x - Vt) + (B- mA Vt) sin m(x- Vt).
The peculiarity of this solution is, that previously to displace-
ment there is no real surface of separation at all. Hence if we
have a thin surface dividing the air, such as a flag whose inertia
may be neglected, it appears from the last equation that (neglecting
changes in the density of the air), the motion of the flag will be
unstable and that it will flap.
Long Waves in Shallow Water.
92. In the theory of long waves it is assumed, that the lengths
of the waves are so great in proportion to the depth of the water,
that the vertical component of the velocity can be neglected, and
that the horizontal component is uniform across each section of the
canal. In 82 we saw that if the depth is small compared with
the wave-length, then U~ gh, provided the square of the velocity
is neglected. We shall now examine this result in connection with
the above-mentioned assumption.
Let the motion be made steady by impressing on the whole
liquid a velocity equal and opposite to the velocity of propagation
of the waves. Let 77 be the elevation of the liquid above the
undisturbed surface ; IT, u the velocities corresponding to h and
h + i) respectively. The equation of continuity gives
u
whence U z -u 2 = U' 2 (2/177 + V 2 )/(h + t)) 2 .
If Bp be the excess of pressure due to the wave motion
When t]/h is very small, the quantity in brackets is U 2 /h g ;
whence if U 2 = gh, the change of pressure at a height h + 77 vanishes
to a first approximation, and therefore a free surface is possible.
If the condition U 2 = gh is satisfied, the change of pressure to
a second approximation is
72
100 WAVES.
which shows that the pressure is defective at all parts of the wave
at which 77 differs from zero. Unless therefore rf can be neglected,
it is impossible to satisfy the condition of a free surface for a
stationary long wave ; in other words, it is impossible for a long
wave of finite height to be propagated in still water without change
of type. If however 77 be everywhere positive, a better result can
be obtained with a somewhat increased value of U\ and if 77 be
everywhere negative, with a diminished value. We therefore infer
that waves of elevation travel with a somewhat higher, and waves
of depression with a somewhat lower, velocity than that due to
half the undisturbed depth 1 .
93. The theory of long waves in a canal may be investigated
analytically as follows 2 .
Let the origin be in the bottom of the liquid, h the undisturbed
depth, 77 the elevation ; and let x be the abscissa of an element of
liquid when undisturbed, the horizontal displacement. The
quantity of liquid originally between the planes x and x + dx is
hdx ; at the end of an interval t, the breadth of this stratum is
dx (1 + d^jdx), and its height is h + 77, whence the equation of
continuity is
(I + d%fdx)(h + ' n ) = h .................. (23).
Let us now investigate the motion of a column of liquid
contained between the planes whose original distance was dx ; and
let us suppose that in addition to gravity, small horizontal and
vertical disturbing forces X and Fact. Since the vertical accelera-
tion is neglected, the pressure will be equal to the hydrostatic
pressure due to a column of liquid of height h + 77, whence
r
J y
Ydy ............ (24).
The equation of motion of the stratum is
Now from (24),
dp _ dtj v dt} C h+r >dY
1 Lord Rayleigh, "On Waves," Phil. Mag. April, 1876.
2 Airy, "Tides and Waves," Encyc. Met.
LONG WAVES. 101
also in most problems to which the theory applies, the last two
terms on the right-hand side of (26) are very much smaller than
the first, and may therefore be neglected, whence (25) becomes
Substituting the value of rj from (23) we obtain
For a first approximation, we may neglect squares and products
of small quantities, and (23) and (27) respectively become
(28),
In order to solve (29) when X = 0, assume ^ = e l(Ma; ~ nt) ,and we
obtain n/m = (gh)^, which shows that the velocity of propagation
is equal to (gh)*.
Stationary Waves in Flowing Water 1 .
94. Let us suppose that water is flowing uniformly along a
straight canal with vertical sides, and that between two points A
and B there are small inequalities, and that beyond these points
the bottom is perfectly level. Let a be the depth, u the velocity,
p the mean pressure beyond A ; b the depth, v the velocity, and q
the mean pressure beyond B : also let f be the difference of levels
of the bottom at A and B.
The total energy of the liquid per unit of the canal's length
and breadth, at points beyond B, is
Cb
% v 2 b + g I ydy + w = ^ (v- + gb) b + w,
Jo
where w is the wave energy, and the density of the liquid is taken
as unity. At very great distances beyond B the wave motion will
have subsided and w will be zero.
The equation of continuity is
au = bv = M (30).
* Sir W. Thomson, Phil. Mag. (5) vol. xxn. 353.
102 WAVES.
The dynamical equation is found from the consideration, that
the difference between the work done by the pressure p upon the
volume of water entering at A, and the work done by the pressure
q at B upon an equal volume of water passing away at B, is equal
to the difference between the energy which passes away at B, and
the energy which enters at A. Whence
fa+f
pau qbv = (^v~b + \gH z + w)v (^u 2 a + g I ydy) u,
which by (30) becomes,
p-q = ^- + ^gb + w/b-^-g(f+^a) ...... (31).
Now p and q are the mean pressures, and therefore since the
pressure at the free surface is zero,
p = \ga, q=igb + w'/b,
where w' denotes a quantity depending on the wave disturbance ;
whence (31) becomes
| M* (a 2 - 6 2 )/a 2 6 2 -g(a-b +/) + (w - w')/b - ...... (32).
If we put
D 3 = 2a 2 6 2 /(a + &)> M=VD-,
D will denote a mean depth intermediate between a and 6, and
approximately equal to their arithmetic mean when their differ-
ence is small in comparison with either ; and V will similarly
denote a corresponding mean velocity of flow. We thus obtain
from (32)
_f-(w-w')/gb
-
If b a were exactly equal to f, and there were no disturbance
of the water beyond B, the mean level of the water would be the
same at great distances beyond A and B ; but if this is not the
case, there will be a rise or fall of level, determined by the formula
V*flgD + (w-w')/gb
l-V'/gD
Let us now suppose that between A and B there are various
small inequalities ; each of these inequalities will produce small
waves whose nature is determined by the form of the functions w,
w'; hence w and w will both be small quantities and the sign of
y will be independent of that of w w'. Now / is positive or
negative according as the bottom at A is higher or lower than the
STATIONARY WAVES IN FLOWING WATER. 103
bottom at B. Hence if V- < gD the upper surface of the water
rises when the bottom falls, and falls when the bottom rises ; and
the converse is the case when V 2 > gD.
Theory of Group Velocity.
95. When a group of waves advances into still water, it is
observed that the velocity of the group is less than that of the
individual waves of which it is composed. This phenomenon was
first explained by Sir G. Stokes 1 , who regarded the group as
formed by the superposition of two infinite trains of waves of
equal amplitudes and nearly equal wave-lengths, advancing in the
same direction.
Let the two trains of waves be represented by cos k ( Vt x)
and cos k' (Vt x) ; their resultant is equal to
cos k(Vt-x)+ cos k' (V't-x)=2 cos \ {(k f V-kV)t-(k'- k) x}
x cos {(k' V' + kV)t-(k' + k) x}.
If k' k, V V be small, this represents a train of waves
whose amplitude varies slowly from one point to another between
the limits and 2, forming a series of groups separated from one
another by regions comparatively free from disturbance. The
position at time t of the middle of the group, which was initially at
the origin, is given by
(k'V-kV)t-(k'-k)x = 0,
which shows that the velocity of propagation U of the group is
U = (k'V-kV)/(k'-k).
In the limit when the number of waves in each group is
indefinitely great we have k' = k + 8k, V=V+SV, whence
n _d(kV}
~~dk
Capillary Waves.
96. Most liquids which are incapable of remaining perma-
nently mixed, exhibit a certain phenomenon called capillarity 2 ,
1 Smith's Prize Examination, 1876; and Lord Bayleigh, "On Progressive
Waves " ; Proc. Land. Soc. vol. ix.
3 The reader who desires to study the theory of Capillarity is recommended
104 WAVES.
when in contact with one another. This phenomenon can be
explained by supposing that the surface of separation is capable
of sustaining a tension, which is equal in all directions, and is
independent of the form of the surface of separation.
The surface tension depends upon the nature of both the
liquids which are in contact with one another. Thus at a tem-
perature of 20 C., the surface tension of water in contact with
air is 81 dynes per centimetre; whilst the surface tension of
water in contact with mercury is 418 dynes per centimetre.
The surface tension diminishes as the temperature increases ;
also a surface tension cannot exist at the common surface of two
liquids, such as water and alcohol, which are capable of becoming
permanently mixed.
97. We shall now consider the effect of surface tension upon
the propagation of waves.
. Let T be the surface tension, and let p and p + 8p be the
pressures just outside and just inside the free surface of a liquid;
then
8p/p + g = ..................... (33).
But if we resolve the forces which act upon a small element
8s of the free surface vertically, and neglect the vertical accelera-
tion, and put 8% for the angle which Ss subtends at the centre of
curvature, we obtain
SpSs = TSx>
whence 8p=T-^.
as
Now -j- = cot Y,
dx
d*r) dy
therefore -r{ = cosec 2 y -f* .
dx* * ds
Since ^ is nearly equal to ^TT, we may put cosec %=1, and
ds = dx, whence
to consult Chapter xx. of Maxwell's Heat ; and also the article on Capillarity in the
Encyclopedia Britannica by the same author.
A table of the superficial tensions of various liquids will be found in Everett's
Units and Physical Constants, p. 49.
CAPILLARY WAVES. 105
Substituting in (33), differentiating the result with respect to
t, and remembering that 17 = dfyjdz, and that dty/dx 2 =
we obtain
_ .................. (34).
p dz 3 y dz dt 2
This is the condition to be satisfied at the free surface.
98. We shall now apply the preceding result to determine
the capillary waves propagated in an ocean of depth h.
Let = A cosh m(z + h) cos (mx nt).
Substituting in (34) we obtain
Tm 3 /p + mg = n 2 coth mh,
whence
U* = n*/m? = (g\f^TT + 2irT/p\) tanh 2irhj\ ...... (35).
Equation (35) determines the velocity of propagation corre-
sponding to a given wave-length.
99. Let us now suppose that the depth h is so great that mh
may be treated as infinite ; then coth mh = + 1 according as m is
positive or negative. Hence it will be sufficient for our purpose
to discuss the equation
Tm s /p+mg = n' 2 ........................ (36),
when m is positive.
When ra = 0, ?i = 0; and as m increases from zero to infinity,
the value of rt 2 is always positive, and consequently periodic
motion is possible for all values of m, that is for all wave-lengths.
If a given value be assigned to n, (36) is a cubic for deter-
mining m. The real root is obviously positive ; and since the
discriminant 1 of the cubic is positive, the other two roots are
complex. Hence there is only one wave-length corresponding
to any given period.
1 It can be shown by means of Taylor's theorem, that, if f(x) be any rational
algebraic function of x, the condition that the equation f(x)=0 should have a pair
of equal roots is obtained by eliminating x between f(x) = and /' (x)=0. The
result of the elimination is a certain function of the coefficients which is called the
discriminant of f(x) ; and it is shown in treatises on Algebra that two of the roots
of a cubic will be real, equal or complex, according as the discriminant is negative,
zero or positive. If the cubic be Ax 3 + Bx 2 + Cx + D, the discriminant is
3 - IS A BCD -
106 WAVES.
If U be the velocity of propagation, n = mU; whence (36)
becomes
Tm?/p-mU 2 + g = ..................... (37).
Hence U = oo , when m = and m = oo ; and therefore U must be
a minimum for some intermediate value of m, which by ordinary
methods can be shown to be given by the equation U 2 =2TrH/p.
By means of this result, the minimum velocity of propagation
and the corresponding wave-length can be shown to be given by
the equations
U=(4Tg/pp, \ = 2ir(T/gp} ............... (38).
A wave whose length is less than the preceding critical value
of X is called by Lord Kelvin a ripple 1 . Now if we write (36)
in the form
U* = gX/2Tr + 27rT/p\ .................. (39),
it follows that when X is given by (38) both terms on the right-
hand side are equal; also for long waves the first term is the
most important, whilst for short waves the second is the most
important. Hence the effect of gravity is most potent in pro-
ducing long waves, and the effect of surface tension in producing
ripples.
100. In 85 we have considered the propagation of waves at
the surface of separation of two liquids, which are moving with
different velocities. We shall now consider the production of
ripples by wind blowing over the surface of still water.
Let V be the velocity of the wind, which is supposed to be
parallel to the undisturbed surface of the water, cr the density of
air referred to water.
Since the changes of density of the air are very small in the
neighbourhood of the water, the air may approximately be regarded
as an incompressible fluid, whence if the accented letters refer to
the water, the kinematical conditions at the boundary give
= Vx + a(U-V) + $ { V- am ( U - F) sin (mas - nt)} 2 - % V 2 = 0,
or &p + acr {g + n ( U F) m ( U V )} sin (mac nt) = 0.
Similarly
Sp' + (g Uri) a sin (mac nt) = 0,
whence (40) becomes
Tm*-mU*-o-m(U-V)* + (l-a-)g = Q ......... (41).
Putting U=n/m this may be written
Tm*-
- -"- ............ (46) -
the value of n will be complex, and periodic motion will be
impossible. Equation (44) gives the minimum value of W ; hence
in order that wave-motion may be possible for waves of all lengths,
we must have
When (47) is satisfied, (46) may be written
A
1 + 0-- (1
We shall now discuss this equation.
108 WAVES.
Case(i).
In this case one of the values of U is positive and the other
negative ; hence waves can travel either with or against the wind.
Moreover since the positive value is numerically greater than the
negative value, waves travel faster with the wind than against
the wind ; also the velocity of waves travelling against the wind
is always less than W.
Case(ii). V> W V(l + W; when V > 2 W this velocity is < W; and
when V=ZW, the velocity of waves travelling with the wind is
undisturbed.
EXAMPLES.
1. A liquid of infinite depth is bounded by a fixed plane
perpendicular to the direction of propagation of the waves. Prove
that each element of liquid will vibrate in a straight line, and
draw a figure representing the free surface and the direction of
motion of the elements, when the crest of the wave reaches the
fixed plane.
2. Prove that the velocity of propagation of long waves in a
semicircular canal of radius a and whose banks are vertical, is
3. If two series of waves of equal amplitude and nearly equal
wave-length travel in the same direction, so as to form alternate
lulls and roughness, prove that in deep water these are propagated
with half the velocity of the waves; and that as the ratio of the
depth to the wave-length decreases from oo to 0, the ratio of the
two velocities of propagation increases from to 1.
4. If a small system of rectilinear waves move parallel to and
over another large rectilinear system, prove that the path of a
particle of water is an epicycloid or hypocycloid, according as the
two systems are moving in the same or opposite directions.
EXAMPLES. 109
5. A fine tube made of a thin slightly elastic substance is
filled with liquid; prove that the velocity of propagation of a
disturbance in the liquid is (XO/ap)^ where a is the internal
diameter of the tube, its thickness, X the coefficient of elasticity
of the material of which it is made, and p the density of the liquid.
6. A horizontal rectangular box is completely filled with
three liquids which do not mix, whose densities reckoned down-
wards are cr l , a 2 , a s> and whose depths when in equilibrium are
l-i, h, 1 3 respectively. Show that if long waves are propagated at
their common surfaces, the velocity of propagation V must satisfy
the equation
/^ + /r, we obtain
112 RECTILINEAR VORTEX MOTION.
Substituting from (2) and (4) in (1), we obtain
_ __^^ = Q (5)
dt dy dx dx dy)
From this equation it follows that the molecular rotation is
not a quantity which can be chosen arbitrarily; for ty must
satisfy (5), and the corresponding value of is then determined
by (4).
When the motion is steady, none of the quantities are functions
of t, and we obtain from (1) by Lagrange's method
2f = *"(*),
where F is an arbitrary function, which agrees with 24. The
pressure is determined by (31) of the same section.
The current function at all points of the rotationally moving
liquid is now determined by the equation
V 2 ^+ J P'(^) = ........................ (6).
At every point of the irrotationally moving liquid which sur-
rounds the vortices, f = 0, and therefore
(7)
da? dy 2
Equations (4) and (7) show, that ty is the potential of in-
definitely long cylinders composed of attracting matter of density
/27r, which occupy the same positions as the vortices.
102. The integral fftdxdy is called the vorticity of the mass
of rotationally moving liquid ; and we shall now show that the
vorticity is an absolute constant.
Draw any closed curve which completely surrounds all the
rotationally moving liquid and does not cut any of it. Then
since is zero at all points of the liquid where the motion is irro-
tational, it follows that if r denote the vorticity,
T = ftfdxdy,
where the integration extends over the whole area enclosed by
the curve. Substituting the value of from (3) we obtain by
Green's theorem,
= f(udx + vdy),
SINGLE CIRCULAR VORTEX. 113
where the line integral extends round the closed curve. By | 27,
this line integral is equal to the circulation K due to the whole of
the rotationally moving liquid within the curve ; and by the same
article, the circulation has been shown to be constant. Hence the
vorticity is constant and equal to half the circulation.
103. We shall now consider the steady motion in an infinite
liquid of a single rectilinear vortex, whose cross section is a circle
of radius a, and whose molecular rotation is constant.
In order that the cross section may remain circular, it is
necessary that -^r should be a function of r alone.
Denoting the values of quantities inside the vortex by ac-
cented letters, equations (4) and (7) become
?=0 ..................... (8),
dr* r dr
which gives the value of ty inside the vortex, and
^ + 1^ = (9)
dr* + r dr
which gives the value outside.
The complete integrals of (8) and (9) are
and ty = G log r + D.
Now T|/ must not be infinite when r= 0, and therefore A = ;
also at the boundary of the vortex, where r = a,
ty' = i/r, dty'/dr = d^Jr/dr ;
whence B- a 2 = (7 log a + D
-fa=0,
and therefore C = - 0? = - O-/TT = - mjir,
where cr is the area of the cross section, and ra is the vorticity of
the vortex. The constant D contributes nothing to the velocity,
and may therefore be omitted, whence
i^ = K(a-r)-(m/ir)loga ............... (10),
ty= (w/7r)logr ........................... (11).
B. H. 8
114 RECTILINEAR VORTEX MOTION.
Now - dtyjdr is the velocity perpendicular to r, whence inside
the vortex
& ........................ (12),
which vanishes when r 0, and outside
- d-^r/dr = m/7rr ........................ (13).
Hence a single vortex whose cross section is circular, if existing
in an infinite liquid, ivill remain at rest and will rotate as a rigid
body. It ivill also produce at every point of the irrotationally
moving liquid with which it is surrounded, a velocity which is
perpendicular to the line joining that point with the centre of its
cross section, and which is inversely proportional to the distance
of that point from the centre.
104. Outside the vortex, where the motion is irrotational, a
velocity potential of course exists. To find its value we have
d(f> d-^r my d _ dty _ mx
dx dy TiT 2 ' dy dx -rrr 2 '
whence d> = - \ y x ~ x y = (mlir) tan" 1 ylx .......... (14).
TT J x 2 + y 2
It therefore follows that > is a many-valued function, whose
cyclic constant is 2m. The circulation, i.e. the line integral
f(udx + vdy), is zero when taken round any closed curve which
does not surround the vortex, and is equal to 2m when the curve
surrounds the vortex ; whence if K be the circulation, m = $K, and
the values of and -fy* may be written
< = (/c/2?r) tan" 1 yjx, ty = (/c/2-Tr) log r.
105. The investigations of the last two articles are kinematical ;
we shall now calculate the value of the pressure within and without
the vortex.
Let the values of the quantities inside the vortex be distin-
guished from those outside by accented letters.
Outside the vortex
p/p=C--$q*,
and since < = 0, and q = m/Trr = /c/2?rr, we obtain
p/p = C- K 2 /87rV 2 ..................... (15),
SINGLE CIRCULAR VORTEX. 115
whence if II be the pressure at an infinite distance
n K 2
p p S-TrV 2
The equation of motion inside the vortex is
(16).
p dr r 4-7T 2 a 4 '
t) /eV 2 P
whence = h (17),
p 87r 2 a 4 p
where P is the pressure at the centre of the vortex.
At the surface of the vortex where r = a, p =p' , whence
P/p = U/p - 2 /47r 2 a 2 (18),
ft II 2 / 1 st \
and therefore = - (1 -(19)-
p p 47r 2 a 2 V 2a 2 /
Hence if II < /c 2 p/47r 2 a 2 ,
p' will become negative for some value of r < a, which shows that
a cylindrical hollow will exist in the vortex, which is concentric
with its outer boundary.
When there is no hollow, equations (16) and (19) show that
the pressure is a minimum at the centre of the vortex, where it is
equal to II /c 2 p/47T 2 a 2 , and that it gradually increases until the
surface is reached, at which it is equal to II 2 p/87T 2 a 2 , and that
it then continues to increase to infinity, where its value is II.
It is also possible to have a hollow cylindrical space, round
which there is cyclic irrotational motion. Such a space is called
a hollow vortex. The condition for its existence requires that p =
when r = a, and therefore by (16)
H = K 2 />/87r 2 a 2 .
This equation determines the value of the radius of the hollow,
when the pressure at a very great distance is given.
106. Kirchhoff has shown that it is possible for a vortex
whose cross section is an invariable ellipse, and whose molecular
rotation at every point is constant, to rotate in a state of steady
motion in an infinite liquid, provided a certain relation exists
between the molecular rotation and the angular velocity of the
axes of the cross section.
82
116 RECTILINEAR VORTEX MOTION.
The current function is evidently equal to the potential of an
elliptic cylinder of density f/271-. Let a and b be the serai-axes of
the cross section, then the value of -fr inside the vortex may be
taken to be
+' = D - (A* + By*)l(A + B},
where A, B, D are constants, for this value of ty' satisfies (4).
Let x = c cosh 77 cos %,y = c sinh 77 sin , where c = (a 2 ft 2 )*, and
let 77 = ft at the surface ; the value of ty' becomes
TJT = D - fc s (A cosh 2 1] cos 2 + B sinh 2 rj sin 2 )/( J.
Also let the value of ty outside the vortex be
ijr = A'e~*> cos 2f +
When 77 = /3, we must have
tyty' = const., d-^rfdr)
Therefore A 'e~* = - i fc 2 ( J. cosh 2 /3 - 5 sinh 2 yS)/(^l + J5)
and A'e-v = $&* (A - B) sinh cos
Whe: ^'fa - bY- - &( A <*
2(A+
Therefore Aa = Bb and
tf = D- Z(bx> + ay*)/(a + b).
Let a> be the angular velocity of the axes ; u, v the velocities
of the liquid parallel to them, then
(a + b),
(a + b).
The boundary condition is
.dF .dF
*a? + *^.-*
where F = (a;/a) 2 + (y/6) 2 -1 = 0. Whence
therefore w = 2ab^/(a + 6) 2 .
We therefore obtain
x = aooy/b, y = bwxja,
the integrals of which are
* = La cos (tat + a), y = Lb sin ( (^ + f) - \M log {f +(as- c) 2 } [f + (a + c) 2 } = A.
109. If two opposite vortices of vorticities ra and m are
present in the liquid, the vortices will move perpendicularly to the
line joining them with velocity M/2c, where 2c is the distance
between them.
1 Greenhill, "Plane Vortex Motion," Quart. Journ. vol. xv. p. 20.
MOTION OF TWO VORTICES. 119
In this case there is evidently no flux across the plane which
bisects the line joining the vortices, and which is perpendicular to
it ; we may therefore remove one of the vortices and substitute
this plane for it. Hence a vortex in a liquid which is bounded by
a fixed plane will move parallel to the plane, and the motion of
the liquid will be the same as would be caused by the original
vortex, together with another vortex of equal and opposite vorticity,
which is at an equal distance and on the opposite side of the
plane.
This vortex is evidently the image of the original vortex, and
we may therefore apply the theory of images in considering the
motion of vortices in a liquid bounded by planes.
110. If there is a vortex at the point (x, y) moving in a
square corner bounded by the planes Ox, Oy, the images will
consist of two negative vortices at the points ( x, y}, (x, y), and
a positive vortex at the point ( x, y)\ for if these vortices be
substituted for the planes, their combined effect will be to cause
no flux across them.
Since the vortex is incapable of producing any motion of
translation upon itself, its motion will be due solely to that pro-
duced by the combined effect of its images ; whence,
M My Mx-
x =
2y 2 (a? + f) 2y (a? + y 2 ) '
M MX Mtf
2x 2(x 2 + y 2 ) 2x (x 2 + y 2 ) '
therefore x/x 3 + yjy 3 = 0,
whence a 2 (x 2 + y-) = x 2 y 2 ,
or r sin 20 = 2a,
120 KECTILINEAR VORTEX MOTION.
which is the curve described by the vortices. The curve in
question is the reciprocal polar with respect to its centre of a
four-cusped hypocycloid ; but it also belongs to the class of curves
called Cotes' Spirals, which are the curves described by a particle
under the action of a central force varying inversely as the cube
of the distance. Since
the vortex describes the spiral in exactly the same way as a particle
would describe it, if repelled from the origin with a force
111. The method of images may also be applied to determine
the current function due to a vortex in a liquid, which is bounded
externally or internally by a circular cylinder.
Let H be the vortex, a the radius of the cylinder, OH = c ; and
let S be a point such that OS=f=a 2 /c, then the triangles SOP
and POH are similar, therefore
SPO = OHP,
OPH = OSP,
also OSP + SPA = OAP = OP A
= OPH + HPA,
therefore SPA = HP A.
Let us place another vortex of equal and opposite vorticity
at S, then the velocity along OP due to the two vortices is
~s
But sin HPO sin HPO
sin SPO ~ sin OHP
= OH/a
= HP ISP,
hence u = and there is no flux across the cylinder.
VORTEX IN A CIRCULAR CYLINDER. 121
Hence the image of a vortex inside a cylinder is another vortex
of equal and opposite vorticity situated on the line joining the
vortex with the centre of the cylinder, and at a distance 2 /c fro m
the centre, and the vortex will describe a circle about the centre
with a velocity
The current function of the liquid at a point (r, 6) within the
cylinder is
r 2 + c 2 - 2rc cos 6
When the vortex is situated outside the cylinder, the image
consists of a vortex of equal and opposite vorticity at H, together
with a vortex of equal vorticity at 0. The latter vortex does not
produce any alteration in the normal velocity at the surface of the
cylinder, and its existence arises from the fact that the circulation
round any closed curve which surrounds the cylinder must remain
unaltered. The circulation due to vortex at H is 2irM, whilst
that due to the vortex at is 2-Tr.lf, so that the two circulations
cancel one another.
112. We have shown that the velocity potential due to a
source is m log r ; hence if we have a combination of a source of
strength m, and a vortex of vorticity m , the velocity potential due
to the two is
< = m log r + M tan" 1 y/x,
where M = m'/Tr. Whence
mac My _ my + MX
''~ '
An arrangement of this kind is called Rankines free spiral
vortex.
In order to find the stream lines let us transfer to polar co-
ordinates, and we find
dr_m d6 _M
dt~r' T dt~^'
whence if m/M = a, we obtain
r = Ae a9 ,
and therefore the stream lines are equiangular spirals.
122 RECTILINEAR VORTEX MOTION.
113. We shall conclude this Chapter by proving three funda-
mental properties of vortex motion.
We have defined a vortex line to be a line whose direction
coincides with the direction of the instantaneous axis of molecular
rotation. If through every point of a small closed curve a series of
vortex lines be drawn, they will enclose a volume of fluid which
may be called a vortex filament, or shortly a vortex.
We have shown that if the forces which act on the fluid have a
potential, and the density is a function of the pressure, the motion
of the fluid constituting the vortex can never become irrotational.
It will now be shown that every vortex possesses the following
three fundamental properties :
(i) Every vortex is always composed of the same elements of
fluid.
(ii) The product of the molecular rotation of any vortex into
its cross section is constant with respect to the time, and is the same
throughout its length.
(iii) Every vortex must either form a closed curve, or have its
extremities in the boundaries of the fluid.
To prove the first proposition, let P and Q be any two adjacent
points on a vortex, co the molecular rotation at P. Then by the
definition of a vortex line, PQ is the direction about which the
rotation co takes place.
Let P', Q' be the positions of P and Q at the end of an interval
Bt ; then we have to show that P'Q' is the instantaneous axis of
rotation at P'.
Let x, y, z be the coordinates of P ; u, v, w the velocities of
the element of fluid which at time t is situated at P.
If PQ = h, the coordinates of Q are evidently
x + h/co, y + hrj/co, z + h^/co ;
also since u = F (x, y, z, t), it follows that if u , v l , w be the
velocities of Q,
u l = F(x + kg/to, y + hy/co, z + h/co, t)
h / ..du du . ^du\
co \ dx dy
hp
co
by 20.
PROPERTIES OF VORTEX MOTION. 123
The coordinates of P' are
x + uSt, y + v&t, 2 + w8t,
and those of Q' are
- -
(p dt
= x + u8t +
hp?
wp
by (20), where />' is the density, and ', ?/, ' are the components
of molecular rotation at P'.
Hence if h' denote the length of P'Q', and X', //, v its direction
cosines, then
\'h' = hpg'ftop', /j.'h' = hprj'/wp', v'ti = hp^'/wp (21),
whence X'/f = pf^' = i/'/f,
which shows that P'Q' is the instantaneous axis of rotation at P',
and therefore P'Q' is the element of the vortex line, which at time
t occupied the position PQ. This proves the first theorem.
To prove the second theorem, square and add (21) and we
obtain
h' = hpw'/wp'.
But since the mass of the element is constant
phcr = p'h'a',
whence cos edS = 0,
where e is the angle between the axis of rotation and the normal
to S drawn outwards.
Now if we choose S so as to coincide with the surface of any
finite portion of a vortex of small section, together with its two
124 RECTILINEAR VORTEX MOTION.
ends, cos e vanishes except at the two ends ; and is equal to + 1 at
one end, and 1 at the other ; hence
&> l rf be the angle which any element 8s makes with the axis of
x, the equation of motion is
pySs = ~ (T, sin >) Bs + pYSs.
Now sin < = dy/ds ; also since the displacement y is small, the
curvature will also be small, and we may therefore put ds = dx.
The tension T^ may also be regarded as constant throughout the
length of the string, whence the equation of motion becomes
d^_T 1 d-y
dt* ~ p dx^
If the motion does not take place in a plane, we may resolve
the displacements and forces into two components respectively
parallel to the axes of y and z, and we shall thus obtain a second
138 VIBRATIONS OF STRINGS AND MEMBRANES.
equation of the same form as (1), in which z, Z are written for y,
Y respectively.
125. Let us now suppose that the length of the string is
equal to I, and that there are no impressed forces ; also let
tf^TJp (2).
Equation (I) now becomes
-
*i- B daf
To solve this equation, assume
y = F(x)e an
Substituting in (3) we obtain
the solution of which is
f = sin mx + D cos mx.
The solution of (3) may therefore be written in the form
y=^(G sin mx + D cos moo) e*"* 1 * ............ (4),
where m is at present undetermined, and C and D are complex
constants.
The value of m will depend upon the particular problem under
consideration. We shall now suppose that both ends of the string
are fixed ; in this case the conditions to be satisfied at the fixed
ends are, that y and y should vanish when x = and x = I. These
conditions evidently require that
D = 0, sin ml = ;
from the last of which we deduce
m = STT/l,
where s is a positive integer. Writing G = A t,B and rejecting
the imaginary part, the solution becomes
00
y = S (A g cos STrat/l + B s sin STrat/l) sin siroc/l ...... (5),
and therefore the period r g of the sth component is given by
_2Z_2^ /p
~m~7V 2V
and the frequency
FREQUENCY OF VIBRATION. 139
The gravest note corresponds to s=l, and therefore its
frequency is
f-L fL
fl ~ 21 V P
From these results we draw the following conclusions.
(i) The frequency is inversely proportional to the length;
and therefore if the string be shortened, the pitch of the note will
rise, and conversely if the string be lengthened, the pitch will fall.
We thus see why it is that in playing a violin different notes can
be obtained from the same string.
(ii) The frequency is proportional to the square root of the
tension, accordingly if the string be tightened the pitch will
rise.
(iii) The frequency is inversely proportional to the square
root of the density ; and therefore if two strings having the same
lengths, cross sections and tensions, be made of catgut and metal
respectively, the pitch of the note yielded by the catgut string
will be higher than that yielded by the metal string ; also the
pitch of the note yielded by a thick string will be graver than
that of the note yielded by a thin string, of the same material,
length and tension.
If s be any integer other than unity, we learn from (5) that
the displacement is zero at all points for which x = rljs, where
r = 1, 2, 3, ... s I ; it therefore follows that, corresponding to the
sth harmonic, there are s 1 points situated at equal intervals
along the string, at which there is no motion. These points are
called nodes.
126. The constants A and B depend upon the initial
circumstances of the motion. Now the motion of dynamical
systems of which a string is an example may be produced either
by displacing every point in any arbitrary manner, subject to the
condition that the connections of the system are not violated;
or by imparting to every point an arbitrary initial velocity,
subject to the same condition. Hence the most general possible
motion is obtained by communicating to every point of the
string an initial displacement, and an initial velocity. We shall
now show that when the initial displacements and velocities are
given, the constants A and B are completely determined.
140 VIBRATIONS OF STRINGS AND MEMBRANES.
Let y , yo be the initial displacements and velocities. Then it
follows from (5) that
y = 2 t A s sin STTX/I ..................... (6),
y = ^ 1 (sTra/l) B t sin STrxfl ............ (7).
ri
Now the integral I sin (STTX/I) sin (s'lrxjl) dx is equal to
Jo
zero
if s and s' are different integers, and is equal to ^l if s = s';
whence multiplying (6) by sin STTX/I and integrating between the
limits I and 0, we obtain
S7TX ,
--dx ..................... (8).
2 f l
8 = |J
Similarly from (7)
2 f l . . STTX ,
I y sin j ax ............ ..... ( ( J ).
Jo I
Since y , y are given functions of x, these equations completely
determine the constants. We notice that B s is zero when the
initial velocity is zero, and that A g is zero when there is no
initial displacement.
127. As an example of these formulae, let us suppose that a
point P, whose abscissa is b, of a string fixed at A and B, is
displaced to a distance 7 and then let go.
From x = to x = b, y = yx/b, and therefore for this portion
of the string
. , 27 f b . STTX , 27 / b s-rrb I . S7rb\
A * = TT I x sm ~T dx = IT --- cos 1- + 1 -, sm -T I-
bl J o I b \ S7T I S 2 7T 2 I /
From x = b to x = l, y = y (I x)/(l b), whence
27 [ l . STTX .
A > *fip%) > s sin STTX/I ..................... (18),
where s is a function of the time which satisfies (14), and whose
value is therefore determined by (17). The quantities denoted
by < g are called normal coordinates ; and we shall now prove
that the expressions for the kinetic and potential energies do
not contain any of the products of the normal coordinates. This
is the characteristic property of these quantities.
If T be the kinetic energy, we have
Cl [I oo .
T = %p I y z dx = ^p I {S (f> s sin S7rx/l} 2 dx.
Jo Jo 1
KINETIC AND POTENTIAL ENERGY. 145
Since all the products vanish when integrated between the
limits, we obtain
T=lpl i & ........................ (19).
The potential energy is equal to the work done in displacing
the string to its actual position. In order to calculate its value,
let the string be held in equilibrium in its actual configuration at
time t by means of a force Y applied at every point of its length.
The value of this force per unit of mass is equal to
_Tidfy
p da?'
by (1). Let SF be the work which must be done by this force
in order to displace every element of the string through a space
By ; then the work done upon an element Bs
and therefore since Bs = 8x, the whole work done is
Integrating by parts, and recollecting that By = at both ends,
we obtain
g r<^ ry^y
'Jo eta dx 'Jo \dooj
ri /v7x + -=- Tex -3- 8x,
dx\ * dxj dy \ dy)
which becomes
( }>
if = Tlp.
135. If the boundary of the membrane consists of a rectangle,
whose sides are the axes and the lines ac = a, y = b, we may assume
as a particular solution of (22),
w = A sin mirx/a sin vnry/b cos pt ............ (23),
where p* = cV 2 (m*/a? + ri*/b 2 ) ........................ (24),
m and n being any integers ; for this expression satisfies (22) and
also makes w vanish at the boundaries. Equation (24) determines
the frequency of the different notes ; and from (23) we see that
the nodal lines (i.e. the lines of no motion) consist of a system
of n 1 lines parallel to x, whose distances apart are b/n, together
with m 1 lines parallel to y, whose distances apart are equal to
a/m.
If the membrane be square, a = b, and (23) and (24) become
w = A sin rrnrx/a sin mry/a cos pt,
2> = C7r(m 2 +w 2 )Va.
The gravest note is obtained by putting m = n = 1, and
corresponding to this note there are no nodes.
In the next place we shall determine the nodal lines cor-
responding to vibrations whose frequency is ^c^/5/a.
Here V5 = V( 2 + w 2 ),
which requires that m = 2, n = l or ra = l, ?i = 2; and therefore
the complete vibration corresponding to this period is,
w = (C sin 27rx/a sin Try fa + D sin TTX/O, sin 2,7ryfa) cos pt.
In this expression C and D depend solely upon the initial
circumstances of the motion, and may have any values whatever
consistent with the boundary conditions. If however we suppose
102
148 VIBRATION'S OF STRINGS AND MEMBRANES.
that the initial conditions are such, that the ratio C/D has an
assigned value, we may obtain a variety of special cases.
(i) Let D = 0. The nodal system now consists of the line
x = $a, which bisects the membrane.
(ii) Let (7 = 0, and we have a nodal line y=^a, similarly
bisecting the membrane.
(iii) Let C = D; then the value of w may be written
w = 4(7 sin TTX/a sin Try /a cos \ir (x + y)/a cos TT (x y)/a cos pt.
This expression vanishes when,
x = a, y = a, sc + y=a, x y = a.
The first and second equations correspond to the edges ; the
fourth must be rejected, because it does not represent a line
drawn on the membrane ; and the third represents one of the
diagonals of the square.
Since a nodal line may be supposed to be rigidly fixed without
interfering with the motion, the preceding solution determines
the frequency of the gravest note of a right-angled isosceles
triangle.
(iv) Let G= D, and we shall find that the nodal line is
y = x, which represents the other diagonal of the square.
For further examples in this branch of the subject, the reader
is referred to Chapter IX. of Lord Rayleigh's treatise.
136. The motion of a circular membrane, which is the best
representative of a drum, cannot be solved by elementary methods.
The simplest case of all, is when the vibrations are symmetrical
with respect to the centre, so that (22) becomes
d?w _ 2 /d-w I dw
~ ~
,dr 2 r dr ,
and if we put w = F(r) &<*, the equation for F is
dW IdF
dr* r dr
This equation cannot be integrated in finite terms. The two
solutions are called Bessel's functions 1 , from the name of their
1 During recent years the ungrammatical phrase Bessel functions has begun to
creep into mathematical literature. The use of a proper noun as an adjective is a
violation of one of the most elementary rules of grammar ; and in cases where it is
not possible to form the corresponding adjective without introducing a cumbrous and
inelegant term, the genitive case of the proper noun ought always to be employed.
EXAMPLES. 149
discoverer; and the investigation of their properties constitutes
an important branch of analysis. Algebraic solutions may how-
ever be invented, by supposing that the density, and therefore c,
is a function of r.
EXAMPLES.
1. A string of length 1 + 1', is stretched with tension T
between two fixed points. The linear densities of the lengths I, I'
are ra, m' respectively; prove that the periods T of transverse
vibrations are given by
m'* tan (^Trlm^rT^ = m* tan
2. Investigate the motion of a string of length I, which is
initially at rest in a straight line, each extremity of which is
subject to the same obligatory motion y = k sin mat. Show
that if a sufficient period be allowed to elapse for the natural
vibrations to subside, the position of the nodes will be given by
the equation
2m# = ml + (2i + 1) TT,
where i is any integer.
3. A uniform string in the form of a circle of radius a, rests
on a smooth plane under a central repulsion, whose value at
distance r is ga n /r n . Show that if the string be slightly displaced,
so that it is initially at rest and in the form of the curve
r = a + 2 the area of the cross
section at P. Also let X, Y, L be the tangential and normal
components of the impressed forces and the couple at P, per unit
of mass, measured in the directions of T, N, 0.
The equations of equilibrium of the wire, are obtained by
resolving all the forces along the tangent and normal at P, and
taking moments about this point ; whence
152 FLEXION OF WIRES.
T - (T + ST) cos 80 + (N + 8N) sin 8 + aa>X8s = 0,
N-(T+ 8T) sin 8-(N+ 8N) cos 80 + - (1) -
= \- N = acoL
as
138. We must now find an expression for the flexural
couple G.
The curve which passes through the centre of inertia of each
cross section, is called the axis of the wire. When a wire is bent
in such a manner that its curvature is increased, the filaments
into which the wire may be conceived to be divided, which lie on
the outer side of the axis, will usually be extended ; and those
which lie on the inner side will usually be contracted, whilst the
axis itself undergoes no extension nor contraction. Cases of course
may occur, in which the axis undergoes extension or contraction,
and when this is the case, the difficulties of the problem are
greatly increased, and cannot be satisfactorily discussed without
a knowledge of the Theory of Elasticity. We shall therefore
confine our attention to the case in which the extension or
contraction of the axis is so small (if it exists), that it may be
neglected.
In the figure, let AE be the axis of the wire, PQ any filament,
whose distance from AE is h, the centre of
curvature at B; also let these points after deforma-
tion be denoted by accented letters.
It can be proved in a variety of ways 1 that
the tension T' at P' due to the action of con-
tiguous portions of the wire is equal to the
product of the extension of the element PQ and
a physical constant called Young's modulus. If
k is the resistance to compression, n the rigidity
of the substance of which the wire is made and q
See Thomson and Tait's Natural Philosophy, 6823.
BENDING MOMENT. 153
Young's modulus, it is shown in treatises on Elasticity that
q = 9nk/(3k + ri).
Hence if a- be the extension
T' = q* .............................. (2).
Now if p, p be the radii of curvature before and after de-
formation
P'Q'_p'
_= _
AB p A'E'~ p'
Since we assume that the extension of the axis may be
neglected, AB = A'B'; whence neglecting A 2 , &c.,
P'Q'-PQ ,/i i
and
p p
Accordingly
where /c 2 , whence the first two of (1) become
and therefore
-*-.
the integral of which is
T= C cos $ + D sin ,
whence N = G sin + D cos <.
Let t be the tension of the string, and a and TT a the values
of $ at the two extremities, then
t cos a = N= C sin a + D cos a,
t sin a = T, = C cos a + D sin a,
therefore (7=0, D = -t.
Writing A = q/c 2 a), and remembering that p = oo , since the
natural form of the wire is straight, we have G = A/p', whence
the third of (1) becomes
Integrating, we obtain
2A
^ tsm = E.
Since G = when > = a, E = t sin a,
whence if A/t = a* - = _ ... (5)>
d(f) (sm > - sin a)*
which determines the intrinsic equation to the curve.
We may also integrate (4) in a different manner, for if the
string be the axis of x, and its middle point be the origin,
cos = dy/ds,
THE ELASTICA. 155
and therefore (4) may be written
ds p
whence p'y = a 2 ; .............................. (6),
no constant being required because p'~ l = when y = 0.
AT , ds dy dy
Now ==ec
whence integrating (6) again
y z = 2a 2 (sin $ - sin a) (7).
The forms of the various curves which the wire is capable of
assuming, are shown in Thomson and Tait's Natural Philosophy,
Part n. p. 148.
If a lies between and TT, the maxima values of y are obtained
by putting = ^TT, and are therefore equal to + 2a sin (?r Ja).
The form of the curve is shown in the figures 1, 2 or 3 of that
work ; and if the curve be bent upon itself and the slight torsion
be neglected, the forms are shown in figures 4 and 5. In all these
cases except the first, in which the wire is bent into the shape
of a bow, the maximum value of y is numerically equal to its
minimum value. If, however, a lies between TT and 2?r, we may
put it equal to TT + /3, in which case (7) becomes
2/ 2 = 2a 2 (sin < + sin /3).
In this case the maximum value of y occurs when < = ^TT, and
is equal to 2a cos (^TT 1-/3), and the minimum when
the form of the curve is shown in fig. 7.
The constants a and a are capable of being determined when
the lengths of the wire and string are given ; and the equation of
the curve in Cartesian coordinates can also be obtained, but to do
this a knowledge of elliptic functions is required 1 . If, however,
a = |7r, the integral in an algebraic form can be obtained; for
since tan = dx/dy, (7) becomes
# =
= _ (4 a 2 _ y2)i + a log {2a/y + (4a 2 /y 2 - 1)*} + 0.
1 Greenhill, Mess. Math. vol. vm. p. 82.
156 FLEXION OF WIRES.
Since y = a^2 when x = 0, C=
whence x = (4a 2 - y^ - a V2 + a log
It will be noticed that this curve is the same as that described
by an elliptic cylinder, in the limiting case between oscillation
and rotation. See page 75.
141. Equation (5) enables us to prove a theorem discovered
by Kirchhoff, and which is known as Kirchhoff's kinetic analogue.
The theorem is, that if a point move along the elastica with uniform
velocity, the angular velocity of the tangent at that point, is the same
as that of a pendulum under the action of gravity.
If V be the' velocity of the moving point, (5) may be written
d V
j7 = 7o ( sm - Sm a )
at a\/2
If we put x i 71 " + $> a = |TT + /3,
dy V
this becomes -r L = TK (cos y cos p)*,
at a\ '2
which is the equation of motion of a common pendulum, whose
length is equal 4ga z /V 2 .
Stability under Thrust.
142. When a thin straight wire or column is subjected to
a pressure or thrust, which is applied at one extremity in the
direction of its length, experiment shows that as soon as the
thrust exceeds a certain limit the wire commences to bend. There
are several methods by which this limiting value can be obtained,
but the following is perhaps the simplest.
Let a thrust P be applied at one extremity of the wire ; and
let P be the limiting value of P which is just sufficient to
produce bending. Then if P is less than P , no bending will take
place ; but if P is slightly greater than P , the wire will assume
a sinuous form which differs very little from a straight line. We
must therefore solve the equations of equilibrium on the hypo-
thesis that the wire is not absolutely straight in its configuration
of equilibrium, but assumes a slightly sinuous form ; and we shall
find that our solution leads to a certain equation of the form
P = a, where a is a quantity upon which the sinuous curve assumed
STABILITY UNDER THRUST. 157
by the wire depends. Now P may have any value we please ; if
therefore we assign a value to P which is less than the least
value of a, the equation P = a cannot be satisfied, which shows
that equilibrium in the sinuous form is impossible. Hence the
minimum value of a is the limiting value of P, which is just
sufficient to produce bending.
143. Let -or denote the curvature of the wire when slightly
bent ; then -sr = p /-1 , p~ l = 0,0 = Ats, and equations (1) become
-*- <*>
T S +T -= Q < 9 >'
A + N =0 . ...(10).
ds
From (8) and (10) we obtain
dT d _
ds ds
whence
rn T) i A 2 /I 1 \
J_ = JD -k -O.TS ^11^.
Since -or is a very small quantity throughout the length of the
wire, the constant B may be put equal to P, where P is the
thrust applied to the end of the wire; accordingly (11) may be
written
T= P ^-A^ ...(12).
Owing to the smallness of -ar, we may neglect its cube ; whence
eliminating N from (9) and (10) we get
A d* r>
A h Pv = 0,
the integral of which is
iy = C cos us + D sin/is (13)>
where fj? = P/A. We have now three cases to consider.
Case I. Let the lower end A of the wire be firmly clamped,
whilst the upper end B is pressed vertically downwards by a force
P, but is otherwise free ; also let I be the length of the wire, and
let the arc s be measured from A.
158 FLEXION OF WIRES.
At the end B, G and therefore is are zero ; whence
C cos pi + D sin pi = ..................... (14).
Also by considering the equilibrium of the whole wire, it
follows that JV=0 at A, whence by (10) d-sr/ds = Q when s = 0,
accordingly D = 0. This requires that cos pi = 0, whence
which gives
(15).
The least value of the right-hand side of (15) occurs when
n = ; and this gives the thrust P which must be applied to
the upper end of the wire to produce an infinitesimally small
deflection ; if, therefore, the thrust is less than this quantity, no
deflection will" take place and the wire will remain straight.
Whence the condition of stability is that
P<^A/l 2 ........................ (16).
Case II. Let the wire be pressed between two parallel planes
which are perpendicular to its undisplaced position. If the planes
were perfectly hard, smooth and rigid (a condition which can only
be approximately realized in nature), the ends of the wire would
tend to slip on the slightest pressure being applied; we shall
therefore suppose that the ends are in contact with mechanical
appliances which will prevent any such slipping taking place,
but are otherwise free.
Under these circumstances the terminal conditions are or =
when s = and s = I. Whence by (13)
(7=0, sin pi = 0, pi = TT,
and the condition of stability is that
PAIP ........................... (17).
Case III. Let both ends of the wire be clamped. The ter-
minal conditions require that the values of N and or at the two
ends should be equal to one another. Consequently
D (1 cos fd) = C sin pi,
C (1 cos pi) = D sin pi.
Eliminating C and D we obtain
and the condition of stability is
(18).
STABILITY OF A COLUMN. 159
The value of tr may now be written
OT = G COS 27TS/I,
which shows that there are two points of inflexion, which occur
when s = %l and s = f I.
The first case corresponds to a column or pillar whose lower
end is cemented into a bed of concrete, whilst the upper end
supports a building which simply rests upon but is not fastened
to the pillar; and we see that in this case the weight required
to cause the pillar to collapse is less than in the other two cases.
The second case corresponds to a pillar or rod both of whose ends
rest on bearings to which they are not cemented. The third case
corresponds to a pillar whose ends are respectively cemented to
the foundations and to the building supported. In the third case,
the force required to cause the pillar to collapse is four times
greater than in the second and sixteen times greater than in
the first case.
144. Another interesting problem is the greatest height of a
rod or column consistent with stability.
Let the lower end of a vertical rod be firmly fixed; then it
can be shown by experiment that the rod will bend when its
length exceeds a certain limit. To find this limit, we shall sup-
pose that the rod when slightly bent is in equilibrium.
Since dx = ds, and p~ l = vs, equations (1) become
dT
-Nv= 9
dN
A + N =
dx
From the first and third we obtain
(19).
where B is a constant. At the free end, where x = l, T and
must vanish ; whence
Substituting this value of T in the second of (19), eliminating
N from the third and neglecting squares of -ar, we obtain
A d ~=a= dv/dx,
and since is very nearly equal to \TT,
$ = dv/dx,
and therefore (24) becomes
d s v Ar , d 3 v ,-..
+ N = J Q ^
149. For the complete discussion of these equations we must
refer the reader to Chapter vm. of Lord Rayleigh's treatise ; but
one or two special cases may be noticed.
If the wire is so long that it may be treated as infinite, we
may neglect the conditions to be satisfied at its extremities. If
therefore the vibrations consist of waves of length \, we may
assume as a solution of (28) that v is proportional to
Substituting in (28) we obtain,
and therefore the frequency is
150. We shall now investigate the flexural vibrations of a
wire 1 of length I.
Taking the origin at the middle point of the wire, we may
assume
v = U exp (ifcbmH/l*),
where U is a function of x, and m is a constant whose value
has to be determined. Substituting in (28) we obtain
dx*~ I* '
To solve this equation assume U exp (pmx/l), and we see
that the values of p are the four fourth roots of unity, viz. 1, 1,
i, i. The solution may therefore be written
U = A sin mxjl + B sinh mac/I
+ G cos mx/l + D cosh mxjl ............... (30).
151. We have now three cases to consider.
(i) Let both ends of the wire be free. The first of (29)
requires that
A sin mx/l + B sinh mx/l G cos mx/l + D cosh mxjl = 0,
1 Greenhill, Mess. Math. Vol. xvi. p. 115; Lord Eayleigh, Theory of Sound,
Ch. vin.
112
164 FLEXION OF WIRES.
when x = 1. This equation of condition may be satisfied in
two different ways ; we may first suppose that C = D = ; and
-A sin w-f -Bsmhra = ............... (31),
or that A = B = 0, and
shim = ............... (32).
The first solution corresponds to the first line of (30), which is
an odd function of x, and may therefore be called odd vibrations ;
whilst the second solution corresponds to the second line of (30),
which is an even function of x, and may be called even vibrations.
We thus see that the odd and even vibrations are independent of
one another.
Taking the case of the odd vibrations, the second of (29)
requires that
A cos \m + B cosh \m = 0,
and therefore by (31)
m = tan |m ..... , ............... (33).
For the even vibrations the second of (29) gives
C sin \m + D sinh \m = 0,
and therefore by (32)
tanh \m = tan \m .................. (34).
Equations (33) and (34) determine the values of m for the odd
and even vibrations respectively, and consequently the frequency
of the different notes can be found.
(ii) Let both ends of the wire be clamped.
In this case the conditions to be satisfied at the ends are, that
U = 0, dU/dx = ..................... (35),
the first of which expresses the condition that the displacement
at each end should be zero, and the second that the direction of
the axis should be unchanged.
The solution for this case may evidently be obtained by
integrating the results of case (i) wire twice with respect to x, and
consequently the values of m for the odd and even vibrations are
given by (33) and (34).
EXTENSIONAL VIBRATIONS. 165
(iii) Let the wire be clamped at x = ^l, and free at x = ^l.
When x \l equations (35) have to be satisfied ; and when
x=\l, the conditions are given by (29). Taking the value of U
given by (30) and writing out the four equations of condition in
full, it will be found that they can be satisfied in two ways, i.e.
either, B =(7 = 0,
A sin \m + D cosh \m = 0,
A cos \m D sinh \m = 0,
which gives tanh \m = cot \m (36),
or, A = D = 0,
B sinh \m + C cos ^ m = 0,
B cosh ra + C sin ^ m = 0,
which gives tanh |ra = cot \m (37).
Equations (33) and (34) are both included in the equation
cos m cosh m=l (38),
and (36) and (37) in the equation
cos m cosh m = 1 (39).
For a discussion of the roots of these equations, we must refer
to Lord Rayleigh's Theory of Sound, Chapter vni. and to Prof.
Greenhill's paper.
Extensional Vibrations.
152. The equation of motion for extensional vibrations of a
straight wire may be obtained immediately from the first of (22).
In this case p = , ds = doc, and therefore
dT_ d?u
u being the longitudinal displacement.
But T**fm&-,
whence putting q/cr = 6 2 , we obtain
cte*~ dp'
1(56 FLEXION OF WIRES.
which is the same equation which we have obtained for the trans-
verse vibrations of a string. The condition to be satisfied at a
fixed end is that u = ; whilst the condition to be satisfied at a
free end is that T = 0, or du/dx = 0.
In the case of a wire of infinite length, which propagates waves
of length \, we must put
^ _ 2ura;/A4- ipt
and therefore p = 27T&/A, = - A / - .
A< y O"
In the corresponding case of flexural vibrations of the same
wave-length,
27TK /g
X 2 V0-'
whence p'/p = K/\.
Since X is usually very much greater than K, which is the
radius of gyration of the cross section, we see that the pitch of
notes arising from extensional vibrations is usually much higher
than that of notes arising from flexural ones.
It will be observed that when the vibrations of a straight wire
are extensional, the displacement is parallel to the length of the
wire ; hence such vibrations are sometimes called longitudinal
vibrations. But if the natural form of the wire is curved, exten-
sional vibrations usually involve the normal as well as the
tangential displacement of the central axis.
EXAMPLES.
1. A naturally straight wire AB, of which the end A is fixed,
is lying on a smooth horizontal plane, and the other end is pulled
with a force F, whose direction is perpendicular to the undis-
placed position of the wire. Prove that the projection of any
length AP on the undisplaced position AB is equal to
(2F/A$ (V(cos ) - V(cos - cos <)},
where is the angle which the normal at P makes with AB, and
$ is the value of < at the end B.
EXAMPLES. 167
2. If a uniform horizontal wire, both of whose ends are fixed,
be displaced horizontally, so that one half is uniformly extended,
and the other half is uniformly compressed, prove that the displace-
ment at time t of any particle whose abscissa is x is
(4wZ/7r 2 ) 2 (2i + 1)- 2 cos (2t + 1) Trat/^l cos (2t + 1) Trx/21,
where 11 is the length of the wire, the middle of which is the
origin, and nl is the initial displacement of that point.
3. The extremities of a uniform wire of length I are at-
tached to two fixed points distant I apart by springs of equal
strength. Show that if the longitudinal displacement of the wire
is represented by Pe imat sin (mxjl + a), the admissible values of m
are given by the equation
(my - Z> 2 ) tan m + Zmqlp = 0,
where p, is the strength of either of the springs, and q the ratio
of the tension to the extension in the wire.
4. An elastic wire, indefinitely extended in one direction, is
firmly held in a clamp at the other end. If a series of simple
transverse waves travelling along the wire be reflected at the
clamp ; show that the reflected waves will have the same ampli-
tude as the incident waves, but that their phase is accelerated by
one quarter of a wave-length.
5. A heavy wire of uniform section is carried on a series of
supports in the same horizontal plane, L r is the bending moment
at the rth point of support, l r the distance between the (? l)th
and the rth support, and m the mass of the wire per unit of
length ; prove that
Lr-Jr + 2L r (l r + l r+l ) + L r+l l r+1 =
6. Prove that if an elastic wire of length I, with flat ends, im-
pinges directly with velocity V on a longer wire at rest, of length
nl and of the same material and cross section, also with flat ends,
the first wire will be reduced to rest by the impact ; and the second
wire will appear to move with successive advances of the ends
with velocity V for intervals of time 2l/a, and intervals of
rest of Z(n 1) I/a, a denoting the velocity of propagation of
longitudinal vibrations.
168 FLEXION OF WIRES.
7. An elastic rod of length I lies on a smooth plane, and is
longitudinally compressed between two pegs at a distance I' apart.
One peg is suddenly removed ; prove that the rod leaves the other
peg just as it reaches its natural state, and then proceeds with a
velocity equal to V (I l')/l, where V is the velocity of propagation
of a longitudinal wave in the rod.
8. A metal rod fits freely in a tube of the same length, but of
a different substance, and the extremities of each are united by
equal perfectly rigid discs fitted symmetrically at the end. Show
that the frequencies of the notes emissible, which have a node at
the centre of the system, are given by xjZTrl, where 21 is the length
of the rod or tube, and n is a root of the equation
2Mx = ma cot x/a + mfa cot x/a' ;
where M, m, m are the masses of a disc, the wire, and the tube,
and a, a' are the velocities of propagation of sound along the wire
and the tube.
9. Two equal and similar elastic rods A G, EG are hinged at
C so as to form a right angle, while their other extremities are
clamped. One vibrates transversely and the other longitudinally ;
prove that the periods are 2P/f 2 0' 2 , where 6 is given by the
equation
1 + cosh 6 cos 6
+ (sin d cosh Q - cos Q sinh d) (gljf z d) cot (0*f 2 /gl) = 0,
where I is the length of either rod, and / g are two constants
depending on the material.
10. The natural form of a thin rod when at rest is a circular
arc, and the rod makes small oscillations about this form in its
own plane. Assuming that the couple due to bending varies as
the change of curvature, and that the tension follows Hooke's
law, prove that if the arc be a complete circle the periods ZTT/P
are given by the quadratic,
p 4 - (6 (n 2 + 1) + an 2 (n? - 1)} p 2 + abn 2 (n 2 - 1) = 0,
where n is any integer, and a, b are two constants which depend
upon the moduli of stretching and bending, and on the radius of
the circle.
EXAMPLES. 169
11. If in the last example the arc be not a complete circle, but
have both ends free and be inextensible, show that it can be made
to vibrate symmetrically about its middle point by suitable initial
conditions in a period 2ir/p, provided the angle 20, which the arc
subtends at its centre, satisfies the equation
2 (f + 1) (V 2 ~ ?" 2 ) cot tf + , Srj be the angles of contingence and torsion at
P, so that POP' 8(j), OP'O' = 877 ; let p, a- be the radii of principal
curvature and torsion at P. Also let T + 8T, Ni + SN^ &c. be
the values of the resultant stresses at P'; and K, "jB, %> ; 1L, Jl,
ffi the components of the bodily forces and couples per unit of
length of the wire.
The equations of equilibrium are obtained by resolving all the
forces and couples which act upon PP' parallel to Px, Py, Pz.
Resolving the forces parallel to Pz, we get
(T + BT) cos 8 < f>-T-(N l + &NJ cos 877 sin 80
+ (N 2 + SN 2 ) sin 8< sin 877 + 5Ss = 0,
or
BT - N<> + 58s = 0.
Treating all the other forces and couples in the same way, we
obtain the following six equations :
as
ds cos 0', where 8&> is the rotation due to bending,
FLEXTJRAL COUPLE. 173
about a line through Q perpendicular to the plane of bending.
Now if C' be the centre of curvature in the plane PCQ after
bending,
So = PG'Q - PCQ = PQ (-, - -},
\P pJ
whence the displacement of q along pq is
r$ = A ( J sin tf>
G v = cos d> = A ( ) cos
\P P
(5),
.(6).
Let QO, QO' be the principal normals at Q before and after
bending ; 0, 0' the centres of principal curvature ; also let
CQO = x, CQO' = x- Let R x , R x ', R y , R y ' be the radii of cur-
vature before and after bending in planes perpendicular to Qx,
Qy, and let p lf pi be the radii of principal curvature before and
after bending. Then
1 1
1 1 1
; = -, cos y ,
P Pi
= cos y.
P Pi
11.
1 _ 1
._
^x pi
Rx~~ P i sli x *r
1 1
i i
"y PI y PI
Since the curvature in the plane through the tangent which is
perpendicular to the plane of bending is unchanged,
.(8).
TORSION AL COUPLE.
From the first and second of (7) combined with (8) we get
11. 1
-p-, = sin q> -- cos Jo J
by (2) and (3).
The work done by torsion is
o o
It therefore follows that if W be the potential energy per unit of
length,
2 (12).
Equilibrium of Naturally Straight Wires.
158. The preceding formulae can be simplified when the wire
is naturally straight. In this case the curvature in every plane
through the axis of the wire is zero before deformation ; and
since the change of curvature in that plane through the tangent
to the deformed wire which is perpendicular to the plane of
bending is zero after deformation, it follows that the curvature in
the above-mentioned plane is also zero after deformation. Hence
the plane of bending is the osculating plane of the deformed
wire.
From this it follows that
whence, by the fourth of equations (1),
_
ds ~
or H = const.,
which shows that the torsional couple is constant throughout the
length of the wire.
This is a very important proposition.
NATURALLY STRAIGHT WIRES. 177
159. We shall now proceed to integrate the equations of
equilibrium of a naturally straight wire.
Since H is constant and O l = 0, it follows that if CT denote the
curvature, so that -& = l/p, equations (1) become
Since G 2 = Ata, we obtain from (13) and (17)
(13),
dN, N 2
_>___ 2+7V = o ..................... (14),
dN 2 N,
-ds + ^ = ..................... (I*)*
(17).
dT dts
-j- + A-ST -=- = 0,
ds ds
whence T = P - \A^ (18),
where P is a constant.
/ A \
From (16) we get N 2 =(H j (19),
and from (13) and (18) N^ = -A~ (20).
ds
From (19) and (20) combined with (15) we get
H^-A(^} = 0,
ds ds V a- J
A
whence = 1>H + (21),
<7 <&*
where Q is a constant ; accordingly by (19)
..(22).
To obtain a third integral, substitute the values of T, N lt N*
from (18), (20) and (22) in (14) and we get
A^ + ^H'-PAjv-^ + lsAW^O (23).
as* TX
B. H. 12
178 THEORY OF CURVED WIRES.
Integrating we obtain
\'= -IAW+^P-IH^^+R^-Q*.. (24),
where R is another constant.
160. From (24) we see that (d^jdsf is a cubic function of or 2 ,
and therefore or' 2 can be expressed in terms of s by means of
elliptic functions of the first kind. Let \A*Z denote this cubic
function; then collecting our results from (18), (21) and (24), we
have the following three first integrals of the equations of equi-
librium, viz.
(25).
The first of (25) merely determines the tension, but the second
leads to important results. If the curve assumed by the wire is
a plane curve, er = oo ; whence if Q is not zero, TX must be constant,
and therefore the curve is a circle. If, however, Q is zero, a is
constant, and therefore the curve assumed by the wire is one 01
constant tortuosity ; and if we suppose the curve to be plane, so
that (30).
T sm a + iv 2 cos a = cos a
a a 2 J
Equations (30) show that the resultant force F which must be
applied to the ends of the wire must be parallel to the axis of the
cylinder on which the helix is traced, and that its magnitude is
H A
F= cos a 7 sin a cos 2 a (31).
a a 2
The resultant couple (ffi is
& 2 = # 2 + ^cos 4 a (32).
The resultant force and couple are therefore to a certain
extent arbitrary, since both contain the torsional couple H, the
only limitation on whose value is that it must not be large enough
to break the wire or to produce a permanent set. We have there-
fore two special cases to consider.
162. Case I. Let H = 0; then the terminal stresses consist
of a pushing force or thrust P, whose value is
P = A\d? . sin a cos 2 a,
together with a flexural couple G 2 , whose value is A/a . cos 2 a.
The pitch of the helix is sin" 1 (Pa/6r 2 ), from which we see that in
order that equilibrium may be possible Pa must not be greater
than # 2 .
122
180 THEORY OF CURVED WIRES.
Case II. Let F=0, then
A A
H= sin a cos a = (33),
a a
whilst (ft = A\a . cos a. The torsional couple is therefore pro-
portional to the tortuosity; also since
H cos a G 2 sin a = 0,
H sin a + G 2 cos a = A /a . cos a,
it follows that the terminal stress consists of a couple whose axis
is parallel to the axis of the cylinder on which the helix is traced,
and whose magnitude is A/a. cos a.
Stability.
163. In 143 we investigated the stability of a naturally
straight wire which was subjected to longitudinal thrust ; we shall
now investigate the stability when the wire is subjected to a
torsional couple as well as a thrust.
In Case L, in which one end is undamped, TS must vanish at
the free end; and in Case II., in which both ends are undamped,
CT must vanish at each end. Now (21) may be written
whence if is vanishes anywhere, the constant Q must be zero. In
Case III., both ends are clamped, and the tangents at the ends of
the wire are consequently parallel ; hence there must be at least
two points of inflexion, and consequently vf must vanish at two
points of the wire. Hence in this case also Q = 0.
Writing P for P in (23), and recollecting that nr 3 is to be
neglected, the equation becomes
where ^ 2= & + z ........................ (34)>
whence -cr = C cos /is + D sin /&s,
accordingly if the right-hand side of (34) is less than the least
value of /A, equilibrium in the sinuous form will be impossible, and
the straight form will be stable.
STABILITY OF A TWISTED WIRE. 181
Case I. Here one end is undamped, whence or = 0, when
s = I ; whilst at the clamped end dur/ds, which is proportional to
N 1} must also vanish; whence D=Q, and cos/i = 0, therefore
yu, = \ir\l, and the condition becomes
H^ P -^
4,A 2 + A < 4,1* '
Case II. Here CT = 0, when s = and s = I, whence the con-
dition is
H* P 7T 2
4,A* + A < I 2 '
Case III. Here OT and d^/ds must be equal when s = and
s = I, and the condition is
H 2 P 47T 2
4,A* + A < I 2 '
All these results agree with our former ones, as can be seen by
putting H = 0.
164. A wire whose natural form is a tortuous curve is first
unbent; secondly, the wire is twisted, and thirdly, the ends are
joined together; it is required to find the condition that a circle
may be a possible figure of equilibrium.
When a circle is a figure of equilibrium none of the quantities
can be functions of s ; we therefore obtain from (1 )
(35)
H = const., 6^2 = const., N^ = H/a,\
The constancy of G 2 and H requires that the changes of
curvature and twist which occur in passing from the natural to
the circular form shall be constant quantities. These conditions
will be satisfied if the natural form of the wire is a helix, which
includes as a particular case a circular coil of fine wire, the radius
of whose cross-section is small in comparison with the mean
radius of the coil.
165. If in the last example the natural form of the wire is a
straight line, the circular form will be unstable when the twist
exceeds a certain limit. To find this limit, we observe that since
the natural form is straight, it follows that in the circular form
G z = A/a; whilst in the sinuous form H will still be constant and
182 THEORY OF CURVED WIRES.
GI zero. But T and Nj, will be small quantities depending on the
change of curvature ts, also we must write
Accordingly equations (1) become
dT N,_
-j u,
as a
_
7 T^ V >
as a,
and p' = a, since the difference between p'~ l and a" 1 may be
neglected when multiplied by T or N. These equations therefore
become, measuring u in the opposite direction, so that u and $
increase together,
dT AT
d^~ N =
^f-r + T =-v} (36),
Na = Q
d(f> +
the rotatory inertia being neglected.
We must now find an expression for the change of curvature
due to deformation.
If R, 3> be the coordinates after displacement, of the point on
the axis which was initially at (a, ), then
R = a + v, 4> = < + u/a.
1 Mess. Math. vol. xix. p. 184.
2 Proc. Lond. Math. Soc. vol. xxm. p. 120 ; American Journal, vol. xvii. p. 315.
3 Crelle, vol. LXIII. ; Lord Kayleigh, Theory of Sound, 233.
184 THEORY OF CURVED WIRES.
Hence if P be the perpendicular from the centre on to the
tangent to the deformed axis, at the point in question, we have by
a well-known formula,
1 = J. dP
p R dR
1 1 r 1 dR
Now
r*
=^n +
.R 2 (1 + du/ady-\ '
also the displacements and their differential coefficients are all
small quantities ; whence expanding and neglecting cubes of small
quantities, the above equation becomes
, D /- 1 d*v\ dv ,,
whence dP =1 --- 7 -^ dd>.
\ a d$>) d(j>
Also dR = -=-; dd>,
d
c 111 /d 2 v /0>7X
'- = -* +V .................. (37) '
which determines the change of curvature in terms of the normal
displacement.
We must next find the condition that the axis undergoes no
extension.
The elementary arc ds' of the deformed surface is given by the
equation
ds' 2 = (dvj + (a + v? (d + dujaf,
and since this is equal to a 2 r&/> 2 , we obtain, neglecting squares of
small quantities,
which is the condition of inextensibility.
Substituting from (37) and (3) of 138 in the last of (36) we
obtain
d s v dv
VIBRATIONS OP A CIRCULAR RING. 185
From the first two of (36) we obtain
d /d 2 N , T \ (d z v dii
U /V I = fTfl.M I - J- -
_
' W
by (38), whence eliminating N we obtain
\
A /or\
= ........... (39).
, 9 ,
a a 4 \d< 2 / d* \d(f> 2
To solve this equation, assume that v oc <-pt+^ i an( j we obtain
qjSs^-YT
o-a 4 (s 2 + l)
If the wire is a complete circle, v must necessarily be periodic
with respect to , and therefore s must be an integer, unity and
zero excluded. We therefore see that there are an infinite number
of modes of vibration, whose frequencies are obtained by putting
s = 2,3,4... in (40).
If the wire is not a complete circle, s is not an integer ; its
values in terms of p are the six roots of (40), but since p is
unknown, another equation is necessary. This equation is obtained
by considering the boundary conditions to be satisfied at the free
ends, which are that T, N and G should vanish there. These
conditions will furnish six additional equations, by means of
which the six constants which appear in the solution of (39) can
be eliminated, and the resulting determinantal equation combined
with (40), will determine the frequency 1 .
1 See Lamb, "On the flexure and the vibrations of a curved bar," Proc. Land.
Math. Soc. vol. xix. p. 365.
CHAPTER X.
EQUATIONS OF MOTION OF A PERFECT GAS.
167. WE have already called attention to the fact, that air is
the vehicle by means of which sound is transmitted ; we must
therefore investigate the equations of motion of a gas.
The general equations of fluid motion, which we obtained in
Chapter I, are of course applicable to elastic fluids such as air and
other gases, as well as to incompressible fluids such as water;
but in order to investigate the propagation of sound in gases,
these equations require modification.
In all problems relating to vibrations, the velocities upon which
the vibrations depend, are usually so small that their squares and
products may be neglected ; also the variation of the density of
the gas is usually a small quantity. If therefore a gas, which is
at rest, be disturbed by the passage of sound waves, we may write
d/dt for d/dt + ud/dx + vd/dy + wd/dz, and also put p = p Q (1 + s),
where s, which is called the condensation, is a small quantity.
The equations of motion therefore become
du
di
dv
v 1 dp^
= A -j
p ax
_ Y ldp
V .
dt
pdy
r
(1),
dw _ I dp |
dt p dzj
whilst the equation of continuity, 6, equation (5), becomes
ds du dv dw _
VIBRATIONS OF A GAS. 187
We shall also suppose that the bodily forces (if any) which act
upon the gas arise from a potential U, and also that the motion is
irrotational ; (2) therefore becomes
We have already shown that when the motion is irrotational,
the pressure is determined by the equation
Now ( is to be neglected, also if we assume Boyle's law to
hold, we shall have
p=kp = kp (l
and therefore
= ks + C',
neglecting s 2 &c. Whence (4) becomes
If there were no forces in action and no motion, the first three
terms would be zero ; whence C' = C, and therefore,
ks+ U+ 4> = ......................... (5),
or if Sp denote the small variable part of p, (5) may be written
= ........................ (6).
Po
Eliminating s between (3) and (5) we obtain
^_V*-
dt*~ ^ ~dt .........
Equation (6) and (7) are the fundamental equations of the
small vibrations of a gas.
168. In almost all the applications of these equations, no
impressed forces act, and therefore 7=0; accordingly (7) becomes
< 8 >-
Let us now suppose that plane waves of sound are propagated
in a gas of unlimited extent. Let I, m, n be the direction cosines
ISS EQUATIONS OF MOTION OP A PERFECT GAS.
of the wave front, a the velocity of propagation of the wave. We
may assume
Substituting in (8) we obtain
k = a? ............................... (9).
This equation determines the physical meaning of k, and
shows that it is equal to the square of the velocity of propagation.
We may therefore write (8) in the form
Let , 77, f be the displacements of an element of fluid, then
^if _ _T A trie" 1 (lx+my+nz-at)
7 , ~ ~ 7 - -iXt/tl/C ,
at Ax
whence = - (A I /a) e llt *+'+-*) ,
with similar expressions for rj and We thus obtain
$1 = r)/m = /n,
which shows that the displacement is perpendicular to the front
of the wave. This constitutes one of the fundamental distinctions
between waves of sound and waves of light, for it is well known
that in a wave of light the direction of displacement always lies
in the wave front. It therefore follows that waves of sound are
incapable of polarization ; they are, however, capable of interfering
with one another and also of being diffracted, since these phe-
nomena do not depend upon the direction of vibration.
169. Equation (9) enables us to calculate the velocity of
sound in a gas, and we shall now show how it may be applied to
obtain the velocity of sound in air.
We have
where p is the pressure corresponding to a given density. Now
it is found by experiment that at C. under a pressure equal to
the weight of 1033 grammes per square centimetre, at the place
where the experiment is made (i.e. a pressure equal to 1033 g
barads 1 ), the density of dry air is '001293 grammes per cubic
1 In the report of the British Association at Bath, 1888, the Committee on Units
recommended the introduction of the following additional units, viz. that
(i) The unit of velocity on the c. G. s. system, i.e. the velocity of one centimetre
per second, should be called one kine.
THERMODYNAMICS OF GASES. 189
centimetre. Hence if we employ the C. G. s. system units, and
take , accordingly (15) may be
written
(16).
171. This equation is the analytical expression of a very im-
portant but somewhat recondite law, known as the second law of
Thermodynamics. For a full discussion of the second law, we
must refer to treatises on Thermodynamics, but a few remarks on
this subject may be useful.
If one substance at a temperature S be placed in contact with
another substance at a lower temperature T, heat will flow from
the hot substance into the cold substance ; and this process will
continue until both substances are reduced to the same temperature.
It can however be shown by means of a theoretical heat-engine
devised by Carnot, that it is possible to transfer heat from a cold
body to a hot body by means of the expenditure of work ; and the
second law asserts, that it is impossible to do this without ex-
penditure of work. The law was first enunciated by Clausius in
the following terms :
It is impossible for a self-acting 'machine, unaided by external
agency, to convey heat from one body to another at a higher
temperature.
Lord Kelvin states the law in a slightly different form as
follows :
It is impossible by means of inanimate material agency, to
derive mechanical effect from any portion of matter, by cooling it
below the temperature of the coldest surrounding objects.
192 EQUATIONS OF MOTION OF A PERFECT GAS.
By means of the experimental law, that the intrinsic energy of
a gas depends upon its temperature and not upon its volume, the
second law of Thermodynamics may be dispensed with in dealing
with gases ; or to put the matter more correctly, the second law
can be deduced as a consequence of the experimental law. But
in the case of substances which are not in the gaseous state, the
first law is not sufficient to enable us to investigate their thermo-
dynamical properties. Moreover, although it is always assumed
that the pressure, temperature and volume are connected together
by a certain relation, which may be mathematically expressed by
an equation of the form F( p, v, 6) = ; yet the form of the func-
tion F is not accurately known, except in the case of perfect gases.
It can be shown that for all substances the second law is mathe-
matically expressed by means of equation (16), and it thus leads
to a certain function , which is capable of being theoretically
expressed as a function of any two of the quantities p, v, 9, and
which specifies the properties of the substance when it is not
allowed to gain or lose heat.
The function $ was called the Thermodynamic Function by
Rankine ; but it is now always known as the Entropy.
172. Returning to 170, let us take p and 6 to be the
independent variables; equation (13) may then be written
dH=Rdp + K p d8,
where K p is the specific heat at constant pressure. Substituting
in (11) and eliminating dp by (12) we obtain
dE = (Rpje + K p )d0-p(l + R/v) dv.
Since the right-hand side of this equation must be identical
with the right-hand side of (14), we must have
whence K p K v = h ........................... (17).
Equation (17) shows that the difference between the two
specific heats is constant ; also since the specific heat at constant
volume has been shown to be a function of the temperature, it
follows that the specific heat at constant pressure must also be a
function of the temperature.
173. The value of the specific heat of air at constant pressure
has been determined by Regnault, and he finds that it is very
nearly independent of the temperature, and is equal to 183'6 foot-
ADIABAT1C LINES. 193
pounds 1 per degree Fahrenheit. It therefore follows from (17)
that K v is also very nearly independent of the temperature.
It also follows from Regnault's experiments, that the value of h
for air is 53*21 foot-pounds per degree Fahrenheit ; we thus obtain
K v = Kp h,
= 183-6 - 53-21 = 130-4
The quantity with which we are most concerned in Acoustics,
is the ratio of the specific heat at constant pressure, to the specific
heat at constant volume, which is usually denoted by 7. We
accordingly find
7 = ^/^ = 1-408.
The specific heats of all perfect gases are so very nearly
independent of the temperature, that they may be treated as
constant. The value of the ratio 7, is also approximately the
same for all gases.
174. We have already proved the equation dH -- dd$. The
quantity (/> is called by Clausius the entropy of the gas, and is a
quantity which specifies in an analytical form, the properties of a
gas which expands or contracts without loss OF gain of heat ; for
when this is the case dH = 0, and therefore = const. If there-
fore we suppose that is expressed as a function of p and v, the
curve = h\og v + K v log + const .............. (19).
By (12) and (17), this may be expressed in the form
= (K p - K v ) log v + K v logpv/h + const.,
whence pw = A**'*' ........................... ( 20 )>
where A is a constant.
This is the equation of the adiabatic lines of a perfect gas.
i This calculation is taken from Chapter XI. of Maxwell's Heat, in which
British units are employed.
B. H.
194 EQUATIONS OF MOTION OF A PERFECT GAS.
If p be the density of the gas, v oc p~ ] ; whence by (20) the
relation between the pressure and density of a gas, which expands
without loss or gain of heat, is
p = k'py ........................... (21),
where k' is a constant.
The equation of the isothermal lines may be written
pv = (K p -K v )6 ........................ (22).
175. The mechanical properties of perfect gases are specified
by two quantities, viz. their densities and their elasticities. The
density, as is well known, is defined to be the mass of a unit of
volume ; but in order to understand what is meant by the
elasticity of a gas, some further definitions will be necessary.
The elasticity of a gas under any given conditions, is the ratio
of any small increase of pressure, to the voluminal compression
thereby produced.
The voluminal compression, is the ratio of the diminution of
volume to the original volume.
Hence if v the original volume, be reduced by the application
of pressure Sp, to v + 8v (8v being of course negative), the elasticity
E is equal to
The quantity E is called the compressibility by Lord Rayleigh
(Chapter xv.), and is denoted by him by m.
The value of dp/dp, and therefore E, depends upon the thermal
conditions under which the compression takes place. The two
most important cases are, (i) when the temperature remains
constant, (ii) when there is no loss or gain of heat. We shall,
following Maxwell, denote the elasticity under these two conditions
by E e and E+.
In the first case p = kp, whence dp/dp k ; accordingly
E e = kp=p ......................... (24).
In the second case p = Wp"*, whence dp I 'dp = k'yp^~ l ; accordingly
E^ = k'ypy = yp ...................... (25).
From (24) and (25) we obtain,
INTENSITY OF SOUND. 195
Velocity of Sound in Air.
176. Having made this digression upon the thermodynamics
of gases, we are prepared to investigate the velocity of sound
in a gas.
From (21) we obtain
jf = 7^T^~ 1
k'y
= ^ /V"" 1 + & o
since p = p (1 + s). Whence (4) becomes
Eliminating s from (3) we obtain
and therefore the velocity of sound is equal to
Now k=p n /p ,
and k' = p /p Q y ,
whence k'pj~ l = k.
The velocity of sound is therefore equal to (7)* and is there-
fore augmented in the ratio <\/7 : 1- In the case of air, the value
of & in feet per second has already been shown to be equal to
918'49, and therefore
= 1083-82,
which nearly agrees with the value 1089 feet per second given
above.
Intensity of Sound 1 .
177. We have stated in 118 that the intensity of sound is
measured by the rate at which energy is transmitted across unit
area of the wave front. We shall therefore find an expression for
this quantity.
Let the velocity potential of a plane wave be
PO + &P be the pressures when the air is at rest and in
motion respectively, the rate dW/dt at which work is trans-
mitted is
dW d
and since Bp = p(j> = p V - sin (x Vt),
A A,
we obtain
dW ( r^ A 27T, T7-.J 27T . 27T,
sin - a; -Fsm.-
dt X
= - v " + periodic terms,
since FT = X.
We therefore see that the rate at which energy is transmitted,
consists of two terms : viz. a constant term, which shows that a
definite quantity of energy flows across the wave front per unit of
time; and a periodic term, which fluctuates in value and con-
tributes nothing to the final effect. The first term measures the
intensity of sound, and shows that it varies directly as the square
of the amplitude, and inversely as the product of the velocity of
propagation in the medium and the square of the period.
CHAPTER XL
PLANE AND SPHERICAL WAVES.
178. WE shall devote the present chapter to the consideration
of certain special problems relating to plane and spherical waves
of sound.
The theory of the vibrations of strings, which was discussed in
Chapter VIL, explains the production of notes by means of stringed
instruments ; but in order to understand how notes are produced
by means of wind instruments, it will be necessary to investigate
the motion of air in a closed or partially closed vessel. The
simplest problem of this kind is the motion of plane waves of
sound in a cylindrical pipe, which we shall proceed to consider.
Motion in a Cylindrical Pipe.
179. Let I be the length of a cylindrical pipe, whose cross
section is any plane curve, and let the fronts of the waves be
perpendicular to .the sides of the cylinder.
We shall suppose for simplicity, that the motion is in one
dimension, whence measuring x from one end of the pipe, the
equation of motion is
d^_ dQ
~dV~ L da? '
where a is the velocity of sound in air.
Since the motion is periodic, we may assume that = 'e tnt ,
whence if
n/a = 2-7T/X = K ......................... (2),
198 PLANE AND SPHERICAL WAVES.
(1) becomes ^ + *y = 0,
the solution of which is
' = (A cos KX + Rsiu KX).
If the pipe is closed at both ends, d/dtv vanishes whenever x = rl/i,
where r is any integer not greater than i. Corresponding to the
iih harmonic, there are therefore i 1 nodes which divide the
pipe into i equal parts.
The increment of the pressure due to the wave motion is given
by the equation
8p = - /?,
and therefore 8p vanishes whenever cos iirxjl = ; i.e. whenever
x = (2r + 1) l/2i, where r is zero or any positive integer less than i.
Points at which there is no pressure variation are called loops.
We thus see that corresponding to the gravest note (i 1, r = 0),
there is a loop at the middle point of the pipe. The loops
corresponding to the overtones, occur at points x = l/2i, 3//2t. . . ;
and consequently the loops bisect the distances between the nodes.
The conditions that a node may exist at any point of the pipe,
can be secured by placing a rigid barrier across the interior of the
MOTION IN A CYLINDRICAL PIPE. 199
pipe at that point. The conditions for a loop may be approxi-
mately realised, by making a communication at the point in
question with the external air ; and consequently it was assumed
by Euler and Lagrange, that the open end of a pipe may be
treated as a loop. This supposition is however only approximately
true, but the error is small provided the diameter of the pipe is
small in comparison with the wave-length. Whenever a disturb-
ance is excited in a pipe which communicates with the air, the
external air is set in motion, and a complete solution of the
problem would necessitate the motion of the latter being taken
into account.
181. Let us in the next place suppose that one end of the
pipe is fitted with a disc, which is constrained to vibrate with a
velocity cos nt.
The condition to be satisfied at the origin, where the disc is
situated, is
-^r- = cos nt. when x = 0.
doc
If therefore we assume
= (A cos KX + B sin KX) cos nt,
we obtain BK = 1.
If the other end of the pipe is closed, d/dx = when x = I,
whence
cos K(I x)
= r^ s ^ cos nt.
K sin Kl
If the other end be open, the condition is that = when
as = I, whence
sin K(IX)
= s-z cos nt.
K cos Kl
The value of K is of course n/a.
Reflection and Refraction 1 .
182. We shall now investigate the reflection and refraction of
plane waves of sound at the surface of separation of two gases.
Let the origin be in the surface of separation, let the axis of
1 Green, Trans. Camb. Phil. Soc. 1838.
200 PLANE AND SPHERICAL WAVES.
x be drawn into the first medium, and let the axis of z be parallel
to the line of intersection of the fronts of the waves with the
surface of separation.
Let i be the angle of incidence, r the angle of refraction ; also
let F, V l} be the velocities of propagation in the two gases, and
p, p 1 their densities when undisturbed. Then in the first medium
we must have
p' = p (1 + s), p' = k'p'y = k'pv (1 + ys)
and in the second medium
p'i = Pi (1 + *i)> P'I = fcVi* = &Vi (1 + 7*i)-
Since the two gases are supposed to be in equilibrium when
undisturbed by the waves of sound, we must have
k' p y=k\p\ ........................... (5).
Again, V* = k'ypir- 1 , V? = k'wpJ-\
whence V*p=Vfa ........................... (6).
The equations of motion in the first medium are
dp
* '
and in the second
*fe_W*fe.>to
dt* ~ J {do? df) '
The boundary conditions are,
(i) That the component velocity perpendicular to the surface
of separation should be the same in both media.
(ii) That the pressure in the two media should be equal at
their surface.
The first condition gives
d$_d$ 1
dx~ dx "
and the second gives P = Pi>
which by (5), (8) and (10) gives
(12).
REFLECTION AND REFRACTION. 201
If we suppose that the velocity potential of the incident
wave is
< = 4 6 KH-&V+) ........................ (13),
the velocity potentials of the reflected and the refracted waves
may be written
...................... (15),
for the coefficient of t must be the same in these three equations,
because the periods 27r/&> of the three waves must be the same ;
whilst the coefficients of y must be the same, because the traces
of the three waves on the surface of separation must move together.
Substituting the value of + (ft' in (7), and the value of fa in
(9), we obtain
G>* = F 2 (a 2 + Z> 2 ) = F 2 (a' 2 + 6 2 ) = V? (a? + 6 2 ) ......... (16),
and therefore a = a. Also if X, \ be the wave-lengths in the
two media
a = (2-7r/\) cos i, b = (2?r/X) sin i = (27r/\ 1 ) sin r) . ^
Ox = (27T/XO cos r, a> = 2?r F/X = 2-rr F^ J ' '
From the equation a = a, we see that the angle of incidence
is equal to the angle of reflection ; and from (17) it follows
that
(18),
.
sin % sin r
which is the law of sines.
To obtain the ratio of the amplitudes, we must substitute the
values of > + 0' and ^ from (13), (14) and (15) in (11) and (12);
we thus obtain
(A -A')a=A l a l
By (17) and (18) these become
(A J/)tan r A^ tant
( A + A') sin 2 T = A! sin 2 i
from which we deduce
tan (i + r)
2 A sin 2 r cot i /2>\
1 ~~ sin (i + r) sin (i r)-"
202 PLANE AND SPHERICAL WAVES.
The first formula is the same as Fresnel's tangent formula for
the intensity of the reflected light, when the incident light is
polarized perpendicularly to the plane of incidence ; and we ob-
serve that the reflected wave vanishes when i + r = |TT, i.e. when
183. When light is reflected at the surface of a medium,
which propagates optical waves with a velocity which is greater
than that of the medium from which the light proceeds, it is well
known that the light will be totally reflected, when the angle of
incidence exceeds a certain value which is called the critical
angle ; and that total reflection is accompanied with a change of
phase. We shall now show that a similar phenomenon occurs in
the case of sound.
Since cos r = { 1 - ( VJ F) 2 sin 2 i}*,
it follows that if F a > V, cos r will vanish when i = sin" 1 F/ V 1 , and
for angles of incidence greater than this value, cos r will become
imaginary; and therefore by (17), = <3>e l(ca{ ,
this becomes
(V^ 2 ) = ........................ (25).
i The remainder of this Chapter is taken from Lord Bayleigh's Theory of Sound,
Vol. n. Chapter xvn. His original investigations are given in the Proc. Land.
Math. Soc. Vol. rv. pp. 93 and 253.
204 PLANE AND SPHERICAL WAVES.
By (12) of 7, it follows that if r, 6,
is a function of r alone, (25) becomes
d*tf) 2 d<& ,
dr 2 r dr
which may be written in the form
the integral of which is
If the motion is finite at the origin, we must have A = B,
in which case
in which A may be complex.
185. This equation may be applied to determine the sym-
metrical vibrations of a gas, which is enclosed within a rigid
spherical envelop of radius c ; for the condition to be satisfied at
the surface of the envelop is
d/dr = 0,
which gives K cos KG c -1 sin KG = 0,
or tan KG = KG (28).
Since the wave-length X = ZTT/K, and the frequency is equal to
/ca/2-TT, (28) determines the notes which can be produced. The
roots of (28) have been investigated by Lord Rayleigh, and he
finds that the first root is KG = 1'4303 x TT. We therefore see that
the frequency of the gravest note is 7151 x (a/c) ; accordingly the
pitch falls as the radius of the sphere increases. This result
exemplifies a general law, that the frequencies of vibration of
similar bodies formed of similar materials, are inversely pro-
portional to their linear dimensions.
The loops are determined by the equation sin KT = 0, which
gives r = mir/tc, where m is an integer.
VIBRATIONS IN A CONICAL PIPE. 205
186. Since any circular cone whose vertex is the origin is a
nodal cone, the above solution determines the notes which could
be produced by a conical pipe closed by a spherical segment of
radius c.
If a conical pipe be open at one end, and we assume that the
condition to be satisfied at the open end is that it should be a
loop, we obtain K = nnr/c, and therefore the value of is
> = 2t Ar~ l tmnatlc sin mirr/c.
The frequency of the gravest note is therefore ^a/c, which is
less than if the pipe were closed.
187. The most general value of in the case of symmetrical
waves is
> = Ar~ l e l * (at+r} + Br~ l e l(C (at ~ r) ............. (29),
the first term of which represents waves converging upon the
origin, whilst the second represents waves diverging from the
origin.
Let us now draw a very small sphere surrounding the origin ;
then taking the second term of (29), the flux across the sphere is
{ (
JJ
r 2 dtl = - B (1 + IKT) e^ at ~^ dtl
dr
when r = 0. The second term of (29) therefore represents a source
of sound diverging from the pole, of strength ^irBe lKat ; similarly
the first term represents a source of sound converging towards the
pole 1 .
188. The general solution of (25) cannot be effected without
the aid of spherical harmonic analysis, but there is one solution of
considerable utility, which we shall now consider.
Let .. (f> = ^e lltat cos 0,
where is a function of r alone. Substituting in (25), we obtain
**+?** **+*..ft
dr* r dr r 2
1 The corresponding problems in two-dimensional motion cannot be investi-
gated without employing the Bessel's function of the second kind Y O (KT). It is
worth noticing, that certain expressions for these functions in the forms of series
and definite integrals, can be obtained by means of the theory of sources of sound.
See Lord Rayleigh, Proc. Lend. Math. Soc., Vol. xix. p. 504.
206 PLANE AND SPHERICAL WAVES.
To solve this equation, put $> = dw/dr and integrate; we at
once obtain
d-w 2dw
j-i + - -J- + ?W = 0,
ar 2 r dr
the solution of which has already been shown to be
w = r- 1 (Ae lKr +B- tKr );
accordingly
4> = ^ (A?* - Be-"") - ^ (Ae r + Be-"") ...... (30).
In order to find the condition that the motion should be finite
at the origin, we must expand the exponentials in powers of tier,
and equate the coefficients of negative powers of r to zero ; we
shall thus find that A= B, whence writing A for 2iK?A, the
solution becomes
,
<>
A ( sin /cr\ .
= - cos/cr -- ................ (31).
KT\ KT J
If gas, contained in a spherical envelop, be vibrating in this
manner, the frequency is determined by the equation
d<>/dr = 0, when r = c ;
. 2/cc
which gives tan KG = ^ -- 2 .
The least root of this equation (other than zero), is found by
Lord Rayleigh to be KC = '662 x TT ; and therefore the frequency of
the gravest note is '331 x (a/c).
This note is the gravest note which can be produced by gas
vibrating within a sphere ; it is more than an octave lower than
the gravest radial vibration, whose frequency has been shown to
be -7151 x (a/c).
Since the motion is symmetrical with respect to the diameter
= 0, every meridional plane is a nodal plane ; but since d<&/d0
does not vanish anywhere except along the diameter in question,
there are no conical nodal sheets.
189. We shall now consider the motion of a spherical
pendulum surrounded with air, which is performing small oscil-
lations.
Since the periods of the pendulum and of the air must be the
VIBRATIONS OF A SPHERICAL REDUCTION. 207
same, we may suppose the velocity of the pendulum to be re-
presented by Ve LKat , and therefore the condition to be satisfied at
the surface of the sphere is
Ve tKat cos6 ..................... (32).
The form of this equation suggests that = 3>e llcat cos 6, where is given
by (30). Since the disturbance is propagated outwards, A = 0,
and therefore
< = - Br~* (1 + tier) e~ lKr .
Substituting in (32), we obtain
3ncc
(88)>
where c is the radius of the sphere.
If X be the resistance experienced by the sphere,
cos
IK (1 + IKG)
where = Ve lKat , is the velocity of the sphere.
Rationalising the denominator, and putting
_2_-KK 2 C 2 K*C*
P ~ 4 + * 4 c 4 ' q ~ 4T+ * 4 c 4 '
and remembering that | = t/caf, we obtain
Z = M' (p% + *
where M' is the mass of the displaced fluid.
The first term of this expression represents an increase in the
inertia of the sphere; whilst the second term represents a re-
sistance proportional to the velocity, which is therefore a viscous
term, and shows that initial energy is gradually dissipated into
space. If M be the mass of the sphere, I the distance of its centre
from the point of suspension, the equation of motion of the
pendulum is
{M(l 2 + fc 2 ) + M'lp] + M'faaqe + (M- M') gW = 0.
208 PLANE AND SPHERICAL WAVES.
By 129, the integral of this equation is of the form
and the modulus of decay is
2 [M (I* + fc 2 ) + M'Pp}/M'l* K aq.
If the wave-length X, of the vibrations of the gas, is large in
comparison with the radius of the sphere, KG will be of the order
cf\, and will therefore be small ; accordingly the value of p will be
nearly equal to ^, whilst the value of /cq, upon which the viscous
term depends, will be of the order c 3 /\ 4 . We therefore see that
in this case the viscous term will be very small, and the motion
will die away gradually ; hence the sphere will vibrate very
nearly in the same manner as if the gas were an incompressible
fluid.
If, on the other hand, c were large compared with \, p would
be nearly equal to unity, and the apparent inertia of the sphere
would be greater than when c/\ is small ; but icq would be of the
order cr 1 and would therefore be small.
190. Another interesting problem is that of the scattering of
a plane wave of sound by a fixed rigid sphere, whose diameter is
small compared with the wave-length.
Measuring 6 from the direction of propagation, the velocity
potential of the plane waves may be taken to be
jA _ e iic(at+x) = gix (at+rcosfl)
the positive sign being taken, because the waves are supposed to
be travelling in the negative direction of the axis of x.
If c be the radius of the sphere, it follows that in the
neighbourhood of the sphere, KT or 2?rr/\ is a small quantity, and
therefore expanding the exponential and dropping the time factor
for the present, we may arrange in the form of the series 1
0=1 -/eV 2 + ucrcos 6 - ^^(3 cos 2 0- 1)...
When the waves impinge upon the sphere, a reflected or
1 The reader, who is acquainted with Spherical Harmonic analysis, will observe
that we have arranged in a series of zonal harmonics. It can be shown that the
solution of (25) can be expressed in a series of terms of the type F (r)S n , where S. A
is a spherical surface harmonic.
SCATTERING OF A PLANE WAVE. 209
scattered wave is thrown off, whose velocity potential may be
assumed to be
<' = A $> + A& cos 6 + %A 2 <& 2 (3 cos 2 d - 1) + . ..
The quantities , ! are given by (26) and (30) respectively ;
but since the scattered wave diverges from the sphere, we must
put A = 0, and take B=l, since the constant B may be supposed
to be included in A , A^.. ; accordingly
4>! = - r~ 2 (1 + tier) -
With regard to 2) it can be verified by trial, that a solution
of (25) is < 2 (3 cos 2 1), where 2 is a function of r alone ; it will
not however be necessary to consider the form of 4> 2 , since it
introduces quantities of a higher order than /c 2 c 3 , which will be
neglected.
The equation to be satisfied at the surface of the sphere is
cto
dr dr
when r = c. This equation must hold good for all values of 6,
whence
l = ,
dr
which determine A , A,. Substituting from (34) we obtain
i 3
v-
approximately, since we shall not retain powers of c higher than
c 3 . We thus obtain
so
At a considerable distance from the sphere, the term Kc'/r 2 is
small that it may be neglected, we may therefore write
or
14
B. H.
210 PLANE AND SPHERICAL WAVES.
Restoring the time factor and putting K = 2?r/X, we finally
obtain in real quantities
+f cos e)cos~(at-r) ......... (35),
corresponding to the wave
..................... (36).
Equation (35) accordingly gives the velocity potential of the
scattered wave, corresponding to the incident wave whose velocity
potential is given by (36). This expression is however only an
approximate one, and the correctness of the approximation depends
upon the assumption, that the radius of the sphere is so small in
comparison with the wave length, that terms of a higher order
than c?/\ 2 may be neglected. We have also neglected tcc 3 /r 2 ,
which is equivalent to supposing, that the point at which we are
observing the effect of the scattered wave, is at a considerable
distance from the sphere. For a more complete investigation, we
must refer to Lord Rayleigh's treatise.
EXAMPLES.
1. If two simple tones of equal intensity and having a given
small difference of pitch be heard together, prove that the number
of beats in a given time will be greater, the higher the two simple
tones are in the musical scale ; and prove that the pitch of the
resultant sound in the course of each beat is constant.
2. One end of a tube which contains air is open, whilst the
other is fitted with a disc, which vibrates in such a manner that
the pressure of the air in contact with the disc is
n (1 - k sin 27r/ T )
where & is a small quantity. Find the velocity potential of the
motion.
3. The radius of a solid sphere surrounded by an unlimited
mass of air, is given by R (1 + a sin nat), where a is the velocity
of sound in air. Show that the mean energy per unit of mass
EXAMPLES. 211
of air at a distance r from the centre of the sphere, due to the
motion of the latter is
%n 2 a?a?R s (1 + 2n 2 r 2 )/?- 4 (1 + nlR 2 ).
4>. Prove that in order that indefinite plane waves may
be transmitted without alteration, with uniform velocity a in a
homogeneous fluid medium, the pressure and density must be
connected by the equation
where p , p are the pressure and density in the undisturbed part
of the fluid.
5. Two gases of densities p, p l are separated by a plane
uniform flexible membrane, whose equation is y = 0, and whose
superficial density and tension are a- and T. If plane waves of
sound impinge obliquely at an angle i, and the displacements of
the incident reflected and refracted waves of sound and of the
membrane, be represented by
(i) A sin {m (x sin i y cos i) nt + a],
(ii) A' sin [m (x sin i + y cos i) nt + a'},
(iii) A 1 sin [m 1 (x sin r y cos r) nt + aj,
(iv) a sin (mx sin i nt),
respectively; find the relations to be satisfied, and prove that
the ratio of the intensities of the reflected and incident waves is
equal to
(Tm z sin 2 i an 2 ) 2 + (p^ sec r pm sec i) 2
(Tm? sin 2 i cm 2 ) 2 + (pm sec r + pm sec if '
6. If sound waves be travelling along a straight tube of
infinite length which is adiathermanous, and no conduction of
heat takes place through the air, prove that the equations of
motion may be accurately satisfied by supposing a wave of con-
densation to travel along the tube, with a velocity of propagation
which at each point depends only on the condensation at that
point, and which for a density p is
where p , p a are the pressure and density at each end of the
wave.
212 PLANE AND SPHERICAL WAVES.
7. Prove that in a closed endless uniform tube of length I
filled with air, a piston of mass M will perform 7?* complete small
vibrations under the elasticity of a spring, if
Mnnrl (v? ,\
tan mirl a = -^-f, - 1 ,
M a \w? /
where M' is the mass of the air in the tube, and a the velocity of
sound, supposing the piston to make n vibrations in a second
when the air is exhausted.
8. Investigate the forced oscillations in a straight pipe, which
will occur when the temperature of air in the pipe is compelled to
undergo small harmonic vibrations expressed by 6 cos m (vt x),
where x is measured along the axis of the pipe.
9. The greatest angle inclination of the adiabatic lines of a
gas to its isothermals occurs, when the slope of the isothermal to
the line of zero pressure is TT cot" 1 7 ; and the locus of all these
points of maximum angle, is a straight line through the origin,
inclined to the line of zero pressure at an angle cot" 1 7*.
10. A sphere of mean radius R, executes simple harmonic
radial vibrations of amplitude a, in air of density p ; prove that
its energy is radiated into the atmosphere in sound waves at the
rate
per unit of time, where X is the length of the waves propagated
in air, and a is their velocity.
CAMBRIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
DEC 1 ?,
EMS LIBRARV
52SJREGIONAI. LIBRARY
A r\rln "' '"'