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Cornell University Library
Dthaca, New York
ROLLIN ARTHUR HARRIS
MATHEMATICAL
LIBRARY
THE GIFT OF
EMILY DOTY HARRIS
1919
Elements of wave motion relating to soun
ELEMENTS
OF
WAVE MOTION
RELATING TO
SOUND AND LIGHT.
_A TEXT BOOK
PREPARED EXPRESSLY FOR THE USE OF THE
CADETS OF THE UNITED STATES MILITARY ACADEMY,
WEST POINT.
BY
PETER 8S. MICHIE,
=
PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE U. S. MILITARY ACADEMY 5
AND BREVET-LIEUTENANT-COLONEL U,. S. ARMY.
THIRD EDITION—REVISED.
NEW YORK:
JOHN WILEY & SONS,
53 East TENTH STREET.
1893. °
tv.
PREFACE.
HIS text-book, as is stated on the title-page, has been prepared
expressly for the use of the Cadets of the United States
Military Academy, and this specific object has therefore wholly
controlled its design and restricted its scope. It is thus in no
sense a treatise. Because of the limited time allotted to the sub-
jects of sound and light in the present distribution of studies at
the Academy, the problem of arranging a fundamental course of
sufficient strength, to be something more than popular, and yet to
be mastered within the allotted time, has been somewhat perplex-
ing. The basis of this arrangement is necessarily the mathematical
attainments of the class for which the course is intended. In this.
respect, the class has completed the study of elementary Mathe-.
matics, as far as to include the Calculus, and has had a four
months’ study of the application of pure Mathematics, in a course
of Analytical Mechanics. With these elements to govern, this
text-book has been designed for a seven weeks’ course, including
advance and review. The fact of being able, through the discipline
of the Academy, to exact of each student a certain number of hours
of hard study on each lesson, is of course an important element
necessary to be stated.
The study of the text is supplemented by lectures, in which the
vi PREFACE.
principles of Acoustics and Optics are amply illustrated by the aid
of a very well equipped laboratory of physical apparatus. Carefully
written notes of the lectures are submitted by each student to the
instructor on the following morning for revision and criticism.
Important errors of fact and misinterpretation of principle are thus
at once detected, corrected, and hence prevented from obtaining a
lodgment in the mind of the pupil. Another element in this
matter of instruction, of sufficient importance to be mentioned, is
the opportunity freely exercised by each student of making known
the difficulties that he has encountered, before being called upon
to exhibit his proficiency in the lesson of the day. It is required
that these difficulties shall be clearly and exactly stated, in order
that the instructor may, by a judicious question or a concise expla-
nation, enable the student to clear up the difficulty as of himself,
and thus complete the elucidation.
The author believes that this method of instruction, taken as a
whole, is in sufficiently intimate accord with the text as to gain the
following advantages, viz.: 1°, the tasks are of the requisite strength
to demand all the study-time allotted to his department of instruc-
tion, and thus is secured the invaluable mental effort and discipline
due to a specified number of hours of hard study; 2°, while the
daily tasks are progressive, they are based on fundamental princi-
ples which require the exercise of a rational faith, and develop a
continual growth of confidence in the mind of the pupil, and a
belief in his own ability to overcome each difficulty as it arises;
3°, when the course is completed, the student finds himself
equipped with a satisfactory knowledge of the essential principles
of the physical science, to which he may add by further individual
study, without the necessity of reconstructing his foundation.
PREFACE, vii
The elements of character developed in the student by this sys-
tem of instruction, viz, confidence in his powers, reliance on
individual effort, and capacity to appreciate truly his sources of
information, are of essential importance in a career where he may
be called upon in emergencies to exercise self-control, and to meet
manfully unforeseen difficulties; and they offer a sufficient reason
for the importance given to these studies in the curriculum of the
Academy.
Text-books are generally compilations. The subject-matter of
this text has been gathered by the author from whatever source
appeared to him best for the purpose in view. And as it is often
desirable to refer to original treatises, for a better conception of the
subject under discussion, a list of authors is appended to this
Preface.
In the arrangement of the matter, the author has been governed
alone by the necessities of the case and the restrictions of the
course. It has therefore seemed advisable to arrive at the deduc-
tion of Fresnel’s wave surface as expeditiously as possible, and on
the way to establish all of the essential principles of undulatory
motion common to sound and light. Sufficient theoretical atten-
tion is paid in the text to the wave surface, and a study of its model
in the lecture-room makes clear its important properties and those
of its special cases. Acoustics is briefly treated, and is indeed made
subsidiary to Optics, by utilizing its numerous illustrations in vi-
bratory motion, so that the laws of this motion may be the more
clearly apprehended in the subject of light. In Optics, while the
essential principles of the deviation of light by lenses and mirrors,
the construction of optical images and the principal telescopic com-
binations, are carried only to first approximations, and are some-
vill PREFACE.
what more condensed than is usual, nothing essential to the
Academic course of Astronomy has been omitted. The part
relating to physical Optics is very concise, but the experiments
performed and illustrations given in the lecture room, especially in
diffraction, dispersion, and polarization, largely remedy this defect.
The figures throughout the text were drawn by Lieut. Arthur
Murray, Ist U. 8. Artillery, Acting Asst. Professor of Philosophy,
U.S. M. A., to whom I desire to acknowledge my great indebted-
ness.
PS. M.
West Point, N. Y., May, 1882.
CPAP oT BR wD
mnmonnvnwn de kee HP Hee een
SOR DPS PSS DN otro wre s
LIST OF AUTHORITIES.
AIRY. UNDULATORY THEORY OF OPTics. 1866.
ANNALES DE CHIMIE ET DE PHysIquE. Tome XXXVI,
ARCHIVES DE Mus&e TEYLER. Vol.I. 1868.
BARTLETT. ANALYTICAL MECHANICS. 1858.
BILLET. TRAIT& D'OPTIQUE PHysIQuE. 1859.
CHALLIS. LECTURES ON PRacTicaL ASTRONOMY. 1879.
CoppiIneTon. SysTEM oF Optics. 1829.
Comptes Renpus. Vol. LXVI. 1863.
Daguin. TRAITS DE PHysiQuE. Vols. I and IV. 1879.
Donxin’s Acoustics. 1870.
ENCYCLOPEDIA BRITANNICA. Ninth Edition, Vol. VII. 1877.
EVERETT’s UNITS AND PHysiIcaL CoNnsTANTs. 1879.
FRESNEL. CiUVRES DE. Vols, I, IL.
HELMHOLTZ. TONEMPFINDUNGEN. 1866.
HELMHOLTZ. PHYSIOLOGISCHE Optix. 1867.
JAMIN. COURS DE PuysiquE. Vol. III. 1866.
Lamé, THEORIE MATHEMATIQUE DE L’ELASTICITS. 1852.
Lioyp. Wave THEORY OF LigHT. 1873.
PaRKINSON’s Optics. Third Edition. 1870.
Pottsr’s Optics. 1851.
Price. INFINITESIMAL CatcuLus. Vol. IV. 1865.
Roop. MopERN Caromatics. 1879.
THOMSON AND Tart, NaTuRAL PuiLosopsy. Vol. I. 1879.
VERDET. COURS DE PHysiquE. Vol. II. 1869.
VERDET. LEGONS DE L’OPTIQUE PuysiquE, Vols. I and II. 1869.
CONTENTS.
PART I. WAVE MOTION.
ART. PAGE
1. General Equation of Energy. ........5. cece eee e eee e cece neces 17
2. Relation of Hypothesis to Theory. ...........eeee seers eeee cece 17
8-9. Molecular Science... ...... ccc cee cece eee teen ene e eee ennee 18
ELASTICITY.
10-15. Elasticity, General Definitions......... ccc cee ee scene ene eeneee 19
16-20. Origin of Theory of Elasticity... ....... 66s cece ee eee eee eee eens 20
21-22, Elastic Force defined.... 2.0.0... ccc eee cece eee eter een eeeeeee 22
23-24, Elasticity of Solids,... .......ccccece cece cer ecte cnet ee seemers 23
25-27. Fundamental Coefficients of Elasticity...... aay baud vas Sraiccaveuw eens 25
28-87. Analytical Expression of Elastic Forces developed in the Motion
of a System of Molecules subjected to Small Displacements... 27
88-43. Surfaces of Elasticity ....... 0... ccc cece eee ee eee eeeneee 35
WAVES.
44-51. General Definitions and Form of Function ...........0eeeeeeeee 38
52. Simple Harmonic Motion........ J Rat Mia wera eee mNe 40
53-05; "The Harmonic Curves esx cists ve ascetitesdtnes cise e ne ee age 41
56-61. Composition of Harmonic Curves..... 0... ccc cece esse cece eees 42
G2EG4, Wiayier FU CHON. s :4 in re cic nneonn c eidte acdrsar sma na isan snnnaida nese ieee oa 45
65-66. Wave Interference, .... 1.2.6... cece ene tee cece recettaceers 46
67-68. Interference of any Number of Undulations..............00.eee 48
69-71. The Principle of Huyghens............ 0. ce ee cece cece ee eeeeeee 49
72. Diffusion and Decay of Kinetic Energy...........2-2. cece eens 52
73-74. Reflection and Refraction... 0.0.00... 0c cece eee et eee eecateetees 53
75-76. Diverging, Converging, and Plane Waves.. ........e0ec0eeeeee 53
ART.
77-80.
81-87.
88-92.
93-96.
97-102.
103-105.
106-110.
111-112.
118-115.
116-125.
126-128.
129-134.
135-140.
141-142.
143-149.
150.
151-154.
155-158.
159-161.
162-165.
166-167.
168-169.
170.
CONTENTS. xi
Reflection and Refraction of Plane Waves.......ceesseseeerees yd
General Construction of the Reflected and Refracted Waves..... 56
Utility of Considering the Propagation of the Disturbance by
Plane Waves ~ Segoe edie sls steam oaame aie vameipunencwaieninuwacmgs 59
Relation between the Velocity of Wave Propagation of Plane
Waves and the Wave Length in Isotropic Media............ 63
Plane Waves in a Homogeneous Medium of Three Unequal Elas-
ticities in Rectangular Directions ............0 eee eee eee eee 66
The Double-Napped Surface of Hlasticity...... 06. . cece see ee eee 69
THO Wave Suriate.. ssc sca caunenccaweeeuweatee S GO ey vie sess 71
Construction of the Wave Surface by Means of Ellipsoid (W).... 75
Relations between the Directions of Normal Propagation of Plane
Waves, the Directions of Radii-Vectores of the Wave Surface,
and the Directions of Vibrations...........6 6. see eeeeeeeee U7
Discussion of the Wave Surface..........e eee eee eee teen ee ees 79
Relations between the Velocities and Positions of Plane Waves
with respect to the Optic AX€S.. 0.6... cece eee cere ee ee ee eee 84
Relations between the Velocities of Two Rays which are Coinci-
dent in Direction and the Angles that this Direction makes
with the Axes of Exterior Conical Refraction...........++-+ 86
PART II ACOUSTICS.
Acoustics defined ; Description of Ear ; Curve of Pressure....... 89
Propagation of a Disturbance in an Indefinite Cylinder.......... 91
Curve of Pressure due to a Tuning-Fork.........sseeee eee eee 93
Illustrations of Motion transmitted. ........ 2. cece cree eee e eres 96
Properties of Sound; Intensity, Pitch, Quality......-eeeseeeees 95
Curves of Pressure for Noises.......-.:ee seer cree erte eer rneet 97
Curves of Pressure for Simple Tones... ..+-2 ++ +r ere rete reerere 99
Properties of Audition ; Definitions of Simple, Single, and Musi-
Cal Tones... cece ce cere cree eter eee teen e teen eee re ee eee 101
Musical Intervals. ..... 0+. cess ee ere ere seen tener see nen ee seees 103
Musical Scales......ccce cece ete e eee renee nen een seen rere eee se 104
Perfect ACCOPdS,. cece eee e cece eet eee e etree ee senrreee eres e ss 106
xii
ART.
171-174.
175-178.
179-182.
183-184.
185-189.
190.
191.
192-194.
195.
196-199.
200.
201-208,
204-209.
210.
211.
212.
213-221.
222-225.
226-228.
229-230.
231-232.
233-234,
235.
236.
237.
238.
239-240.
241-242,
243-245.
246-247.
248-249.
CONTENTS.
PAGE
The Diatonic Scale ; Harmonics.......-+.sesseceeereeeeeteers 106
Sympathetic Resonance....... +s sees sce eee eect ee eee eee ees 109
Separation of Vibrating Bodies into Component Parts........... 111
Velocity of Sound in any Isotropic Meéditths:.cicwcs.anewweedens 112
Velocity of Sound in Gases...... 2... secs eee eee eee eee eee 118
Pressure of a Standard Atmosphere.......... cece ee eee cree renee 116
Height of the Homogeneous Atmosphere...........0 sees ee ee ees 116
Final Velocity of Sound in Air. 2. 1.6... ce eee eee cee eee eee 117
Formula for Velocity of Sound in any Gas.....-....6-- see ee eee 118
Velocity of Sound in Air and other Gases as Affected by their not
being Perfect Gases... ... 6... cee e ee cette eee eee eee 119
Velocity of Sound in Gases independent of the Barometric Pres-
SUPE iced wedties aii ao Read naan aee eI Se MORES Tawa 121
Velocity of Sound in Liquids .......... see etc eer eee eee ween 122
Velocity of Sound in Solids........ 0... cece ec ee cece eee e ee eeees 124
Reflection and Refraction of Sound.......... 00. cess eee eee sees 126
Consequences of the Laws of Reflection................. eee eee 127
Refruction Of SoUnd iiss si sccen etki ee ewacocan ia vadi aie es 8 128
General Equations for the Vibratory Motion of a Stretched String 128
Application of Equations to the Longitudinal Vibration of a Rod. 136
Vibrations of Air Columns; 1°, Closed at One End.............. 138
2°, Open Air ColumMsSi.s iscckecees veces ics es as eewaotwaw anaes 140
Relative Velocities of Sound in Different Material............... 142
Transversal Vibration of Elastic Rods..............-000eceeeee 143
Harmonic Vibrations of Elastic Rods...............ce cece eeeee 145
Dunne MORK ss :5 ai05.3 cas aessierasiney Ranier eb ae as aewe ae Hee aay nn 146
VIDPALLON OL PILATES: ox: sccie cnaceicrauestaalenace ena haclasetee esc ardinure Auten 147
Vibration of Membranes............. 0. cece eee e cree ereeeeees 148
Subjects referred to by Lecture... 0... cece cece eee cence eens 143
PART III. OPTICS.
Light and Optics defined ; Geometrical and Physical Optics...... 150
Luminiferous Ether and its accepted Properties................ 150
DOHC O MSs wire sce. Sh TS. 6 estes, Senin encided Sealer ease waongave s WED Oe DES 151
ART.
249-250.
251-252.
253-256.
257.
258.
259-261.
262-263.
264-265.
266-267.
268-270.
art.
272.
273.
274-275.
276-277.
278.
279,
280.
281.
282-284.
285.
286-287.
288.
289.
290-291.
292.
293.
294-295.
296.
297.
298.
299.
800-308.
CONTENTS. xiii
Medium defined ; Opaque and Transparent.,.........000seee0e% "153
Shadows and Shade............... ccc ce cecececeescuceceanees 153
PHOtOM etry cctiecssantgele elle saw aea¥a de ania see 154
Velocity of Light by Eclipse of Jupiter’s Satellite..............5 157
Velocity of Light by Aberration...........-....cceseeeseues oe 158
Velocity of Light by Actual Measurement........ vars sseiakine 158
GEOMETRICAL OPTICS.
Index of Refraction ; Radiant and Focus..........e.ccceenveves 160
Deviation of Light by Plane Surfaces............cceeceecuue oe 161
Refraction by Optical Prisms... 10... ccc cee cece ee eee eeeeee 163
Deviation of Light by Spherical Surfaces ..............000.005 165
Application of Eq. (805) to Reflection at Plane Surfaces.......... 168
Multiple Reflection—Parallel Mirrors,........... 0. seseeee eens 169
Multiple Reflection by Inclined Mirrors............ccseeeeeeees 169
Angular Velocity of the Reflected Ray..............ceeeeeeeeee 170
Deviation of Small Direct Pencils by Spherical Surfaces......... 170
LiGTISES! veisers ius coin Deiootade wba eheacal Reda oc wine: vase bilevel olele caves ainNy 172
Principal Focal Distance of a Len8...........ce cece eee eee eee 178
Discussion of the Properties of a Lens............ 0... eee eee eee 173
Relative Velocities of the Radiant and Focus.............eeseee 175
Discussion of Spherical Reflectors.......... ccc cece ewes sees 175
Power of'a Lens: ss vcciisneduus vase eee ss aa ceca $5 ey awenee wares 177
To find the Principal Focal Distance of a Lens and a Reflector.... 178
Deviation of Oblique Pencils by Reflection or Refraction at Spher-
ical Surfaces.........+ 000s. e eee fae tases aes whe eee 180
Circle of Least Confusion............ cece cece eee eet e eens 182
The Positions of the Foci........ cc ccc e ees eee etree ee eeeeees 182
Particular Cases of Caustic Curves...........cee ete e eee ee eens 184
Critical Angle for Refraction... .....-...¢. css ee eee ene eeeeeees 186
Spherical Aberration...... 0... ccc cree eee cect ener e eee eeeeens 188
Optical Centres sass ecco sis segs selene nice sieieeriamigerew gi sie swe Oe vis 8s 191
Focal Centres of a Lens........ 0c cece eee e cent e cece ene eneens 192
The Byes» ssi seed stas ai ed x 0 50444 w wepniweattnayedaberens eae'e wie Sees See glee 193
VisiOns 4c sans jase Ves ee reser es tes SAMAR CREW Mae TY SOK 195
Mav
ART.
304.
305-306.
307.
308.
309.
310.
311,
312.
313.
314,
315.
316.
317-319.
3820.
321,
322.
323,
324.
825,
826-828.
329.
330,
331-333.
334,
335.
336.
837-338.
339.
340-341.
342-345,
346.
347.
348.
CONTENTS.
PAGE
Optical Images discussed for a Concave Lens........ vip Sai siaerecs 199
Optical Images discussed for a Convex Lens..........0.0e0e000, 200
Optical Instruments... 2.6.0.6... cece eee eens gis wlesuls seahenins 201
Camere sWcidalscs0cs eases enn au leuntaarnien te wala ea 5 aie aes ae 201
Camera Obscura. < 3.34.6 ec eus bees data Seueae seney estas mewde 202
Solar Microscope so%s g04 600 reins 44 tise ieneewed anaes ie so4 445 202
Microscopes. + seis cee ahs ee eee niatuedeay See caawesws 203
Magnifying Power of a Microscope......... 00. c cece eee e eee eee 204
The Field of View of a Microscope...... att Revie Maree ae ee ee go eee 205
OGUIBIS: 25 2 saa ga aOR 454 TRE ASS OAS BNR IAG en eases 206
Compound Microscope... 2... ccc ccc e ees eee ete n eee eeeeees 208
Me] OSCOPER 5 isis selena le ae lauls dle ded Boa asa eed sate re x nia Wa oeersrmipage te eed eS 209
The Astronomical Refracting Telescope...........0. see cece eee 209
The Galilean Telescopes s..sics ove essa vitere Somsdlasaveeeseeey 211
The Terrestrial Telescope i.e: cise eesassies weavieeewen tea es 212
Reflecting Peléscopess;..., cig y's Ses es ews eedageemiiedeun mee 212
The Newtonian Telescope. ...........0. ce eeeececeessereceeues 213
The Gregorian Telescope......... cece ec ecceeee sence sanneeees 218
The Cassegrainian Telescope... ....... ccc cece eee cece ces ceeees 214
The Magnifying Power of any Combination of Lenses and Mir-
NOUS iar Wa eietalattee sree SOS eRe o8: 6:3: 410,5 8 alte peels ad Siepeadlacolate 215
Brightness Of Tmages s....cseciadiesd veges acre Sh Abas sue Have mew elelelecaves 218
Magnifying Power of the Principal Telescopes............0000% 219
PHYSICAL OPTICS.
Solar Spectra ya's ieavisic e's spiniaeridates seneagenddeeseayce nes 219
Dispersion sc. cnceceaue Gouge wow sa Yes ews 44 VRAAR Ged odaeuneaey 221
Frauenhofer's Linesiie. sic csccseis eavaavbine cavuan enV esvens 221
Normal /Spectrum,: oisa ics eves s cdes og.ou acca drnoslanneane vv 222
Absence of Dispersion in Ether of Space..............-eeeeeees 222
Irrationality of Dispersion... .......eccceecceceucecceeeueees 223
COLOR Adios inaniureciettiiheldlinhad Souls eswlona Dawes tea das aoa uewece ae¢
A DSOMpUOM ss 95.4 cidiyiadinieatd) whtae heasateig lew Slee lylotea wt: aw aerate 225
BMUISS101) o.05 2a cori nosentameeeugahwurde eur yee eae a vatroun 227
Spectrum Atta lysis’ Baap. (8)
The same general theory thus comprises both liquids and solids,
in admitting the coefficient 2u of the former to be zero. The varia-
tion of this coefficient from zero marks the departure of the body
from the perfect liquid state and its approach to that of the solid.
28. Analytical expression of the elastic forces developed
in the motion of a system of molecules, solicited by the forces
of attraction or repulsion, and subjected to small displace-
ments from their positions of equilibrium.
Let a, y, % and +Az, y+Ay, z+ Az, be the rectangular co-
ordinates of the two molecules of the system, whose masses are
respectively m and p, and whose distance apart is r. The intensity
of the reciprocal action of the molecules, being exerted along the
right line joining them, is
mp f (7);
J (r) being an undetermined function of the distance. If the sys-
tem is in equilibrium, we have the relations,
m3 f(r) = 0,
mu f(r) “2 = 0, (9)
mm Bu f(r) = 0.
At a certain instant, let us suppose that the molecules of the
system are displaced from their positions of equilibrium by a very
small distance, and let &, 7, ¢, be the projections of the displace-
ment e¢ of the molecule m on the axes; let 4+ Aé, n+ An, 6+6,
be the projections of the displacement of the molecule » on the
axes; and r+p the new distance between the molecules. Repre-
senting the components of the elastic force parallel to the axes
exerted upon the molecule m by all the molecules » within the
28 ELEMENTS OF WAVE MOTION.
sphere of molecular activity, by Ye, Ve, Ze, so that X, Y, 3 Z, are
the components of the elastic force for a displacement wnity in the
same directions, we have
A A
He = m3y sor p) ES,
Ye = m3p f(r +p) CE (10)
A A
Ze = mp f(r +p) MER.
29. Developing f(r+p), and neglecting the terms of a higher
order than those containing p, since the displacements are regarded
as very small, we obtain, recollecting that Ag, An, A¢, are of the
same order of magnitude as p, while Az, Ay, Az, may be of any order
whatever,
Xe = mulf(r) + of (LE) (1 —9),
= m 3p E (ry + f(r) — pf (r) 1, (4)
= miro s[ro -£ |pmt;
we also have r= Av? + Ay? + Az, (12)
(7 + p)? = (A®+A5) + (Ay+A7) + (Az+46)?; (18)
= BEAR yy BeOS, (14)
r
from which
Substituting this value of p in equations (11), we obtain
nanel (2s [0 22)
+ re = fe actt an ¢ (18)
+[ Fm — £0 | 22 at.
RELATING TO SOUND AND LIGHT. 29
Similarly, for the axes Y and Z we get,
Ye=m su lr (r) — Ler) | oa: ec Aé )
+ (£0 (r) [re ee £0) af) a
+(? (y—Z Ltn) |Aw: aw act,
Z= mau ro) = £0) oe Ag
+ |r )—£ £0 | ou! ated. L (17)
+(+[ro- 128)
ie ¢(r) for fr) ) wir) for f(r) — f ae 3; and m—
mo! ae
pe? aR?
(16)
aeaik
PE
di?”
for their as Xe, Ye, Ze, we oe
Fe =m OF = map| [om +o )
+ 9(0) eon + wor) eMac
Ye =m! = ma Jr) a ag
> (18)
+[o@+em PF) ant vo Weast,
Ze = mot = m =p \¥ (n) eas
+90) tet an + [oe + woe as ks
which give the values of the component elastic forces developed in
any molecule of the medium, when the displacements are small.
30 ELEMENTS OF WAVE MOTION.
30. If the displacement is only in the direction of each axis in
succession, we have the following groups of equations.
Of x: X, = mp oe) + vy |as |
¥, = mie wy eRe Ag, r (19)
2, = m3] vo) Se Jars |
Of y: X, = m3p wins oY | an nN, )
Y, =m zu fem +o | an, a (20)
Ay A
Zy = ME wi atg 18 | an Ui) J
Of z: X, =m Sp v (EE | as ;
Ay A
Y, = m3p E (r) syst AS, r (21)
3 en
Z, = m3p| (r) + (7)
31. Combining the above equations, we have
= TX, eX,
Ve=Y¥,+Y,+ V3, (22)
Ze= 2,4 42,4- Zs.
From Eqs. (19) we see that the total intensity X24 Ye+Z?
of the elastic force developed is proportional to the relative displace-
ment Ag, and since the axis has been assumed arbitrarily, it can be
said, in general, that the total intensity, e/Y? + Y? + Z?, devel-
oped, is directly proportional to the general relative displacement,
Vas? + An? + AG? = «.
RELATING TO SOUND AND LIGHT. 31
From Eqs. (22) we conclude that the component intensity of the
elastic force developed in the direction of any axis, due to any dis-
placement, is equal to the sum of the three component intensities
developed by three successive displacements along these axes, equal
to the respective projections of the general displacement on these
axes. f
32. Of the nine coefficients of Ag, An, Ag, given in Eqs. (18),
six only are distinct. Representing these by
A=msy| oo +92 |, |
B= msn or) + 90) °E |,
C= mm) oo) tHe) |,
- (23)
[ Ay A
D=msn| 4) UY |,
Aa Ay é
E=mX*p | ee
Pa m| oo |, ;
we can write Eqs. (22),
rm Tart nan Ds |
Ve = FaE+ Ban+ DAS,
Ze = Faki+ Dan+ Ca;
(24)
from which we conclude that the component elastic force developed
along any axis, x, for example, by a displacement ¢ along any other
axis y is equal to the component elastic force developed along the
axis y by an equal displacement along the axis 2.
33. From Eqs. (19-21) we see that when a displacement is made
in any direction, the resulting elastic force is not, in general, in the
same direction. ‘To find whether we can refer the system to rectan-
gular co-ordinate axes, so that when a displacement is made along
such an axis, exceptional elastic forces will be developed, whose
total resultant will be in the direction of the displacement, let «, 8, y,
32 ELEMENTS OF WAVE MOTION.
be the angles which the direction of the displacement makes with
the axes; A, , and v, the angles which the resultant elastic force
makes with the same axes; then we have
x 7}
cos A — ————— th
VxX?24 24 22
cose = ees
OT VEEL YE +
cos v = ape, =
~ fF Ere Ze
Ag (25)
COS GS) aS aaa SSS SS
VAP + Ar? + AS?
An
cos 8 = —————_*+ —-—__
E Vag? + An? + AS?”
cos y = ——— 22 ———.
1S WME + op + ae |
Since the resultant intensity of the elastic force is proportional
to the displacement, we may let A represent the intensity of the
elastic force corresponding to a displacement equal to unity.
& varying with the direction of the displacement, we can then
and the first of Eqs. (25) will become
Xe
cosa = SSS 7
K VAs? + An? + Ag?
cos # = ae 26
KV aE + ap + ae? ee)
Ze
cos ¥ =
EK VaE £ Ane + AG?
Substituting the values of X, Y, Z, aé, An, Aé, derived from
these equations in Eqs. (24), after omitting the common factor e,
we have
Kcosp = Ecos a + Bcos B + Deosy,
K cosa = A cosa + Ecos B + Fos y,
t an
Koos v = Foosa+ DcosB + Ccosy.
RELATING TO SOUND AND LIGHT. 33
Applying the conditions
A= 4, w= 8B, wa Y,
we have the equations of condition,
(A — £) cosa + HeosB + Foosy = 0,
Bone + (BK )on 9 + Deon = 6,| (28)
F cos « + Doos B + (C— K) cosy = 0;
together with cos? « + cos? B + cos? y = 1, (29)
which make four equations containing the four unknown quantities
a, B, y, and K.
34. In order that Eqs. (28) may be true for the same set of
values of cos «, cos 8, cos y, we must have the determinant
A—K, E, Ff,
E, B—R, D, = 0. (30)
F, D, C—K,
Multiplying Eqs. (28) respectively by D, F, and H, we get
(AD — KD) cosa + DE cos 8 + DF cosy = 0,
EF cos « + (BF — FE) cos B + DF'cos y = 0, (81)
EF cos « + DE cos B + (CE — HK) cosy = 0.
Placing AD — EF = aD,
BF—DEHE=da'F,
CE — DF =a'E,
(32)
we have
(a—K) Deos« + FF cosa + DE cos B + DF cosy = 0,
(a —K) FcosB + HF cosa + DE cosB + DF cosy = 0, tap
(a"—K) Eoosy + EF cosa + DE cosB + DF cosy = 0;
from which (a— K)D cos a = (a — K) Fos B } (34)
= (a'— K)Eocsy = P;
h (egos, |
W. ence, cos a2 = (a == K)D’
P
Bia aos 35
OER = ra ye (35)
P
a aie ke
34 ELEMENTS OF WAVE MOTION.
Substituting these values in the first of Eqs. 33, we obtain
EF DE DF
1+G-mDt@—xyF twee -* 4)
Clearing of fractions, we have
DEF (K—a)(K—«')(K—a")—E?F? (K—a') (K—a")
— DE? (K—a") (K—a) t 0. (37)
— DPF? (K—a) (K—a«')
If DEF be positive, supposing
1°, that a)
The non-periodic term A, is introduced to make the theorem
conform to the most general case. If the function is capable of
expression in periodic terms only, then A, = 0; this fact can only
be determined by considering each special case.
The equation which expresses the mathematical statement of
Fourier’s Theorem is ,
Qa
Y¥=Yot aD 1 1G; sin ex + «i), (66)
in which y, is the mean value of y, and each of the variable terms
represents, by itself, a harmonic vibration of which the period is an
aliquot part of the whole period r.
62. Wave Function. Resuming Eq. (51),
d= $(2, 1) = 9(V#—2),
we see that, since the displacement d passes through all of its values
while the undulation advances a distance equal to its wave length
A, it has the properties of simple harmonic motion, and, therefore,
may be written
8 = asin" (Vi—a), (87)
This is called the wave function. By making ¢ vary continu-
ously through all values from ¢ = ? tot= ——,
from zero to an «, decrease then to — a, and finally return to
6 will increase
as which is evidently the interval of time
required for the undulation to pass over the wave length 4. Again,
supposing ¢ to remain constant and 2 to vary through all values
from Vt —A to Vt, we obtain again all possible values of the dis-
placement, which values will evidently belong, at the same instant,
to all molecules in the wave length. The following diagram illus-
trates the two cases:
zero, during the time
eS
46 ELEMENTS OF WAVE MOTION.
By the addition of an arbitrary arc we can cause the displace-
ment to take any one of its values, at any time ¢, and thus change
our origin at pleasure.
63. The corresponding expression for the velocity of the mole.
cule in its rectilineal orbit, sometimes called the velocity of the wave
element, in contradistinction to the velocity of wave propagation, is
given by
a @ cos =" (Via). (68)
64. The principle of the coexistence and superposition of small
motions is shown in Mechanics to be applicable to planetary per-
turbations. It is, for similar reasons, applicable to the determina-
tion of the resultant displacement of a single molecule, arising
from the concurrent effect of many disturbing causes acting sepa-
rately. The acceptance of this principle is equivalent to assuming
that the several displacements are so small that their products and
powers higher than the first are negligible with respect to the dis-
placements themselves; and it embodies the primary supposition
that the intensity of elastic forces developed varies directly with the
degree of displacement.
65. Wave Interference. If we apply this principle to de-
termine the displacement of a molecule by two disturbing causes,
giving rise to two undulations of the same wave length, we will
have for the first,
od’ = a’ sin i (Vi —2) + 4'|; (69)
for the second,
6" = alsin] = (ve 2) 4.4" |. (70)
The total displacement will be
é' 46" = 6 = (e@' sin A’ + @" sin A”) cos Ei — 2) |
(71)
+ (@' cos A’ + &” cos A") sin EE Vt — 2 |,
which may be put under the form
ean | 2 re— 2) + |, (72)
RELATING TO SOUND AND LIGHT. 47
by placing «cos A = a’ cos A’ + @” cos A”, "3
@sin A = @’ sin A’ + &” sin A”. (73)
Whence, = = a’? 4 a”? +. 2ar' ee!’ cos (.4' — A"), (74)
__ @’ sin. A’ + @” sin A”
tan A = end a ws A (75)
By Eq. (72) we see that the resultant undulation is of the same
wave length as the components; that the maximum displacement
of the resultant undulation is not, in general, equal to that of either
of the components, and that it does not occur at the same time nor
place with either of them.
66. Taking the square root of Hq. (74), we have
@ = Val? + a"? + 2a’ a” cos (A’— A"); (76)
from which it is seen that, when A’— A” = 0, @ = @' +";
and, when A’ — A” = 180°, @ = «e'— a". Hence, in Eq. (75),
A = A' = A” in the first case, and 4A = A’ = 180° + A” in
the second. The maximum displacement, then, of the resultant
undulation may vary between the sum and difference of the maxi-
mum displacements of the two component undulations, depending
upon the difference of phase.
Tf, in the two component undulations, a’ = «”, « will be equal
to 2«’ when A’ = A", and vary from this value to zero as the dif-
ference of phase A’ — A” passes from zero to 180°.
Substituting, in the expression for the displacement, A’ + 180°
for A’, we will have
_ | 27 / Ar j
o! sin | (Vi—z) +4! | = «sin [* (Vt—2 + ‘) +A | (77)
which is exactly the same as
af! ain | 2 (r% —2) 4+ 4'|,
A
when for z we put 2 + 5:
Therefore, if we suppose that two undulations of the same wave
length, starting in the same phase, meet after travelling over routes
which differ by one-half the wave-length, there will be no displace-
48 ELEMENTS OF WAVE MOTION.
ment of the molecule at the place of meeting, and complete inter.
ference will result.
The diagrams of Figure 9 illus-
trate the composition of two un-
dulations of equal wave length,
having the same phase in the first
case, and opposite phases in the
second and third cases. In AB,
the amplitude of the resultant un- fe
dulation @ is equal to the sumof ,
the amplitudes of the component
undulations, a’ and a”; in A'B’ Noy f)
and A”B”, equal to the difference
of the amplitudes. In A”B”, the oe a
displacement of the molecules is Ke i a
zero, and the two components mu- Seach ee gs ge
tually destroy each other’s action. Figure 9,
67. Interference of any Number of Undulations.
1° Case. When the component undulations have the same
ware length.
Let dc =a’ sin| (rea) + 4" |,
Mo ee gs QT a, ”
oe" =a sin | 22 (via) + 4 il (78)
6” = a" sin [2 (Vt —«x) + a,
etc., OCs eo oy
be the values of the several component displacements. By addition
we have
46% 48" 4 ete,
=(«' sin A’+e" sin A” +e'" sin A’” + etc.) cos | (rt—2) | (79)
+(e’ cos A'+e" cos A" +e!" cos A’ + ete.) sin | (vt—2) |
RELATING TO SOUND AND LIGHT. 49
The second member may be placed under the form of
: 2
asin A cos" (Vi—2) + «cos A sin * (Vt—2) ]
(80)
= «sin | (rea) + 4 | 6
J
From which we conclude that the resultant undulation will
have the same wave length as that of the components, but that in
general the maximum displacement and the phase at the time ¢ will
be different from those of its components.
68. 2° Case. Component undulations of different wave lengths.
If the wave lengths are different, the displacements are of the
form
zB
a! sin Ft) 2 a |,
ec” sin | (Vi —2) + a,
which cannot be combined into a single circular function of the
same form. If in addition the wave velocities also differ, they may
,
be combined if z = z. ‘Hence, undulations of different wave
lengths cannot destroy each other, and the combined effect of sev-
eral undulations upon a single molecule will be equal to the alge-
braic sum of their separate effects. If this sum should reduce to
zero for a given molecule, it will differ from zero for the molecules
immediately preceding and following it.
69. Phe Principle of Huyghens. Since the displace-
ment of any molecule is the cause of the subsequent displacement
of other molecules, we may regard the displacement of the mole-
cules upon any wave front as the cause of the subsequent displace-.
ment of the molecules upon any other front which the wave
afterwards reaches. We may therefore consider each molecule of,
the wave front in any of its anterior positions as being the origin,
and its displacement as the cause of secondary waves, each of which
proceed with the same velocity. The aggregate effect of all these
4
50 ELEMENTS OF WAVE MOTION,
secondary waves upon any other molecule beyond, or its resultant
displacement, will evidently be the same as that due to the primary
wave itself. This principle is known as that of Huyghens, and,
together with the principle of interference, is exceedingly fruitful in
-explaining many of the phenomena of wave motion in sound and
light.
By ly
ss—P
\
\
A \
\
|
° iP
: 7 i
Figure 10,
Let O be the origin of disturbance, and BAC the great wave in
any of its anterior positions before reaching a molecule P’; let
AP’ = I, AB = AC = 3; let dz be any indefinitely small part of
the wave front, and @ the angle made by the wave front with any
right line 7 drawn from P’ to any point of the wave front, at a dis-
tance z from A; then
v= AP’ = V(P — 2lzcos 9 + 2) (81)
2
= 1—2c0s 0 + 5 sin? 6 + ete. (82)
= 1— 20038, (83)
2
for all points of BAC near to A, and for which oi is insignificant.
The displacement at P’, due to the secondary waves originating in
dz, will therefore be
: adz . 7
6 = ap sin 3 (Vi — T+ #008 8). (84)
Replacing AP’ by /, and integrating, we have for the resultant
displacement of P’ due to the great wave,
= 38 =F fodesin = (Vt 1 + 2 008 6)
(85)
a an
= — 85T eos 9 88 GZ (Vt — 1 + 2 008 6) ;
RELATING TO SOUND AND LIGHT. 51
and between the limits corresponding to + b and — 42,
_ GA , &nbcosé . Viti —1
= ap (86)
The maximum displacement is therefore
at . rb cos O
ee en)
70. 1°. The above value of the displacement will vary with d,
6, 4, and 2. When, as in sound, A is very great as compared with 3,
6...
amb aes will be so small that the arc may be substituted for the
sine without material error, and
= = (88)
which is independent of 0.
2°. When 4 is much smaller than @, as in the case of light, we
have, when cos @ is very small, and 6 therefore differs but little
from 90°, again
c= ne (89)
At other points, where @ is not great and cos 6 not small, the
resultant displacement becomes equal to zero when
eae cose =+tn, +427, +37, ete;
‘i a 2a 3A
that is, when cos@6= + 3H’ + 3B’ + ap” etc.
The greatest resultant displacement, other than that indicated
above, will be found by making in Eq. (87),
ane ser St 1, (90)
oA,
ml cos 0’
sin
and it will be equal to
and, since the intensity of the sensation is directly proportional to
52 ELEMENTS OF WAVE MOTION.
the square of the maximum displacements, we will have the rela-
tion of the intensities,
a2)? 4a? A
ee. 6 Fe, er Ji 22 22>] : If
me cosd° FP SEO 4726? cos? 6 1. (91)
71. In acoustics it will be shown that the wave lengths corre-
sponding to audible sounds will vary from wal = 57’ to oi
20 40000
= + of an inch, and therefore there will be no point exterior to an
aperture where the displacement will not occur, and hence the cor-
responding sound be heard. In light, the wave lengths vary
between .000026 and .000017 of an inch, and there will be, accord-
ing to the 2° case, alternations of light and darkness surrounding
the central line drawn from the place of original disturbance to the
centre of the aperture. These zones are called Huyghens’ zones,
and will be again referred to in the subject of diffraction.
72. Diffusion and Decay of Kinetic Energy. The
displacement of any molecule due to wave motion of a given wave
length is independent of the periodic time, and, since the orbits of
the molecules are described in equal times when they arise from a
given periodic motion, they will be directly proportional to the dis-
placements or any other homologous lines. The velocities, then,
of the moving molecules being represented by v, their kinetic ener-
2
gies will be represented by > Then, because these energies are
transmitted without appreciable loss from the molecules of one sur-
face to those of another, we will have the energies of the molecules
of the two homologous surfaces,
2 12 2 12
dn? 7 = nr’? 7, or re rs (92)
2 19
that is, : *S :: 72: 7%, or varying according to the law of
the inverse square of the distance. Similarly, we will have
6" = dr, (93)
or the maximum displacements inversely proportional to the dis-
tances to which the disturbance has been propagated.
RELATING TO SOUND AND LIGHT. 53
73. Reflection and Refraction. It is difficult to con-
ceive, satisfactorily, in what manner the molecules belonging to
two media of different elasticity and density are arranged with re-
spect to each other in or near the bounding surface which separates
them. When they occupy positions of relative rest, the elastic
forces must be mutually counterbalanced and must be equal to those
affecting the molecules within the media. We may assume the two
media to have different densities and elasticities, and the relative
positions of the molecules near the separating surface to be deter-
mined by the action of the equilibrating molecular forces. But
when a disturbance arising in one of the media reaches the surface,
the molecules of the second medium must, in general, have motions
and displacements different from those of the first. If we consider
alone the difference in density of the molecules of the media, we
perceive that the energy in the incident wave will not be wholly
given up by the molecules to their neighbors in the new medium.
In either case, whether the molecules have greater or less density,
a return waye will originate in the incident medium, analogous to
the reflected motion in the impact of elastic balls. Again, if the
elasticity of the media be different, the elastic forces for equal dis-
placements will be different, and thus cause a return wave in the
incident medium. We may therefore assume, for the present, that,
owing to the different elasticities or densities, or both, there will be,
in general, a separation of the incident wave whenever it meets a
surface separating two media of different density and elasticity.
The fact of such a separation is experimentally demonstrated in the
phenomena of sound and light. The velocity of wave propagation
will be shown to be a function of the elasticity and density of the
medium, and therefore the waves, in general, will proceed in the
two media with different velocities.
74, The plane of incidence is that plane which is normal to the
deviating surface and to the wave front.
The plane of reflection is normal to the deviating surface and
to the reflected wave front; it coincides with the plane of incidence.
The plane of refraction is normal to the refracted wave front and
to the deviating surface.
75. Diverging, Converging, and Plane Waves.
When the energy of molecular disturbance is distributed among
54 ELEMENTS OF WAVE MOTION.
the molecules, upon an increasing wave front, the wave is said to
be diverging ; when among those of a decreasing wave front, a con-
verging wave; and when among those of an unchanged wave front,
a plane wave. An indefinitely small portion of the front of any
diverging wave, taken at a correspondingly great distance from the
origin, may, without sensible error, be considered as coinciding
throughout with the tangent plane to the wave front, and consid-
ered as a plane wave. The molecules of a plane wave at any
assumed position are animated by equal parallel displacements, and
undergo all their phases while the plane wave advances a distance
equal to the wave length, measured in a direction perpendicular to
the plane.
76. Differentiating Eq. (67), we have
- = Ts. (94)
Multiplying both members by m, the mass of the molecule, and
replacing mo by its equal Ud, the intensity of the elastic force
developed by the displacement 0, we have
PS An®m
U8 =n = ag Vis (95)
2
whence, —— — ve (96)
Hence, when a plane wave is propagated without altera-
tion in a homogencous medium, its velocity of propagation
is directly proportional to the square root of the elastic force
developed by the displacenvent of its molecules.
77. Reflection and Refraction of Plane Waves.
Let the incident plane wave AC (Fig. 11) meet the deviating sur-
face at all points, in succession, from A to B. Let V and A be the
velocity of wave propagation and the wave length in the medium of
incidence, and V' and A’ those in the medium of intromittance.
Let AB = ds, and CB = Vdt. While the disturbance in the in-
cident wave is moving from C to B, the disturbance from A as 4
centre will proceed in all directions in the medium of incidence,
RELATING TO SOUND AND LIGHT. 55
and be found, at the instant considered, upon the hemisphere whose
radius is AD = CB = Vdd, and in the medium of intromittance
on the hemisphere whose radius is AD’ = V' dé.
Each point in the line AB
will, in like manner, become in
succession a new centre of dis-
turbance, sending secondary
waves into the media of inci-
dence and of intromittance,
whose radii will, at the instant
the incident wave reaches B, be
equal to V and V' multiplied
by the interval of time elapsing
between the instant of arrival of Figure I.
the wave front at the centre
considered and that of its arrival at B. The surface through B,
which is tangent to all the reflected pulses, may be taken as the
front of the reflected wave, for it will contain more energy than
any other surface of equal area in the incident medinm. Similarly,
the surface through B tangent to all the refracted pulses will con-
tain more energy than any other of equal area in the medium of
intromittance, and may be taken as the front of the refracted wave
at this instant. These surfaces are readily seen to be planes; hence,
denoting the angle CAB = ABD by ¢, and ABD’ by ¢’, we wilh
have
dssinog = Vdt, dssing' = V'dt; (97)
from which we obtain:
sin @ = a sin ¢’ = psin ¢, (98)
which is known as Snell’s law of the
sines; is called the index of refrac-
tion.
78. The angles ¢ and ¢’ made by
the wave fronts with the deviating sur-
face are, respectively, equal to the
angles made by the normals to the in-
cident and refracted waves with the |
normal to the deviating surface, and Figure 12. ;
56 ELEMENTS OF WAVE MOTION.
are called angles of incidence and refraction. The angles of inci-
dence and refraction are measured from the normal to the deviating
surface on the side of the medium of incidence to the normal of the
incident wave, and to that of the refracted wave produced back
into the medium of incidence.
The angle of reflection is measured from the normal to the de-
viating surface to the normal to the reflected wave front, and is
therefore negative. In the reflected wave, since the velocity of
waye propagation is unchanged, » is equal to unity, and Kq. (98)
becomes
sin @ = — sin ¢’. (99)
79. These principles may, in ordinary cases, then be summa-
rized as follows:
1°. The planes of incidence, reflection, and refraction are coin-
cident.
2°. The sine of the angle of incidence is equal to the index of
refraction or of reflection multiplied by the sine of the angle of re-
fraction or of reflection.
The modifications which take place m polarized light will be
referred to hereafter in physical optics.
80. We see from Art. 77 that the reflected and refracted waves
are plane when an incident plane
wave meets a plane deviating sur-
face. It is evident also, from the
construction, that the reflected
rays are all normal toa plane NN’
symmetrical with MM’ with refer-
ence to OX; and that the incident
and reflected rays are directed
from their corresponding planes
towards the deviating surface.
“The refracted rays are normal to
a pane RR’ on the same side of
the deviating surface as the incident wave, and are also directed
towards that surface.
Ss.
Figure 13,
81. General Construction of the Reflected and
Fefracted Waves, Let the deviating surface AB (Fig. 14) be
any whatever, and the rays proceed from any origin O; take, in
RELATING TO SOUND AND LIGHT. 57
the medium of incidence, any spherical surface SS’, with centre at
O, as the incident diverging wave ; then, from all points I, I’, I’,
etc., of AB, describe spheres, whose radii are equal to the intercepts
of the rays between SS’ and AB.
If, now, tangent planes be drawn
to the deviating surface at I, I’, 1”,
etc. and to the surface SS’ at the
corresponding points s, s’, s”’, etc.,
each pair of tangent planes will
determine, by their intersection, a
right line, through which if a
plane be passed tangent to the cor-
responding sphere on the other
side of the deviating surface, it
will be symmetrical with the in- > Se oS =
finitesimal surface of SS’ at s with poy —
respect to that of AB at the point
I; and similarly for the other
points. Bycontinuity, these points
of tangency may be considered as
forming the envelope of the re-
flected wave. The direction of the
reflected rays is found by joining Figure (4,
these points with I, I’, I”, etc,
and extending the lines toward and beyond the deviating surface.
82. By the proper modification of the radii due to the value of
, the index of refraction, the envelope of the refracted wave and
the direction of the refracted rays may be constructed.
83. Considering the reflected wave as a new incident wave, the
new reflected wave, by another deviating surface, can be constructed
by an application of the above principles; and since reflection may
be considered as refraction whose index is — 1, the principle may
be generally stated, that any number of reflections and refractions
may be replaced by a single refraction at a supposable deviating
surface with a properly modified index of refraction.
84, Let DEF (Fig. 15) be any incident wave whose rays are not
necessarily parallel; MNP any deviating surface. At some subse-
quent time ¢ the incident wave will occupy some position such as
ABG, FG being equal to EB = DA = Vé._ By the principle
sO
58 ELEMENTS OF WAVE MOTION.
established above, aby will be the enveloping surface of the reflected
wave corresponding to ABG, and a,,g, that of the refracted wave,
and both will be concurrent, that is, the phases of the molecular
motions on them will be
similar; gPG’, OdNB’,
aMA' will be the re-
flected, and g,PG,,
b,NB,, a,MA, the re-
fracted rays.
85. Prolong the con-
secutive rays of either
the reflected or refract-
ed waves, say the re-
flected wave abg, untu
they meet two and two:
they will be tangent to
the surface «By, which
is the evolute of adg.
Since the reflected rays
are all normal to adg,
this evolute will corre-
spond to any other po-
sition of the reflected
wave, also. The surface
of which «By 1s a generatrix is in optics called the caustic surface.
It is evident that the points of this caustic are not concurrent,
because their distances, being equal to the radii of curvature of adg
from the reflected wave, are themselves unequal; and points, in
order to be concurrent, must be at equal distances from the wave
surface. Whether the caustic be real or virtual, the displacements
of its molecules being either due to that of two rays, or apparently
so, the energy of the molecules, and hence the resulting sensation,
will be greater than that due to but one ray.
86. When the evolute «By is known, the various possible posi-
tions of the reflected wave can readily be determined. In the ordi-
nary cases considered in optics, the surfaces abg are those of
revolution ; the caustic is then also a surface of revolution. Sup-
pose aby to be one of the generatrices of the reflected wave, consid-
ered as a surface of revolution, and «@y to be its evolute; then, by
Figure 15,
RELATING TO SOUND AND LIGHT. 59
the property of the evolute, if the tangent aca’ be caused to roll on
ay, each point of this tangent will describe one of the sections of
the reflected wave. Thus, a’'b'g’, a''Bg", and abg are such sections;
the second of these being of two nappes, tangent to each other and
normal to the evolute at the point £.
87, The principle that the rays, after the wave has been sub-
jected to any number of reflections and refractions, are all normal
to a theoretically determinable surface, and consequently to a series
of surfaces, of which any two intercept the same length on all the
rays, is principally applicable to tie determination of caustic sur-
faces, and to the formation of optical images, and will therefore be
further discussed in that branch of the subject.
88. Utility of Considering the Propagation of the
Disturbance by Plane Waves. Inahomogeneous medium,
the arbitrary displacement of a molecule gives rise to elastic forces
whose intensities depend on the degree and tiie direction of the
displacements, and whose directions are not, in general, those of the
displacements. In Art. (35) we have seen that the displacements
must be made only in exceptional directions, in order that the elas-
tic forces varying directly with the degree of the displacement
should be wholly in those directions. Should the orbit of the dis-
placed molecule be curvilinear, it is evident that, at each point of
its path, the elastic forces developed would vary both in direction
and intensity, and thus the general problem becomes one of extreme
intricacy.
89. If, however, it be possible to limit’ the discussion to that of
molecules in the same plane, all actuated by equal and parallel dis-
placements, the variation as to direction of the elastic forces may,
perhaps, be eliminated. It has been shown, Art. 76, that when a
plane wave is propagated without alteration in a hemogeneous me-
dium, the velocity of propagation is directly proportional to the
square root of the elastic force developed by the displacement.
Hence the importance of deducing from the general equations (18)
the corresponding equations applicable to the vibratory motions
propagated by plane waves.
90. At the time ¢ let r be the distance of the plane wave, in a
homogeneous medium, from the origin of co-ordinates; ¢ the dis-
placement of the molecules whose co-ordinates are x, y, 23 5, 1; S,
60 ELEMENTS OF WAVE MOTION.
the projections of « on the rectangular co-ordinate axes ; and a, 8, y,
the angles made by the displacement with the axes, respectively.
eR
We then have g = dsin = (Vé—1); (100)
» 20
§ = dcosa sin = (Vi—7);
. aT
n = 5 cos B sin = (Vi — 1); (101)
= 8 cos y sin =" (Tt),
Let 7 + Ar be the distance of the plane at a subsequent instant
from the origin, and 7, m, n, the angles made by the normal to the
plane with the axes, then
r= axcosl+ ycosm + z2c08 7, (102)
Ar = Ax cosl + Ay cos m + Az cos n. (103)
From Eq. (101) we have
£4 AF = 6 cos «sin =" (Vé—r — Ar)
= dcosa sin “a (Tt — 1) cos “nr Ar (104)
ar . 27
— cos -~ (Vi —r) cin ar |;
from which, and similarly for the axes y and z, we have
7
Ag = dcosea sin - (Tt—r) (cos = Ar — 1)
2 ’ 4
— cos > (Vi — 71) sin a | ;
An = 6 cos B [sin 2 (Vt —») (cos ™2 ar —1)
a a‘ \ (105)
Qn Me el
— cos = (Vt —r) sin aot ds
[ . Q0 a7
Ag = dcosy | sin a, (Vt —71) (cos zr 1)
Qqr peeps Fal
— cos a (Vier) sin —- Ar | J
RELATING TO SOUND AND LIGHT. 61
Substituting these values in Eqs. (18), and, since the medium is
homogeneous, the sums arising from the substitution of the second
part of the values of Ag, Ay, AZ, and which are of the form
=u (r) sin a Ar,
Ay . ar
Zp w(r) sa sin = Ar,
Ay Az . an
Ze yp (r) = sin = Ar,
n = Ar, - (106)
A 5
Zu w (r) “- sin = Ar,
A® . ar
=p yp (r) 2 n>
Aw Az . 20
=p wp (r) —a_ ‘sin = Ar,
all reduce to zero, because they are formed of terms which, two and
two, are equal, with contrary signs; for, to the values of Aw, Ay, Az,
equal, with contrary signs, correspond values of Ar ee are also
equal and have contrary signs. oe
equal, 1 — 2 sin? 7 An, and 6 sin oT (Vt—r) by ita aa g,
Eqs. (18) become, for plane waves,
2
X= = = COS & Su oe +904 | ae
+ cos B Ey p = Y sin = 5 Ar + cos y Sup ej sin? = 5 4n
ep aes ot 2
= 5 = cos e Eu p(n) a sin 5 Ar
;
+ cos B Ep e(r) + v(r) af | sin? sar
Ay Az . 4%
+ cos y Zu w(r) “i sin? ae
02 ELEMENTS OF WAVE MOTION.
Z AzAz . 9
£2 = cos « Tu p(r) pa sin? 5 Ar
Ay Az .
+ cos B Sy wp (7) a sin? 5 Ar
+ cos y =p | ° (*) + o(r) = | sin? 5 Ar,
(107)
91. The conditions for the propagation of the plane wave with-
out change are
ei Sr ea
cosa cos cosy Uy. (f08)
Substituting, in Eq. (107), for Av its equal,
Az cos 1 + Ay cos mm + Azcosn = Ar, (109)
and substituting in Eqs. (108) the values of X,, Y,, Z,, thus ob-
tained, we will have two relations which, with
cos? « + cos? B + cor y = 1, (110)
will enable us to determine the angles «, 8, y, which the displace-
ment should make with the axes, in order that the propagation of
the plane wave may be possible.
92. Because of the equality of the coefficients of cos 6 and
cos « in the first and second of Kqs. (107), and of cos @ and cos y
in the third and second, and of cos « and cos y in the third and
first, we can, by substitutions and reductions similar to those em-
ployed in Art. 33, deduce corresponding principles, and hence
determine that, for each direction of the plane wave, there corre-
spond, for the molecular displacements, three rectangular directions
such that the plane wave may be propagated without change, and
that these three directions are parallel to the three axes of an ellip-
soid whoge equation is,
RELATING TO SOUND AND LIGHT. 63
2
2 Sp E (r) + P(r) a | sin? : Ar
Ay |. om
+y Su} o(r) + (7) aE sin? 3 Ar
+2 3u[ 40) + 90) 22 | sine Zar
a A $=. (111)
Ax Ay ang T
72 sin 3 Ar
Av Az . ,7
pas
7a sin x47
+ 2yz Sup (7) a a sin? : Ar
+ 2ay Ey +p (r)
+ az Ly w (r)
J
This is called either the inverse ellipsoid or the ellipsoid of polar-
vation. Having also the relation expressed in Hq. (109), we see
that the coefficients of Eq. (111) depend upon the angles /, m, n,
which determine the direction of the plane wave, upon certain con-
stants which define the constitution of the medium, and upon the
wave length. The velocity of propagation is inversely proportional
to the length of that axis of the ellipsoid to which the molecular
displacements are parallel.
93. Relation between the Velocity of Wave Propa-
gation of Plane Waves and the Wave Length in
Isotropic Media. All directions being identical in isotropic
media, we will assume the plane wave normal to the axis of z. We
then have Ar = Az, and
Sp (n) 4 sin? ~ Ax = 0,
2 A
AB ios 7
ze w (r) a sin? ; Az = 0, (112)
Az.
=p wp (r) ws sin? 5 Aw = 0;
and Eqs. (107) reduce to
64 ELEMENTS OF WAVE MOTION.
Tv
Az |,
X, = cos a Lp 6 (r) + (r) 2 | sin? 5 Ae,
7
Y, = cos B =p E (7) + ¥(r) as | sin? 7 Ae, (118)
Tv
Z, = cosy Lp 6 (r) +4 | sin? 7 Aes
and the equation of the ellipsoid to
Av |. ot
a Sp E (r) + ¥(r) S| sin? 5 Aw
Ay? | 2, 7 =e
+y Sh] o(r) + o(r) - sin? zoe = 1; (114)
+ 2 3p E (7) + P(r) sf | sin? ; Az
and, since all directions perpendicular to the axis of 2 are identical
with reference to the plane of the wave, we have Ay = Az, and
Eq. (114) of the ellipsoid becomes one of revolution about the axis
of z Whence, we conclude that, in an isotropic medium, a plane
wave normal to a given direction can be propagated without change,
whenever the molecular displacement is parallel or perpendicular to
this direction. ‘To any one direction of normal propagation in such
a medium, there corresponds an infinite number of waves with
transversal vibrations, having the same velocity, and but one wave
with longitudinal vibrations whose velocity is different from those
with transversal vibrations.
94, For the wave with longitudinal vibrations, we have
c= 0, B = Y — 90°,
. Ae | .
U, =X, = Sn le (r) + (7) = | Be Ok aie)
ry = Z, =a 0;
and, from Eqs. (96) and (107), we have
v= i Zu E (r) + ¥(r) a | sin? ; Ag. (116)
RELATING TO SOUND AND LIGHT. 65
In Acoustics, it will be shown that sound is due to longitudinal
vibrations of the medium. This equation will then be applicable in
all cases of sound arising from such vibrations, and will be referred
to, in that branch of the subject. In Optics, it will be shown that
transversal vibrations only are efficacious in producing light.
95. For waves with transversal vibrations in isotropic media,
the velocity is independent of the direction of the displacement.
We can then suppose the displacement parallel to the axis of y, and
thus have
«=y = 90°, B=),
Kk, = Zz, a
(117)
Y, = te) o(r + vin sin? — pie Oy;
and = V? = ae Poueat | sine ac. (118)
This equation is applicable in light, for the determination of
wave velocity in isotropic and homogeneous media, and will be used.
‘hereafter in determining the velocity of light propagation.
By developing sin? : Az into a series, we find
7 7 1. x4 2 76 1
ine SAG 8
sin’ x Ae — 73 Av — 3°74 Ag! + 45° po 315 a + ete.
Substituting this in Eq. (118), we obtain
b c ld
a 72 + 3 + 36 + ete, (119)
in which a, 0, c,.... have for values,
1 Ay?
a= }mlom+ven on Aa’
(eee [em + ve 1
c= Zu lomive en aa
SS Re EARS
66 ELEMENTS OF WAVE MOTION.
These constants depend only on the constitution of the medium,
and decrease very rapidly in value, for Az is always a very small
quantity. If the wave length be not excessively small, if it sur-
passes a certain value which observation only can determine, the
terms of the second member of Eq. (119) will have very rapidly
decreasing values, and we will obtain an expression approximately
near to 1? by taking only the first few terms. Hence, a must be
positive, and, since observation shows that the most refrangible
rays are those of the shortest wave length, and that, as a conse-
quence, J” decreases with A, 4 is necessarily negative.
96. Hence, in isotropic media, the elasticity beg uniform in
all directions, the form of the wave surface will be spherical, and
when the displacements are longitudinal, its radius at the unit time
from the epoch will be the value of V obtained from Eq. (116);
when the displacements are transversal, the radius will be the value
of V in Eq. (118). The former relates wholly to waves of sound,
and the latter to those of light.
The subsequent discussion will now apply to transversal vibra-
tions alone, and the conclusions derived belong therefore to the
transmission of light undulations.
Experiment shows that the media which transmit the waves of
light are not in general isotropic, and as a consequence the form of
the wave surface will not be spherical. We will, therefore, now
seek the form of this surface in the general case, and make use of
the properties of plane waves for this purpose.
97. Plane Waves in a Homogeneous Medium of
Three Unequal Elasticities in Rectangular Direc-
tions. In the plane wave, the following conclusions have been
deduced :
1°. The displacements of the molecules, in each position of the
same plane wave, must be rectilineal and parallel to each other and
to their original directions.
2°. The elastic forces developed by these displacements must be
either in the directions of the displacements or alone efficacious in
these directions.
3°. The propagation of the plane wave unaltered igs then
‘possible.
These conclusions involve, as consequences, a constancy of ve-
RELATING TO SOUND AND LIGHT. 67
locity of propagation when the plane wave is unchanged in direction,
and a variation in the velocity as the direction is changed. Hence,
if the elasticities of a homogeneous medium differ in all directions,
and we suppose plane waves, having all possible positions, originate
at any point m of an indefinite medium, these plane waves, at the
end of a unit of time, will be at different distances from m. The
surface which is the envelope of all these plane waves at this instant
is called the wave surface.
98. Let a>b>e
be the principal axes of elasticity of such a medium. Then a?, 2%
c, will measure the elastic forces developed in these directions by a
displacement equal to unity, and any of the surfaces of elasticity
heretofore determined can be used to obtain the elastic forces devel-
oped by an equal displacement in the direction of the corresponding
radius vector of the surface. The velocity of wave propagation
being proportional to the square root of the elastic force, Eq. (96),
its value can be found when the elastic force due to the displace-
ment in any direction is known.
99. Fresnel made use of the single-napped surface of elasticity
whose equation is
Ce + bye + ee = vr; (121)
but for plane waves, the inverse ellipsoid of elasticity or first
ellipsoid,
ae 4- y+ cv? = 1, (E)
together with its reciprocal ellipsoid,
x y gL om
@ + R + a= 1, (W)
can be more readily used, because of its better known properties.
The squares of the semi-axes of (W) and of the reciprocals of (E)
are the principal elasticities of the medium.
100. There are two cases to consider:
1°. The plane of the wave contains two of the principal
axes, and hence is one of the principal planes of the medium.
The plane cuts the ellipsoids in ellipses whose semi-axes are either
two of the principal axes. Whatever be the direction and amount of
68 ELEMENTS OF WAVE MOTION.
the displacement, it may be replaced by its components in the di-
rection of the axes proportional to cos @ and sin «, « being the
angle made by the displacement with either axis.
Considering these separately, we see: 1°, that each will commu-
nicate to the molecules in the adjacent plane analogous rectilineal
motions which will be propagated without alteration of direction;
2°, that the elasticities, and hence the velocities of propagation
which belong to these two, are different, and that after a time there
will be two series of molecules situated in parallel planes, parallel
also to the primitive plane, which will contain all of the original
energy; 3°, that the vibrations of the molecules :n these two plane
waves will be at right angles to each other.
101, 2°. The plane wave is any whatever. The sections
of the ellipsoids will be ellipses, but will not in general contain
either of the axes of the ellipsoids. There will then be no direction
of the displacement that can give a resultant elastic force in the
direction of the displacement. It is, therefore, essential for a rec-
tilineal oscillation of the molecule and for a consecutive transmis-
sion of this oscillation, that there should be no tendency of the
rectilineal displacement to be deflected on either side, but that the
line of the resultant force should be projected upon the displace-
ment. As it is not in general in the plane, but oblique to it, it can
be resolved into two components: one normal to the plane, which
is not effective in light undulations ; and the other, which is alone
efficacious, in the direction of the displacement. In each elliptical
section there are two such directions, which are named singular
directions, and which are perpendicular to each other. Assume any
plane section through the centre of (E); the elasticity measured by
the squares of the reciprocals of the radii-vectores is the same to the
right and left for the two axes of the section, and is the same only
for them. Through either of the axes pass the normal plane to the
section ; it will cut all the parallel plane sections in their homolo-
gous axes. With reference to this normal plane, the radii-vectores,
and therefore the elasticities of each section, are symmetrical.
Hence, if the displacement be along ons of the axes of the section,
the total elastic force will be in the normal plane, and will be pro-
jected on the axis of the section. And since the ellipsoid semi-
diameters are inversely as the velocities of propagation, the recipro-
RELATING TO SOUND AND LIGHT 69
cals of the axes will measure the velocities of wave propagation.
Hence is established the fact that for each section there are two of
these singular directions, and that they are rectangular. These two
singular directions perform the same function for the vibrations of
the plane wave as do the axes of elasticity themselves when the
plane wave contains them. Each vibration is replaced by two others
in the direction of the singular directions, and these two compo-
nents proceed in the medium without change of direction, but with
different velocities, so that there are then, in the general case, two
plane waves parallel to each other and to the original plane wave.
If « be the angle made by the displacement with one of the axes,
the component displacements will be proportional to cos « and
sin «, and the elastic intensities to cos? « and sin? « Whatever
may be the direction of the original supposed vibration in the plane
wave, the two plane waves which replace it are always the two above
designated.
102. If the plane of the wave coincides with either of the circu-
lar sections of the ellipsoid, the plane wave will be propagated
withont alteration, whatever be the direction of the displacement,
with a velocity equal to 4, the reciprocal of the mean semi-axis of
the ellipsoid.
108. The Double-Napped Surface of Elasticity.
If through the centre of (E) we pass any plane, and on the normal
to the section at the centre set off distances inversely proportional
to the semi-axes of the section, the locus of all these pairs of points
is called the doudle-napped surface of elasticity. For, each radius
vector measures the velocity of propagation of one of the plane
waves, arising from a displacement in the plane of section, and the
square of each of these normal velocities is the measure of the elas-
tic force developed by the component displacement along the axes
of the section.
104, If through each of the points so determined planes be
passed parallel to the corresponding plane of section, the envelope
of all these planes will be, by definition, the wave surface. Hence,
the latter can be constructed by points from this surface of
elasticity.
105. To get the polar equation of the latter surface, let us take
for co-ordinate axes the principal axes of the medium ; let 1, m, n,
70 ELEMENTS OF WAVE MOTION.
be the angles made by the normal to the plane wave with these
axes, wv, y, z, respectively ; «, B, y, those which one of the axes of
the ellipse of section make with the same axes ; then we have
cos « cos 1 + cos B cos m + cos y cos 2 = 0. (122)
The elastic force developed by a displacement parallel to the
axis of section is projected on the plane of the wave parallel to this
displacement, and its components are
Y=ecse, Y=PcosP, Z=c cosy. (128)
The cosines of the angles which this elastic force makes with the
axes are then proportional to these values. An auxiliary right line
perpendicular to the direction of the elastic force and to the dis-
placement will lie in the plane of the wave, and if w, v, w, be the
angles which it makes with the axes, we will have
@ cos « cos u + 8? cos B cos v + c? cos y cos w = 0,
cos & cos u% + cos B cos v + cos y cos w = 0, } (124)
cos 7 cos wu + cos m cos v + cos 2 Cos w = 0.
Representing the velocity of propagation of the plane wave by
V, we have
V2 = @ coe « + & cos? B+ C cos? y. (125)
Combining the above equations, and eliminating the quantities
«, B, Y, u,v; W, We will have an equation containing V, J, m, n,
which will be that of the surface required. To eliminate w, , 2,
we will make use of the method of indeterminate coefficients; thus,
multiply Eqs. (124) by B, A, and unity, respectively, add the three
resulting equations, and from the conditions for B and A that the
coefficients of cos v and cos w shall reduce to zero, we will have
(4 + Ba’) cose + cosd = 0,
(4 + Bd?) cos B + cos m= 0, (126)
(1 + Be?) cosy + cosn = 0.
Multiply these by ‘cos «, cos 8, and cos y, respectively, add,
and reduce by Eqs. (122), (125); we will have
A+ BV?=0, (127)
RELATING TO SOUND AND LIGHT. 71
Substitute this value of in Eqs. (126), and we have
cos?! = B(V? —a*) cosa,
cos m= B’V? — 2) cos B, (128)
cos » = B(V? — @) cos y.
From which we get
cos J cos m COS 2
Ve_-@ Rr_—R y2—
cose ~ cosB ~ cosy (129)
= \/ cos? J cos? m cos? 7
= (apt (V2? —PpP - (2-2 J
Replacing cos «, cos 8, cos y, in Eq. (122), by their propor-
cos 7 cos m COs 22
tional quantities, Powe? PoP Poe wv have
cos? J cos? m cos? 72
Ve * Woe Vise
= 0, (130)
the polar equation of the double-napped surface of elasticity, in
which V is any radius vector.
106. The Wave Surface. Through any point of the sur-
face of elasticity pass a plane perpendicular to the radius vector at
that point, and let r, 4, u, v, be the polar co-ordinates of any point
of the plane. The equation of the plane will be
cos J cos A + cos m cos + cos mtos v = ¥. (131)
We have also, as equations of condition,
cos’ J + cos? m + cos? n = 1, (132)
cos? 7 cos? m cos? 72
oe wp) woe ee)
The wave surface is the enveloping surface of the planes given
by Eq. (131), and its equation can be determined by eliminating V,
l,m, m, and finding an equation between r, A,, ¥. To do this,
differentiate Eqs. (131), (132), (133), regarding cos? and cos m as
independent variables, and we will have
92 ELEMENTS OF WAVE MOTION.
dcosn 1 : d - )
Basa B08 dcosi 7 dcoosl’
7 . io cos 0
cos 7 + cos 7 aoe
cos 1 COs 22 dos n eA )
V_—e&' V2—e deol
dv r cos? 1 cos? m cos? 2 |;
=Jost' |\(eoar (eae Cae
d COs 1 d i,
Sa eS dcosm vr dcosm’
2 oe d cos 0
cos m + cosn 7 = 0, ”
COs 72 cosx dcosn > (135)
PRT Ve des m
_ av cos? 7 4 cos? m 4 Cos? 72 :
~ deosm (V?—a@pPe* (V2—Be (V2—epF] J
107. To eliminate g — anes 5 av ; ay
dcosl’ dcosm’ dcosl’ dcosm
ply Eqs. (134) by 1, A, and — B, respectively, add the resulting
equations together, and perform the same operations on Eqs. (135).
Supposing the indeterminate quantities A and B to have such
: : ‘ dcosn dcosn dV
values as will make the coefficients of aur eee ee
dV
d cos m
>» multi-
—, equal to zero, we will have
cos 7
roe
cosA+ Acos] =B
cos » + 4 cos m = Ba
(136)
cos
cosv + A cosn = Barre
1 cos? / cos? m Cos? 72
ro Bv| ay — @) a (V?— P,P + (V2— “a3
Multiply the first three of the above equations by cos J, cos m,
RELATING TO SOUND AND LIGHT. 73
and cos #%, respectively, add the resulting equations, and reduce by
Eqs. (131), (182), (133) ; we will have
v
At = 0. (137)
Squaring the first three of Eqs. (136), and adding, we get, after
reduction
V B
1+ 2d—-- + A =; (138)
whence, we have
A=—4, Bal ie—r, (139)
Substituting these values in Eqs. (136), we obtain
r cos A V cos J )
foe oo
7 COS |b V cos m
PR Vea
r COs v V cos 2
f=6 Ving r (140)
_-7 ois Z cos? m cos? 2 i
2. 2 (= Wray (FE -- Be (V2 — ep
~ cost A cos? cosy |
=r E — ey = (P— BP Pep j
The first three equations (140) can be placed under the form,
yee cos J
Tr
V
cos # ——— cos m = (? —
¥ 2
cos » — —— COB = (rF—
A
Vy) oe
cos
— V2) E
= (7? —
Cos Ye ?
Va A —s
pe BP
(141)
Adding these, after multiplying them y cos A, COS f, COS ¥,
respectively, and reducing by Eq. (131), we have
v4 ELEMENTS OF WAVE MOTION.
i-E= ry
cos? A cos? ps cos? ”) . ;
2 A_@ eae ee (142)
whence, dividing by 7? — V?, we have
cos? A cos? cos? v 1
Pee! PORT PH a 7?
the polar equation of the wave surface.
108. A more advantageous form for discussion can be obtained
by subtracting the identical equation,
1 cost A cos? ys Cos? v
a a a noe
from the equation above, by which there results
corti’ Pcost ye cov _ 0 (145)
— & v2 — rr — i
109. To obtain the equation of the wave surface in rectangular
co-ordinates substitute for cos A, cos #, cos v, and 7, their equals,
=, ‘ <, and’ /a* + y? + 2; whence, we have
(2+ +2) (ee + PY + 22) —@ (PR Pe) e can
—B(C+0)P—C(@+P)24 PPE = 0. Aas
This equation being of the fourth degree, the surface is of the
fourth order, and, as will be shown hereafter, consists of two dis-
tinct nappes, having but four points in common.
110. If two of the velocities become equal, as, for example,
6 =, the equation gives
C+ p+eP=B, (147)
C2 +P (y+ 2) = &P, (148)
which shows that the wave surface, under this supposition, consists
of a spherical surface and that of an ellipsoid of revolution tangent
to the sphere at the extremity of its polar axis.
RELATING TO SOUND AND LIGHT. 75
Finally, if the three principal velocities become equal, or a =
= ¢, as in isotropic media, Eq. (146) becomes
+P t+ A a, (149)
and the wave surface becomes spherical, as has been heretofore
shown.
111. Construction of the Wave Surface by Means
of the Ellipsoid (W). Let us suppose that the ellipsoid (W),
ae poe :
ee ee ta (150)
be cut by any plane through its centre, and that distances be laid
off on the normal equal to the semi-axes of the elliptical section.
Referring to the construction of the double-napped surface of elas-
ticity by means of the ellipsoid (E),
e+ y+ ee = 1, (151)
we see that in designating the polar co-ordinates of the points con-
structed by the aid of (W) by r, a, », », the equation of their loci
can be obtained from the equation of the double-napped surface,
cos? 7 Cos? m cos? 2
Tet RR woat® (152)
by substituting for Fe, a, B, C, d, m,n, respectively, a = 7 =
A, #, v. We thus obtain re a Cc
cos? A. Cos? pu cos? v a
fog ee oe oe (153)
Ag PR Re
a cos? 2 Bb cos?* wc cos? v
ae a) pe R Pua 0, (154)
which is the equation of the wave surface. Hence, points of the
wave surface can be constructed from the ellipsoid (W) in precisely
the same manner as points of the surface of elasticity from the
ellipsoid (EH), except that in the former, distances equal to the semi-
axes are laid off on the normal, and in the latter the distances are
equal to the reciprocals of the semi-axes.
76 ELEMENTS OF WAVE MOTION.
112. Direction of the Vibration at any Point of
the Wave Surface, Let us consider any plane wave tangent
to the wave surface; the displacement propagated by this plane
wave makes the angles «, 8, y, with the axes; the radius vector of
the wave surface at the point of tangency makes with the axes the
angles A, #, v; therefore the angle between these two lines will de-
termine the required direction.
Eliminate in Eq. (129) the angles J, m, , by means of the first
three of Eqs. (140), which, with the last of Eqs. (136), will give
cos A COS ft cos v
rae PP ree
cosa cosB ~~ cosy
cos? A Cos? fu cos? v
7 Ge Gap! oF (155)
a cos* Z cos? m cos? 2
(= a) 1 (r= ~ 6)? + (V3 Vy2— -~
: fa cal
oe BVr ~~ pa/p? — 2”
cos A COs @&
whence, 23 ae
°
cose cos a
Aap rv — o
cos v cos re
rH ey palpi
Substituting in Eq. (148) of the wave surface, we have
V2
cos « cos A + cos B cos # + cos y cos ¥ = ft (157)
In the figure, let M be any point of g
the wave surface, OM the radius vector,
and OP the perpendicular V on the tan-
Vv.
gent plane to the wave surface; — is
Tr Figure 16.
RELATING TO SOUND AND LIGHT. 17
: pe
then the cosine of POM, and 1— ny is its sine; hence, OMP
is complementary to POM, and therefore the vibrations at M
are directed along the line PM. We conclude, therefore, that the
direction of the vibrations of the molecule at any point of the wave
surface is along the projection of the radius vector on the tangent
plane at that point.
When the tangent plane is normal to the radius vector, as is the
case at the extremities of the axes, this determination is not appli-
cable, but the direction is in these cases easily found. The plane
OMP, which contains the radius vector and the direction of the
corresponding vibration, is called the plane of vibration.
113. Relations between the Directions of Normal
Propagation of Plane Waves, the Directions of
Radii-Vectores of the Wave Surface, and the Direc-
tions of Vibrations. By the preceding theorem we have seen
that, in any plane wave whatever, the normal to this plane, the
direction of the vibrations in this wave, and the radius vector drawn
to the wave surface at the point of .tangency, are all contained in
the same plane. Besides, for each normal direction of propagation
of a plane wave, there correspond for the vibrations, two directions
parallel to the axes of the elliptical section of the ellipsoid (E)
made by a parallel plane. These directions, therefore, being per-
pendicular to each other, the planes which contain, at the same
time, the same direction of normal propagation, the two vibrations,
and the two corresponding radii-vectores, are rectangular.
114, Since the wave surface has two nappes, each radius vector
will give two directions for the vibrations. We will now show that
the planes which contain a radius vector and the directions of the
two corresponding vibrations are also rectangular; and for this pur-
pose we shall show that the two vibrations which correspond to the
same radius vector are contained in the two planes passing through
this radius vector and the axes of the elliptical section, that a plane
perpendicular to the radius vector cuts out of the ellipsoid (W).
115. Let ¢, , xv, be the angles made by one of the axes of this
elliptical section with the co-ordinate axes; it is then necessary to
demonstrate that the three lines («, 8, y), (A, #, ), (%, ~, x), are
all in the same plane.
78 ELEMENTS OF WAVE MOTION.
Eq. (129), which gives the relations existing between the angles
«, B, y. made by one of the axes of ellipsoid (E) with the co-ordi-
nate axes, and the right line 2, m, , perpendicular to the elliptical
section, can be applied to the analogous case of the elliptical section
of (W) and the normal radius era by replacing in this equation
ko od
rT «, 3 y, Lm, n, o, B, &, by ‘ Ps Py X> Ay My V > GR? Be? ew
respectively, which will give
«cos A b? cos jt C cos v
re ——_ ae 72 eee be _ re —s Ce
cos@ cos ~ cosy (158)
a cos? A 4 cos? c cos® v
apt mt ar
Let the auxiliary right line defined by the angles (4, B, C) be
drawn perpendicular to the two right lines (A, w, v) and (4, ¥, x);
we will have
cos A cos 2 + cos B cos # + cos pee
cos .1 cos ¢ + cos B cos p + cos Ccos y = 0. (te2)
Replacing, in the last equation, for cos ¢, cos y, cos x, the
quantities proportional to them in Eq. (158), we have
roe a
1
a s B+ <5 cos C= 0. (160)
fe =F
Adding this to the first of Eqs. (159), we have
ae d
we
pod + 5 see
—_ FR
and recollecting that the relations
cos B + < a “008 C = 0; (161)
\
cos 2 Cos pL cos v
7 — PB r—e (162)
cose cosB cosy
exist, we have finally
cos « cos A + cos 8 cos B + cos y cos C= 0. (163)
Hence, the three right lines («, B, y), (A, 4»), (¢, , x), being
perpendicular to the right line (A, B, C), are all contained in the
RELATING TO SOUND AND LIGHT. 79
same plane, and therefore we conclude that the planes which con-
tain, at the same time, the same radius vector, the two vibrations,
and the two corresponding directions of the normal propagation,
are rectangular.
116. Discussion of the Wave Surface. Resuming
Eq. (146),
(2+ P+ 2) (Ce + By + 2) —a(P + 2) 2
; —P(e@+e@)P—C(P+P)24 RE = 0,
and making in succession <= 0, y=0, z=0, we get for the
sections made by the co-ordinate planes yz, xz, and xy, respectively,
(y? + 2 — a?) (By? + e? — Pe) = 0, (164)
(2 + 2 — B) (ax? + ee? — ac?) = 0, (165)
(2 + P— &) (ee + By? — ab?) = 0. (166)
Remembering that a >>> ce, we see that the section in the
plane yz will be a circle whose radius is a, entirely outside of an
ellipse whose semi-axes are 6 and c. The section in the plane zy
will be a circle with radius ¢, entirely within the ellipse whose semi-
axes are a and 6. That in the plane 2z will be a circle with radius
, Y. Oo xX iO x
6, intersecting at four points the ellipse whose semi-axes are @ and ¢.
The axis of x pierces the surface at distances equal to + 6 and +¢
from the centre, that of y at distances of + a and + ¢, and that of
gat taand +0. ,
117. The surface of elasticity of two nappes cuts the axes in the
same points. These principal axes of elasticity have in turn repre-
sented the square roots of the elastic forces developed along the
Figure {7.
80 ELEMENTS OF WAVE MOTION.
three axes of elasticity, the principal velocities of wave propagation,
the axes of the ellipsoids, and now serve to fix points on the surface
of elasticity and on the wave surface.
118. Let Eq. (146) be represented by Z = 0, and the angles
which a tangent plane to the surface at any point makes with the
co-ordinate planes zy, 22, yz, by A, B, C, respectively; then we will
have
1 dL _ 1 4b
cosd =o, cos B= © dy
1 aL
i cos C= Fas (167)
in which, i => ee ee (168)
io dL\? (dLy* (Ay
é \/ (c) Bs (7) + Ve
Taking the differential coefficients of Z with respect to 2, y, 4,
we will have
“ = 2a (x? + PY + F) + Rox (e+ ¥+?—B—C),
wv
dL 22 27/2 292 2a (42 4-42 4+ 22—a2 169
ge + Py? + 22) + Wy (?@+y+2—e—C), ¢ (169)
dl
= 2z (a2? + By? + CB) + 2% (P+y+2—e—P),
dz
For y = 0, the point of tangency is in the plane 22, and we
have
7 = 2a (aa? + 2) + 20a (P+ 2 — Pe),
dL
ee 1
a (170)
aL
== 22 (Ca? + 2) + 20% (2+ 2— a — Dd),
the second equation showing that the tangent plane is normal to
the plane zz.
For y = 0, the equation of the surface gives
P+2—hR=0, av + eg — ee = 0; (171)
whence, for the co-ordinates z and z, we have
RELATING TO SOUND AND LIGHT. 81
FoR Roe
cS + 04/54, = +a\/a—%, (172)
which are real so long as a >> c. There are then four points
of intersection in the plane az. Substituting these values in
Kgs. (167), we obtain
cos d = : csB=-, cos C = g (173)
0’ 0
119. The interpretation of these indeterminate values of the
cosines is, that at the points considered, a tangent plane to the
wave surface may have any position whatever with respect to the
co-ordinate planes. This property shows that these points are the
vertices of conoidal cusps, each having a tangent cone. These
points, called wmdilics, belong to the exterior and interior nappe of
the wave surface, just as the vertex of a cone is common to its
upper and lower nappes.
120. The equation of the right lines joining these points, OI,
OI’, through the centre in the plane «2 is
colo
a /P—@
c= 2 eae” (174)
which shows that the lines Zz
are normal to the circular :
sections of the ellipsoid ys NM P
M. \
(W). The lines themselves ‘ /
are called azes of exterior ‘
conical refraction.
121. If tangent lines
be drawn to the ellipse and
circle, as MN, M'N’, they
will be parallel to each
other, two and two, and
symmetrically placed with
respect to the axes OX and
OZ. The equations of
these lines can easily be shown to be
s=44/; a (175)
Figure 18,
82 ELEMENTS OF WAVE MOTION.
and hence the equation of the line drawn from O perpendicular to
the tangent to be
poe
a= = ape (176)
~which shows that these lines are normal to the circular sections of
.the ellipsoid (E),
122. From the properties of this ellipsoid, we see that a plane
wave perpendicular to one of the right lines MM,, M’M,’, and at the
same time perpendicular to «z, can be propagated without altera-
tion, whatever may be the direction of the displacement inits plane,
and that the velocity of propagation of this plane wave is independ-
ent of the direction of the displacement. The lines MM,, M’M,’,
are called the optic axes of the medium, or aves of intertor conical
refraction.
123. We see, by comparing Eqs. (174) and (176), that the lines
OI and OM differ by the factor < in their tangents. This ratio is
always very nearly unity, and therefore the lines have nearly the
same direction.
124, The planes drawn through the four tangents MN, M'N’,
etc., perpendicular to the plane az, are tangent to the wave surface
along the circumferences of circles, which are projected in the lines
MN, M'N’, etc. To show this, let
F(a, y, 2) = 0 (177)
be the equation of the wave surface ; then, for points in the plane
perpendicular to xz, we have
a = ¥(CE+ BYP + C2) +By (P+ Pt 2A)—P(+ ely ae
which can be satisfied by placing
y = 0, (179)
and = (@ 4B) a? + 2By + (P+2)2—B(+e) =0. (180)
The first of these equations gives the points of contact in the
plane zz; the second represents an ellipsoid. If we combine the
equation of the ellipsoid (180) with the equation of the wave sur-
‘face, eliminating y’, the resulting equation will be the projection on
‘the plane zz of the intersections of these surfaces, and since the co-
RELATING TO SOUND AND LIGHT. 88
ordinates of the points projected satisfy the condition, i = 0, all
the points of the wave surface in the tangent plane which is per-
pendicular to zz, will be obtained from this intersection and projec-
tion. The resulting equation after reduction can be put in the
form of
—#,
(+ 4/faat + 4/3
a) _
« (:-4/5=8 Es
eo ee Ce
x (:+4/Z=4 -14/S=4 )
( 7, n= |
BN, ae ly ey
This equation can be satisfied by placing each factor separately
equal to zero, and each will then be the equation of a plane passed
through one of the tangent lines MN, M’N’, M,N,, M,'N,‘; hence,
each of the four planes touch the surface in those points determined
by its intersection with the ellipsoid, and it is readily seen that
these curves are the circular sections of the ellipsoid, Eq. (180).
The four planes
aos 2
p= 04/Z— # toy /E—S (182)
are called the singular tangent planes of the wave surface.
125. The circles are, in fact, the
edges of the conoidal or umbilic cusps,
determined by the surface of the tan-
gent cones, reaching their limits by
becoming planes in the gradual in-
erease of the inclination of their ele-
ments, as the tangential circumference
recedes from the cusp points.
It thus appears that the general
wave surface consists of two nappes,
the one wholly within the other, ex-
cept at four points, where they unite.
ty
=0. (181)
84 ELEMENTS OF WAVE MOTION.
Fig. 19 represents a model of the wave surface, with sections made
by the co-ordinate planes, so as to show the interior nappe.
126. Relations between the Velocities and Posi-
tions of Plane Waves with respect to the Optic
Axes. For each direction of normal propagation, two plane waves
travel with different velocities, determined, as we have seen, by the
equation of the surface of elasticity,
cos? J cos? m cos? 2
Par ViaP eae
This equation can be put under the form
V4— [(P+e) cos? 1 + (a?+¢*) cos? m + (a?+8) cos? 2] cc (183)
+ &e cos? 7 + ac? cos? m + ab? co? n = 0;
and representing the two square roots by V’? and V'2, we have
V4 V'? = (P+ e) cos?1 + (a + c) cos? m (184)
+ (a + B) cos? n,
Vey? = Be cos? 1 + a2? cos? m + ab? cos? n. (185)
Let 6’, 6”, be the angles that the direction of normal propagation
makes with the optic axes, and @ and 180° — ¢ the angles that the
optic axes make with the axis 2, the axis of greatest elasticity, then
we have
a — : Pe
cos $ = fe sng = (es (186)
cos 6’ = cos¢cosl + sin ¢ cos n, ! (187)
cos 6" = — cos ¢ cos J + sin ¢ cos 2;
whence, we get
_ cos 6’ — cos 6" _ cos 0’ — cos 6" /# — 2
a 2cosp 2 /§ — (es)
__ cos 6’ + cos 6” __ cos 6’ + cos 6” ec
Gere 2 sin = 2 \/% —e oo
Substituting these values of cos 7 and cos m in Eqs. (184) and
(185), and replacing cos? m by 1 -- cos? 1 — cos? n, we obtain
RELATING TO SOUND AND LIGHT. 85
(cos 6’ — = oe 6" (a?
V240%=@4e— —¢)
ap Seon 6’ s + cos 6")? (a — ot) (190)
=S=P+e+ (e— cos 6’ cos 6”,
Vey? = ee — eae (cos 6’ — cos 6")?
+ eae (cos 6’ + cos 6’)? ;
+ (191)
2 p2)\2
= e+ eS (cos? 6" + cos? 6’)
(@-2) (#42)
2
+ cos 6’ cos 6"; J
whence,
(V?2—V'% 2? = (V24V'22_ 4ye2yr
= (# + 2) + (a@&—c?)? cos? 9’ cos? 6" — 4a?
— (@ — c*)* (cos* 6' + cos? 6") > (192)
= (@ — )? (1 — cos? 6’) (1 — cos? 6”)
= (a2 — &) sin? 6’ sin? 6”; J
and finally, V’'®?—V"? = (@—@) sin 6’ sin 6", (193)
This equation establishes the relation between the velocities of
the two plane waves which belong to the same direction of normal
propagation, and the angles that this direction makes with the
optic axes.
127. The directions of the two vibratory motions can be deter-
mined by means of the optic axes. These directions are parallel to
the axes of the elliptical section of (E) made by the plane normal
to the direction of propagation ; but the elliptical section is cut by
the planes of the two circular sections of the ellipsoid in two equal
diameters of the ellipse, since they are equal to the radius d of the
circular section; they are therefore equally inclined to the axes of
the ellipse. The optical axes being normal to the circular sections,
are projected on the plane of the ellipse in two diameters which are
86 ELEMENTS OF WAVE MOTION.
perpendicular to those just spoken of, and are therefore also equally
inclined to the axes of the ellipse. But these projections are the
traces of the planes containing the directions of the normal propa-
gation and each optic axis. We therefore conclude, that the bi-
secting planes of the diedral angle formed by the planes
containing the direction of any normal propagation and
each of the optic axes, are the planes of vibration of the two
plane waves corresponding to this normal propagation.
128. The plane zz being the plane of the optic axes, any direc-
tion of normal propagation in this plane will make the diedral
angle 0° and 180°, and hence the planes of vibration will be the
principal plane zz and a plane containing y and the direction of
propagation.
129. Relations between the Velocities of Two Rays
which are Coincident in Direction, and the Angles
that this Direction makes with the Axes of Exterior
Conical Refraction. The expressions
roe e—e
toe ™ Vere
being the cosines of the - that the optic axes make with the
axes of z, z, and making use of the analogy existing between the
ellipsoid (E) to the surface of elasticity, and the ellipsoid (W) to the
wave surface, we will have, by substituting for a®, &, c*, in the
above, the expressions
1 1
a BR ¢ 72?
a ee
ae oy /5—% —@ b fe —
for the cosines of the angles that the axes of exterior conical refrac-
tion make with the axes of 2, z.
If then r and 7’, the two coincident radii-vectores of the wave
surface, represent the ray velocities propagated in the same direc-
tion, and «’ and w” be the angles made by this direction with the
two axes of exterior conical refraction, a discussion in every way
analogous to that above for the optic axes will determine the re-
quired relation. This relation may be oa once determined by
replacing V', V", 6’, 6”, in Eq. (193), by 5 ss u', w'', respec-
tively; we then have
RELATING TO SOUND AND LIGHT. 87
I 1 1 1 * 1 og "
a a (—5) sin w' sin w”. (194)
130. The axes of exterior conical refraction being normal to the
circular sections of ellipsoid (W), by a similar course of reasoning
asin Art. (127), we will arrive at the theorem, that the bisecting
planes of the diedral angle, formed by the planes containing
any radius vector of the wave surface and each of the axes
of exterior conical refraction, are the planes of vibration of
the two rays corresponding to this radius vector.
131. Thus, from the wave surface we can determine:
1°. The position of the refracted plane waves by its tangent:
planes.
2°. The direction of the two corresponding rays by the points:
of contact of the two parallel tangent planes.
3°. Tne velocities of the two rays by the lengths of the radii-
vectores drawn to the points of contact.
4°. The velocities of the two plane waves by the normals from
the centre upon the tangent planes.
5°, The interior directions of the molecular vibrations by the
projection of the radii-vectores on the tangent planes.
6°. The plane of vibration by the plane of the normals and vi-
brations.
132. We have now shown that when any arbitrary displace-
ment is made in any homogeneous medium, a disturbance is
propagated in all directions from the origin, and that it is materially
affected and controlled by the elastic forces developed. In accept-
ing the conclusions which result, the limitations which have been
primarily established must be kept in mind, to avoid the danger of
accepting these results other than as exceptional and governed by
the admitted hypotheses and by the accuracy of the mathematical
processes employed. Observation and experiment are essential to
ascertain to what extent the corresponding physical phenomena
conform to these deductions. They are to be used, when at vari-
ance, to modify the hypotheses, and ultimately through this modi-
fication to approach nearer and nearer the true theories of the
physical science.
133. The fundamental hypotheses upon which the foregoing
discussion is in part based are as follows:
1°. The admission of such a constitution of the medium that
88 ELEMENTS OF WAVE MOTION.
while it is variable around any molecule, it is similarly variable
around all the molecules. The propagation of the disturbances
without change of direction of the vibrations, when the latter are
excited along the singular directions depends on this assumption.
This inequality of elasticity is unquestionably exhibited in the phe-
nomena of crystallization.
2°, That the excursions of the displaced molecules are so small
that the resultant elastic forces in any direction are proportional to
the displacement. This implies that the distances separating the
adjacent molecules are very great in comparison with thei dis-
placements.
3°. The principle of the coexistence and superposition of small
motions, by which any vibration can be replaced by others equiva-
lent to it which are rectilineal.
4°. The inefficacy of the longitudinal component of the elastic
force in light undulations, and the fact therefore of ttansversal
vibrations. (The grounds of this assumption are to be given sub-
ae
- The correlation of the total intensity of the elastic force to
mr velocities, and its identity with that expresed by the
equation
V= Js
6°. The principle of interference, by which the motion is en-
tirely destroyed everywhere, except upon certain surfaces, which
may be regarded as the loci of first arrival.
134. The agreement of the results obtained by experiment and
from observation with the deductions from the theory is almost
complete, while the crucial test of prediction in several noted in-
stances leaves but little doubt of the truth of the undulatory theory.
The utility of the determination of the wave surface and of its
thorough discussion is thus happily verified, by its almost complete
capability of satisfactorily explaining most, if not all, of the phe-
nomena of physical optics. While in the limited course prescribed
for the Academy we are unable to undertake the complete solution,
we have, in the short and elementary discussion here presented,
obtained sufficient data to prosecute the study of sound and light
to the extent necessary for our purposes, and in this study we will
have frequent occasion to refer to the foregoing analysis.
PART II.
ACOUSTICS.
135. The investigations of physical science show that all sensa-
tion has its origin in the state of relative motion of the molecules
of some medium with which the organ of sensation is in sensible
contact. Each sensation has its peculiar organ, which, with its
nerve system, receives and transmits molecular kinetic energy to
the brain, where it is transformed into sensation. The motions of
the molecules are, in general, vibrations, which are conveyed by
undulations from the source of disturbance in all directions through-
out the medium.
136, Acoustics is that branch of physical science which treats
of sound. The sensation of sound usually arises from the commu-
nication of a vibratory motion of the tympanic membrane of the
ear, due to the slight and rapid changes of the air pressure upon its
exterior surface, the vibratory motion of the air being caused by the
vibration of other bodies.
137. The ear consists essentially of two parts, one being in com-
munication with the external atmosphere, the other with the brain.
The first consists of an irregularly formed tube, beginning at
the orifice of the external ear and ending at the pharynx. Nearly
midway, the tympanic membrane, or drum-skin, of the ear crosses
this tube obliquely, separating the external portion, called the
meatus, from the part immediately within, called the tympanum.
That portion of the tube leading from the tympanum to the pha-
rynx, or cavity behind the tonsils, is called the Eustachian tube.
The orifice of this tube at the tympanum is generally closed ; but
the act of swallowing opens it, whereupon the air on both sides of
the tympanic membrane becomes uniform in density. These three
portions of the first part of the ear generally, however, contain air
differing in density. In the meatus the air responds to all changes,
90 ELEMENTS OF WAVE MOTION.
however slight and rapid, taking place in the external atmosphere;
while the air in the tympanum and Eustachian tube is not so
affected, unless communication with the external atmosphere be
made as above described.
138. The other part, sometimes called the internal ear, is sur-
rounded by bone, except in two places, called the round and oval
windows. The cavity thus formed is called the bony labyrinth.
The windows are closed by membranes which separate the tympa-
num on the one side from the fluid contained in the labyrinth on
the other. Connecting the tympanic membrane with the oval win-
dow is a series of small bones, whose function appears to be to
transfer the vibrations from the former to the latter. The laby-
rinth is filled with liquid, having suspended in it many membrane-
ous bags, also filled with liquid. Upon the surface of these bags are
spread the terminal fibres of the auditory nerves, which, by special
arrangements, are enabled to take up the energy communicated to
the liquid in the labyrinth. The membrane of the round window
readily yields to the pressure of the liquid, moving out and in as
the oval window is moved in and out by the transfer of motion
through the bones of the ear.
Thus the energy communicated to the air in the external ear is
conveyed from the tympanic membrane, through the series of small
bones in the tympanum, to the membrane of the oval window,
thence to the liquid of the labyrinth, and finally to the auditory
nerves. How this energy is transformed into sensation is unknown.
139. To represent, graphically, the variations of air pressure,
we will make use of the curve of pressure, in which the abscissas
correspond to the ¢imes and the ordinates to the excess of the pres-
sure above its mean or average value. The pressure of the air, at
any point, is assumed to be measured by the pressure of air of the
same density and temperature upon a unit of area. Then take
yar (195)
to represent any curve of pressure as
a
of ee ae TS
a
Figure 20,
in which y = 0 represents the standard or mean pressure, and
RELATING TO SOUND AND LIGHT. 91
y = +>), a pressure above or below the standard pressure. When-
ever the pressures are strictly proportional to the corresponding
densities, as by the law of Mariotte, the same curve may also repre-
sent the curve of density. If we now assume that a curve similar
to the above represents the slight and rapid changes of pressure of
the air in contact with the tympanic membrane while the sensation
of a particular sound exists, we see that these changes do not in
general affect the average pressure of the air, for the areas above
and below the axis of the curve are equal. A curve is said to be
periodic when it consists of equal and like parts continuously re-
peated. The wave length of a periodic curve is the projection upon
the axis of the smallest repeated portion.
140. The ear clearly distinguishes between a musical sound and
a noise. The former is a uniform and sustained sensation, unac-
companied by any marked alteration, save that of intensity; while
the latter is more or less varied and ununiform. When a sonorous
body is sounding, the most ordinary examination is sufficient to
show that itis in a state of vibration. The vibrations or oscilla-
tions of its parts set in corresponding motion the adjacent air-parti-
cles, which in turn transmit similar motions to the next following
particles, and so on. The air, then, is ever passing through alter-
nate states of condensation and rarefaction. When these vibrations
are regular, periodic, and sufficiently rapid, the resulting sound is
uniform in character and is called a musical tone. If the resulting
sound arises from vibrations which are non-periodic, it is called a
noise. Ordinary observation shows that few, if any, noises are per-
fectly nnmusical; and few, if any, sounds are absolutely unmixed
with noise.
141. Propagation of a Disturbance in an In-
definite Cylinder. Let us suppose the indefinite cylinder MN
filled with air, and at the origin a piston
p, capable of rapid to-and-fro motion. In M N.
the first place, let the piston be moved a ii
distance ds from p to p’, in the time dt. bat
If the air were incompressible, it would be Figure 21,
moved bodily over the distance ds. But
being compressible, the air yields to the motion of the piston, and
at the end of the time dé the compression will have reached a posi-
92 ELEMENTS OF WAVE MOTION.
tion m, so that the stratum of air, being condensed from pm to p'm,
will exert an elastic force in excess of that due to its normal state.
Call this excess 6. The increased elasticity of p'm will cause it to
expand in the only direction possible, towards the next stratum
mn, which in turn becomes compressed. This second stratum
reacts in both directions; on the side towards mp’ it brings the
molecules of mp’ to rest, their acquired velocity having a tendency
to cause them to pass beyond their positions of equilibrium ; and
on the side 27 it compresses the next stratum, increasing its elas-
ticity ultimately by 6. In this manner the compression is trans-
mitted from stratum to stratum, throughout the waole length of
the cylinder.
142. Let V be the velocity of propagation of the condensation,
v the velocity of the piston; then we have
pp’ =ds=vdt, pm= ds' = Vat,
pm — pp’ = p'm = (V —») di.
Supposing Mariotte’s law applicable, and P to represent the
normal pressure, we have
P:P+6:: p'm:pmn:: (V—v) dt: Vdt;
u .
V—v
Let p’ now return to its primitive position p in the next succes-
sive dt. The first layer of the stratum will be dilated, occupying
the new space p’p, and its pressure P will become P—d. The
elastic force of the next layer P will become, by its expansion to
6
the left, P = increasing that of the first also to P — PY But
the velocity acquired by the molecules of the second layer will
cause them to pass beyond their positions of equilibrium, so that its
elastic force will diminish until it becomes P — 4, at the instant
the elastic force of the first layer, continually increasing, becomes
P, its normal value. The third layer will, in turn, act on the sec-
ond as the second has acted on the first, so that the dilatation cor-
responding to 6 will travel the distance pm in the time df, during
which the piston is retracing its path p'p. The magnitude of 6
will evidently depend on the value pp’ and the time dt. If dt be
constant and d be varied, the condensations will vary with 6. The
or fa P (196)
RELATING TO SOUND AND LIGHT. 93
analysis shows that the compressions and dilatations are propagated
with equal velocities, and that these velocities are independent of
the degree of condensation or of rarefaction, when the medium is
the same and the amplitude is very small.
143. Let the prong of a tuning-fork p....p' (Fig. 22) be dis-
placed a very small but finite distance from its neutral position a.
By its elasticity it will vibrate with cqual displacements on each
side of its position of equilibrium. Its velocity increases from zero at
p to a maximum at a, and decreases in an exactly reverse manner
to zero from a to p’. Let the duration of its motion from p to p’
be divided into equal parts, each represented by dé, the epoch cor-
responding to the position ». From p the prong describes unequal
but increasing distances during the successive dé’s to the position a,
and unequal but. decreasing distances from ato p’. Each corre-
sponding compression can be found from Eq. (196) by the substi-
tution of the proper value of v, and these compressions or conden-
sations will be propagated with a constant velocity V. While the
prong is returning from p’' to p, the rarefactions will increase from
p’ to a, and decrease from a to p, and their values may be deter-
mined from the same equation. The condensations will be sym-
metrically distributed with reference to the maximum condensation,
neglecting the very small amplitude vp’. Likewise the rarefactions
will. be symmetrically distributed with respect to the maximum
rarefaction.
144. The positive ordinates of the curve p'rs represent the suc-
cessive condensations, s being the position of the layer reached by
the first condensation when the prong has arrived at p’; and the
negative ordinates pr’s will represent the successive rarefactions
when the first condensation has reached the position «, and the
prong has returned to its primitive position y. The ordinates of
the other curves represent either condensations or rarefactions, as
indicated in the figure corresponding to the particular state and
position of the prong.
94 ELEMENTS OF WAVE MOTION.
145. In the figure, pu, the length of the wave is the distance
traveled by the disturbance while the prong is making a complete
vibration, and hence we have, 2 being the vibrational number and
z the periodic time,
A=— = Vr. (197)
146. The mean velocity of the air molecules is evidently the
same as that of the vibrating prong, and therefore this will vary
with the vibrating body. In the example given, the mean velocity
of the molecules is 2mm. x 256 = 0.512 m. The actual velocity
of the air molecules continually varies, and at any time is propor-
tional to the ordinates of the curve a quarter of a wave length in
advance of the molecules considered. When the vibrating body has
simple harmonic motion, the molecular velocity is given by
v = a cos Qt t, (198)
147. The value of a, the amplitude of the vibration, diminishes
(Art. 72) according to the law of the inverse distance from the cen-
tre of disturbance; and for each value of « taken as constant within
the wave length we have, by the above equation, sensibly exact
yalues for the molecular velocity at any time.
148, When the vibrations of the body are sufficiently frequent
during the unit of time, and of sufficient amplitude, the sensation
of sound arises in the ear, which, however, we unconsciously refer
to the vibrating body. A sonorous wave comprises the series of
condensations and rarefactions arising from one complete vibration
of the sounding body.
149, The sum of all the condensations in the condensed portion
of the wave is represented by the area of the curve p’rs, and if it be
divided by the duration of half the vibration, the mean cordensa-
tion will result. Thus, take the amplitude of the oscillation of the
tuning-fork, making 128 vibrations per second to be 1 mm., and the
velocity of propagation to be 340 m.; then, from Eq. (196), we
will have
J
256 1
= P abs — P = P oN
V—v 340000 — : 3840000 — 256 1327
Tee (199)
RELATING TO SOUND AND LIGHT. 95
Hence, the change of density in the air, measured by the
barometric height due to the mean condensation, is not greater than
that due to 0.0226 inches of mercury, when a sound corresponding
to 128 vibrations per second, and caused by the fork under the sup-
posed conditions, is passing.
150. From the preceding discussion we see that we can neglect,
in general, the absolute displacements of the air molecules, and
consider the change in pressure and density as being alone propa-
gated. Therefore, a file of elastic balls transmitting motion prac-
tically illustrates the state or condition of a series of air molecules
during the propagation of a sonorous wave. An excellent illustra-
tion is also given by means of a chain cord. If it be attached at
one of its extremities to a fixed point, and be held stretched at the
other, the successive rings or spirals will assume positions of stable
equilibrium with respect to each other, determined by the tension.
These rings, for the purpose of illustration, may be taken to repre-
sent the contiguous air strata or particles in an indefinite tube, or
upon any line along which sound is supposed to be propagated. If
any ring be plucked, it will, when released, oscillate about its place
of rest while the disturbance is being propagated in both directions
to the points of support. Upon reaching these points the dis-
turbance will be divided, a part proceeding in the new medium,
and the remainder, being reflected, will retrace its path, to be again
subdivided at the other end. This will continue until the whole
energy of the original disturbance has been dissipated. By increas-
ing the tension the disturbance will be more quickly propagated,
and conversely. Now suppose, from the point of plucking, lines
be drawn in all directions, and the same phenomena occur on these,
then the behavior of each ring and the progressive motion of the
disturbance illustrates what takes place in air during the passage
of a sound wave along every right line drawn from the origin of the
sounding body. In an isotropic and homogeneous medium, the
disturbance moves with constant velocity, aud the volume whose
surface bounds the disturbed particles at any instant is a sphere
whose radius is Vé.
151. The general properties of any sound are intensity, pitch,
and quality. :
Intensity is that property by which we distinguish the relative
loudness of two tones of the same pitch and quality. We can also,
96 ELEMENTS OF WAVE MOTION.
in general, determine which of two tones of different pitch and
quality has the greater intensity. The air particles have but small
displacements from their positions of relative rest, when the dis-
placement is caused by the passage of a sound wave. The forces
which urge them back to their positions of rest are assumed to vary
directly with the degree of displacement. In Analytical Mechanics,
it is shown that the periodic time of the air particle depends only
upon its mass and the intensity of the force of restitution; and
therefore, in the same medium, with given pressure, density, and
temperature, for the same exciting cause, the periodic time will be
constant, but the mean velocity of the air particle will vary with
the size of the orbit. The kinetic energy in the moving particle,
varying as the square of the velocity, will therefore, for the same
exciting cause and the same medium, under the same circumstances
of pressure, density, and temperature, vary directly as the square of
the maximum displacement. By the law of the decay of energy,
the intensity of the sound will therefore vary inversely as the square
of the distance from the origin of the exciting cause. (Art. 72.)
152. Pitch is that property by which we distinguish the posi-
tion of two tones in the musical scale, and thereby recognize which
is the more acute and which the more grave. The pitch depends
upon the frequency of the vibration; the greater the number of
vibrations produced by a sounding body in a given time, the more
acute will be the resulting sound. The siren is an instrument used
to illustrate this fact. It consists essentially of a disk pierced with
a number of equidistant holes, through which air is forced when it
is put in rapid rotation. ‘As the rotation increases, the sound grad-
ually rises in pitch, and as it diminisbes the pitch falls correspond-
ingly. Ifa coin with a milled edge, or a cogged wheel, be put in
rotation, and a card be held against it, the same changes in pitch
will be observed. In these cases the single puff, or stroke of the
card against the coin, or wheel, is essentially a noise, and when
these strokes are multiplied sufficiently in a given time, the result-
ing effect is a note of definite pitch. So that a clearer distinction.
than that heretofore given should be made between a noise and a
musical tone. To this distinction we will again refer.
153. The quality of a musical tone is that property by which
we can distinguish whether two sounds of the same pitch, of either
equal or unequal intensities, arise from the same or different sono-
RELATING TO SOUND AND LIGHT. 97
rous bodies. This property enables us, within certain limits, to
distinguish voices and the various sounds peculiar to different musi-
cal instruments. We have as yet only exacted that a musical sound
shall be periodic and regular ; that is, that during any vibration the
successive states of motion of the particle shall recur in the same
order as in each of the previous vibrations. But it is evident that
we may have an infinite variety of periodic motion, and it will be
shown that the quality of the sound will vary with each variation
of the periodic motion, the wave length remaining constant.
154, Every one has experienced the fact that more than one
sound can be heard at once. Our attention can be, for the mo-
ment, fixed upon any one of the many sounds that are constantly
occurring, and at the same time we may be conscious of the exist-
ence of the others. Therefore, the meeting of sound waves in the
external ear does not, in general, result in mutual destruction, or
in essential modification; while, at the same time, we must ac-
knowledge that the air in contact with the tympanic membrane, at
any given instant, can possess but one determinate pressure and
density. The changes in pressure and density due to many exciting
causes must, then, result from the superposition and coexistence of
those arising from each separate cause, and, in general, without.
destruction or modification. We have here the application of the
principle enunciated in Art. 204, Mechanics.
The more general statement of the law of the composition of
displacements would be that demonstrated in the principle of the
parallelogram of forces, but when the displacements are infinitely
small, we can take, rigorously, the resultant displacement to be the
algebraic sum of the component displacements. The diameter of
the meatus at the tympanic membrane does not exceed 0.25 inch,
and therefore, for sounds whose sources are at ordinary distances,
the wave fronts at the position of the tympanic membrane coincide
sensibly with their tangent planes, and the changes of density and
pressure may be compounded by the law of small motions, without
appreciable error.
155. Let the broken line, in the following diagram, represent
the changes of pressure upon the tympanic membrane while a con-
tinuous noise, in which the ear recognizes no definite pitch, is
sounding for a small part of a second, and let the dotted line repre-
sent another noise of the same duration.
7
gs ELEMENTS OF WAVE MOTION.
Figure 23,
Then, if both noises sound together, the resultant variation of
‘pressure will be represented by the full line obtained by joining the
extremities of the ordinates found by taking the algebraic sum of
the ordinates of the separate curves.
These two noises do not, in general, unite into one, but are
heard distinctly and simultaneously, except in the case where the
two sounds are nearly alike, and the two curves nearly similar.
Again, there is nothing in the resultant curve to suggest to the eye
the nature of the two component curves. Hence, the ear possesses
the property of separation ; while the eye, according to this method
of combination and representation, does not.
156. Let the component curves be periodic, two periods of O'X’
being equal to three of O"X".
Figure 24,
The resultant curve OX will be a periodic curve, whose repeated
portions are represented above. An examination of this curve by
the eye gives no clue to its components, and we may resolve it into
an indefinite number of pairs of components; but one of which
would represent the two notes which sounding together will give us
the resulting effect upon the ear. But if the ear resolves the com-
posite note represented by OX, it must resolve, in like manner,
‘O'X' and O"X"". Observation confirms this deduction.
157. The only note the ear is incapable of resolving is that of
the simple musical tone, and this incapability arises from the fact
that such a tone is in reality perfectly simple, and not compound.
The tones which are ordinarily called simple, are, in reality, com-
pounded of.a series of simple tones theoretically unlimited in num-
ber. Very few of them have sufficient intensity to be heard; but
RELATING TO SOUND AND. LIGHT. 99
these few form a combined note which is.always the same under
the same circumstances, and we habitually associate them together,
and perceive them as a single note of a special character. But it is
possible, with certain appliances, to partially analyze the composite
note by an attentive study of the separate constituents.
Whenever two sounding bodies give notes whose tones form con-
sonant combinations with each other, the difficulty of analysis is
increased; when the combinations are dissonant, the analysis is less
difficult.
158. A noise may therefore be defined to be a combination of
musical tones, too near in pitch to be separately distinguished by
the unassisted ear, or to be a combination of noises, each of which
is made up of sounds so near-each other in pitch as to be undistin-
guishable ; the separate noises may be near or far apart in pitch.
It is so complex, that its analysis is beyond the power of the un-
assisted ear. A simple musical tone, on the contrary, is incapable
-of resolution by reason of its absolute simplicity. Hence, strictly
speaking, only simple tones have pitch. A simple musical tone
will have a single determinate pitch. The pitch of a musical note
must then be taken to mean the pitch of the gravest simple tone in
its combination. If the higher simple tones be successively stopped
out, the pitch, as defined, will remain unaltered, but the quality of
the note will undergo variations until the single musical simple
tone corresponding to the gravest tone is reached, beyond which no
further modification can take place.
159. We will hereafter assume, as the fundamental simple tone,
that component of any note which corresponds to the regular pe-
riodic curve of the given pitch. This distinction is important; for
it is evident that there may be many periodic curves of the same
pitch, and each may correspond to musical notes differing in quality.
Figure 25,
100 ELEMENTS OF WAVE MOTION.
The preceding curves (Fig. 25) represent notes of the same pitch,
but of different quality. Helmholtz has shown that while every
different quality of tone requires a different form of vibration, the
converse is not necessarily true; 7. ¢., that different forms of vibra-
tion may not correspond to the same quality.
160. We have seen, page 45, that any physical condition, such
as density, pressure, velocity, etc., which is measurable in magnitude
or intensity, and which varies periodically with the time, may, by
Eq. (195), be expressed as a function of the time. Hence, every
periodic disturbance of the air, and particularly such disturbances
as excite the sensation of a musical tone, can be resolved into its
harmonic vibrations.
A single simple tone being represented by the simple harmonic
curve
y =a sin (= + «'), (200)
and another of half wave length by
y" = a" sin es f 1 (201)
2
the resultant curve will be represented by
y = a’ sin 7 + «') + a” sin (72 + a"), (202)
which has the same wave length, but a different amplitude and
phase. This change in the amplitude and phase may be varied at
pleasure, by conceiving the second curve to be shifted along the
axis any distance from zero to 4, and again to pass through all
values of the amplitude between any two limits. The resultant
curve, in all cases, will, however, be a periodic curve of constant
wave length.
161. Considering the simple musical tones which they represent
then to be sounded together, with the same modifications, it has
been found that the ear can distinguish the components when the
attention is cultivated and directed to this effect. With a variation
in phase only, the effect on the ear is constant and invariable, and
hence we see that many different resultant curves may represent
RELATING TO SOUND AND LIGHT. 101
Phase differs 90° at O
Figure 26.
essentially the same sensation. Thus, the two curves above, repre-
sent the same compound tone made up of the two simple tones,
although the forms of the curves are quite different. The resultant
tones are the same, both in quality and in pitch, but differ in inten-
sity. By combining in the same way other simple tones of one-
third, one-fourth the wave length, and so on, the quality will be
changed, without affecting the pitch, as can be seen from the
graphical construction, and heard by audible experience. In all
these cases the untrained ear, by the aid of certain appliances, can
always analyze the resultant sound into its component simple tones,
and when trained, often without this assistance. When but one
simple vibration of sufficient frequency and intensity to produce
sensation alone exists, no such analysis takes place.
162. The investigations of Helmholtz have shown that the ear
possesses the property of analysis of a single musical tone into its
simple musical tones, each of which is distinctive in character, but
which blend harmoniously into the single tone when sounded to-
gether. The wave lengths of these components are aliquot parts of
the wave length of the fundamental, and the simple tones are called
the upper partials of the fundamental or prime tone. Hence, from
Art. 64 and these facts, we conclude that, when several sounding
102 ELEMENTS OF WAVE MOTION.
bodies simultaneously excite different sounds, the variations of air
density and the resultant displacements and velocities of the air
particles in contact with the tympanic membrane are each equal to
the algebraic sum of thé corresponding changes of density, the dis-
placements and the velocities which each system of waves would
have separately produced, had it acted alone.
163. This analysis-by the ear clearly shows, then, that the sep-
arate effects of the simple vibrations are, in general, neither modi-
fied nor destroyed, but actually exist, and it_xemains to be proved
that such is really the case, independent of the peculiar sensation
which is the result of their action upon the ear. Since Fourier’s
Theorem mathematically demonstrates that any form of vibration,
no matter how yaried its shape, can be expressed as the sum of a
series of simple vibrations, its analysis into these simple vibrations
is independent of the capacity of the eye to perceive by examining
its representativé curve whether it contains the simple harmonic
curves or not, and if it does, what they are. All that the curve
indicates is that the more regular its form, the greater the effect of
its deeper or graver tones in comparison with its upper partials.
Before proceeding to show that these component vibrations actually
exist together, and that each can affect the ear or other sensitive
vibrating body, let us now establish clearly the definitions pertain-
ing to the subject.
164. A simple or pendular vibration is that which corresponds
to the complete oscillation of a simple pendulum, and is graphically
represented by the simple harmonic curve.
A simple musical tone is that effect produced upon the ear when
a sonorous body is executing simple vibrations only, of sufficient
frequency and amplitude to be heard. According to this definition,
simple tones do not in reality exist ; but in the vibrations of such
bodies as tuning-forks, the component vibrations which simulta-
neously exist with that of the gravest period, are generally non-
periodic with it, and so deficient in intensity that their influence is
negligible, and we may regard such bodies as producing simple
vibrations alone without sensible error.
165. A single musical tone may be either simple or compound.
When compound, it is made up of its fundamental simple tone,
together with its upper partial simple tones, each of which has a
frequency of either twice, three times, or so on, that of its funda-
RELATING TO SOUND AND LIGHT. 108
mental. It is due to the vibration of a single sonorous body which,
during its motion, vibrates as a whole, and divides also into parts
which vibrate twice, three times, and so on, as rapidly as the whole.
One or more of these upper partials may be wanting during the
vibration ; when this occurs, the quality of the single musical tone
is correspondingly affected.
A composite musical tone is composed of two or more single
musical tones.
166. Musical Intervals. The extreme range of the hu-
man ear lies between 20 and 40000 simple vibrations per second.
The corresponding wave lengths are obtained by dividing the veloc-
ity of sound by these numbers, and are approximately 54.6 feet and
0.0273 feet respectively, assuming the velocity of sound to be 1092
feet at 0° C. The ordinary sounds heard by the ear have a much
less range; their vibrational numbers lie between 40 and 4000, cor-
responding to wave lengths of about 27.3 feet and 0.273 feet,
respectively. When a stretched wire is put into vibration, and the
tension continuously undergoes variation, the pitch of the sound
passes by continuity from lower to higher, or the reverse, and we
therefore experience the sensation of a musical interval between any
two limiting tones. We may, then, define a musical interval by the
ratio of the vibrational numbers of the two limiting tones. Thus,
if the two tones correspond to the vibrational numbers 256 and
384, the name of the interval is the fifth, and it is expressed by the
fraction .. Considering the simpler ratios that lie between two
tones whose vibrational numbers are as 1: 2, we obtain the follow-
ing musical intervals:
Consonant. Dissonant.
Unison, 141 = | Major second, 9:8 = 3
Minor third, 6:5 = & | Minor second, - 10:9 =12
Major third, . 5:4 = £& |One-half major tone, 16:15 = 48
Fourth, - 4:3 = 4 One-half minor tone, 25:24 = $3
Fifth, 3:2 = $|Comma, . 81:80 = §t
Major sixth, hig = & |
Octave, - 2:1 = 2 |
The first are called consonants, because the effect is pleasing to
104 ELEMENTS OF WAVE MOTION.
the ear when the tones of either of these intervals are sounded to-
gether. All other intervals within range of the octave are called
dissonants.
167. The measure of the musical interval represented by the
ratio 2 is the log 2. This arises from the fact that if we consider
any three tones whose vibrational numbers are p, g, and r, the
musical interval between p and 7 must be equal to the sum of the
two intervals between p and g,and g and r. If the ratios of the
vibrational numbers were taken to measure the intervals, we would
have, for the same interval, the expressions
r gr
‘ and a + 3
which are not equal to each other. But since
Feaf,? (203)
p pg
r q r
we have log — = log 4 + log ~ 204
5 Bt Oey (204)
and we may therefore take the logarithm of the ratio of the vibra-
tional numbers as the measure of the musical interval. The name
of any interval, then, is the ratio of the vibrational numbers, and
its measure is the logarithm of that ratio. The logarithms are
usually taken in the common system.
168. Musical Scales. or < a the second impulse would reach
B after or before its molecular orbits had been completed. Under
these circumstances, succeeding impulses would in a short time
reduce the displacements of the molecules to zero, and never permit
them to attain an appreciable value, and therefore the vibration of
the air column would not give a sound of appreciable intensity.
But if, on the contrary, the impulses were of the same periodicity
as the air molecules, each successive impulse would add to the first
displacement, and this addition would continue until the work of
the resistances developed was exactly equal to the increment of
energy caused by each impulse. The displacements of the mole-
cules would then have attained their maximum value, and the
resulting sound a fixed intensity.
177. Each confined mass of air has a particular periodicity, and
each of the resonators of Helmholtz is carefully contrived to respond
to a given periodicity of vibratory motion. If, then, by the rule
above given, any composite sound exist, and one of these resonators
be applied to the ear, the resonant effect will indicate whether the
simple tone corresponding to the resonator is present or absent in
the composite sound. This and analogous experiments show that
sympathetic vibration is not due to any property peculiar to the ear,
but that it is a mechanical effect separate and distinct from the
sense of audition.
178, The energy of motion depending upon the mass and ve-
locity, we see clearly that of two sounding bodies, vibrating with
the same amplitude, the smaller mass will more quickly give up its
energy to the surrounding air and sooner cease sounding. Tuning-
forks being generally made of steel, will, when put into rather
strong vibration, continue sounding for a reasonable length of time.
When mounted upon their resonant boxes, the latter containing a
mass of air capable of vibrating in unison with it, they affect larger
masses of air than when not so mounted, and come more quickly to
rest; but the sound will have greater intensity, and can the more
readily be used to study the phenomena of sympathetic resonance.
If such a tuning-fork be in the vicinity of a vibrating sounding
body whose sound contains the tone of the fork, the latter will in
RELATING TO SOUND AND LIGHT. 111
time indicate the fact by coming into sympathetic vibration. The
analysis, then, of any composite note can be practically made by
means of a sufficient number of such forks, whose vibrational num-
bers embrace all the simple notes of the composite sound. Con-
versely, the synthesis of a composite note can be effected by setting
in vibration all the forks, with proper amplitudes, which the analy-
sis indicates belong to the note in question.
179. When plates, bells, strings, etc., are put into vibration,
they may either vibrate as a whole, or separate into parts which
vibrate two, three, four, or more times as rapidly; or both of these
conditions may occur simultaneously. Each of the simple periodic
vibrations has an actual existence, and corresponds to a single
musical tone of definite pitch, which may be recognized as above
described. :
180. In listening for any simple tone in the composite note, it
is important to clearly fix the attention upon the special tone whose
existence is to be determined, and for this purpose the tone should
be sounded alone before listening for it in the composite note.
When sufficiently practiced in this manner, the ear can readily
acquire the faculty of detecting them without the use of resonators.
181. By means of the monochord, which consists essentially of
a string stretched over two bridges on a sounding-box, we can verify
the simultaneous existence of the prime and upper partials, and
estimate the influence of the latter in affecting the quality of the
sound. The theory of vibrating strings shows that the frequency
of vibration of the same string under the same tension is inversely
proportional-to its length. Plucking the string at its centre, the
resulting tone will be that of its prime, modified by some of the
upper partials, those of the latter being absent that require the
middle point as a point of rest. By a movable bridge, the string
can be divided into its aliquot parts, which being set in vibration,
will give the upper partials in succession. Becoming thus acquaint-
ed with these simple tones, we can verify their presence or absence
in each special case. For example, if the string be plucked at one-
fourth its length, theory requires the presence of the first upper
partial with the prime, and the fact will be made manifest by damp-
ing the string at-the’middle point immediately after plucking, when
the octave will sing out, no longer encompassed by the prime.
112 ELEMENTS OF WAVE MOTION.
182. These and the facts of sympathetic resonance show that
the analysis of all resonant motion into simple pendular vibrations
is real and actual, and that any other analysis is highly improbable.
The analogous property of the ear is expressed by the law of G.S.
Ohm, viz. that the human ear perceives pendular vibrations
alone as simple tones, and resolves all other periodic motions
of the air into a series of pendular vibrations, hearing the
simple tones which correspond to these sinvple vibrations.
We may therefore conclude, that in all cases whenever any motion
of the air caused by a sounding body contains a simple vibration of
the same periodicity as that of any other body, the latter will in
time take up a vibratory motion which, if of sufficient intensity,
will affect the ear with a simple musical tone of a definite pitch ;
and the mechanical effect of vibration will ensue, whether it be of
sufficient amplitude to produce a sonorous effect or not.
, 183. Velocity of Sound in any Isotropic Medium.
‘The air is the medium of transfer to the ear of the vibratory mo-
tion of a sounding body. Under a given temperature and density,
its elastic force is constant in all directions, and it is therefore an
| isotropic medium. Being compressible, the motions of its mole-
‘cules, during the passage of a sound wave, are to and fro along the
‘line of wave propagation. They are then longitudinal vibrations,
‘and Eq. (116), for the velocity of wave propagation, for waves with
' such vibrations in an isotropic medium, is
2
v= 5 =p E (r) + P(r) “| sin? : Ag. (206)
184, The wave lengths of sound in air can never be greater
than 54.6 ft., nor less than 0.027 ft. ; for the usual sounds the limits
are 27.3 ft. and 0.273 ft., at 0° C. In the above equation, Az is the
: distance separating two adjacent molecules, and without knowing
its absolute value for any degree of pressure, we may say that A,
even in the minimum sound wave, is very great with respect to Az.
Therefore the arc is approximately zero, and may be substituted for
. TAL
sin Zz
without appreciable error. We then have
RELATING TO SOUND AND LIGHT. 1138
a2 Ee Aa? | nA:
VP = 53H | Or) +40) | bie
(207)
ag ee | eis
= 5 | o(r) + ¥(r) =e | Ae’,
a
Hence, with the supposition of small displacements, etc., the
velocity of wave propagation of sound in air is theoretically inde-
pendent of the wave length, and all sounds, whether grave or acute,
will travel, in air of constant pressure and temperature, with equal
velocities.
Omitting the term containing Az! as being small compared with
that of which Az? is a factor, and replacing ¢(r) by its equal
f(r)
2+", we have
r
yo ; Sp = Age, (208)
185. Let # represent the modulus of longitudinal elasticity of
air, P the barometric pressure, / the length of the air column with-
out pressure, and 4 the compression due to P. Then, by Eq. (1),
we have
Z
Since, if the pressure P be removed, the expansion would be
indefinitely great, the compression 4 is sensibly equal to /, and
therefore 5
E=P; (210)
that is, the elastic force of the air is that due to the barometric
pressure on the unit area.
186. In Eq. (208), 5H f(r) is the acceleration due to the
aggregate elastic forces developed in the molecules by the arbi-
trary displacement of the molecule m, and reciprocally is the elastic
acceleration of m; hence we have, by multiplying by m, the inten-
sity of the elastic force acting on m,
m Sp f(r) 5
Cole
1i¢ ELEMENTS OF WAVE MOTION.
= is the cosine of the angle made by this force with the axis of 2,
which, since the medium is isotropic, is equal to unity; multiply-
ing the elastic intensity on m by the factor ee we have the elastic
intensity on the unit area, or
1 1
] E Av
whence, 3 =H fy) = a (212)
Substituting in Hq. (208), we have
E Ax
Ps ;
Vom ee (218)
Aa’ is the volume of the molecule, and replacing it by its equal
and extracting the square root, we will have, finally, for the
velocity of wave propagation in air or any gas, subjected to the law
of Mariotte ee
> a eee alll
v= >? (214)
or directly proportional to the square root of the ratio of the elas-
ticity of the medium to its density.
187. This conclusion is deduced on the hypothesis of the direct
ratio of the elastic force to the density, and if the law of Mariotte
were true for all circumstances of pressure, temperature, and den-
sity, this theoretical velocity and the actual velocity determined by
experiment would perfectly accord. But this relation is true only
for a perfect gas and for constant temperature.
188, The relation of these two parameters of air, considered as
a perfect gas, are given by the following formule :
pd = pd, (215)
pid = pd' (1 + 46), (216)
' d'\y
of =(5), (211)
‘in which p and p’ are respectively the old and the new pressures or
RELATING TO SOUND AND LIGHT. 115
the corresponding elastic forces, d and d’ the old and the new den-
sities; « the coefficient of expansion, a constant, and equal to 45
for Centigrade scale; and 0, degrees of temperature Centigrade.
, The first of these equations is the mathematical expression of
Mariotte’s law; the second, of that of Charles or Gay-Lussac; and
the third, of that of Poisson. The gas to which these equations are
applicable is supposed to be a perfect fluid, devoid of friction, and
to have the pressure at each point uniform in all directions.
The temperature is supposed constant during all changes of
pressure and density in Mariotte’s formula, while in that of Charles
the gas takes the pressure and density determined by the change of
temperature. The formula of Poisson supposes the gas subjected
to sudden changes of density, and that the heat developed, whether
considered positively or negatively, is not conveyed by radiation or
conduction to other bodies, or, in other words, that the quantity of
heat in the gas is constant. Remembering that sudden condensa-
tion in air or gas produces heat, and sudden rarefaction cold, and
assuming that these alternations are so rapid that neither the heat
nor the cold is conveyed to the other particles, within the volume
considered, much beyond the point at which they originate, we see
that this heat and cold will produce an elastic force of greater in-
tensity than that in either of the other two cases; therefore the
value of the velocity of propagation will be greater than that given
in Kg. (214), which was deduced under the supposition of the sim-
ple ratio of the elastic force to the density expressed by i It
might be supposed that the influence of the heat produced in the
condensation of the sound wave would be neutralized by the cold
produced in the rarefaction, and that therefore the resultant effect
would be zero. This, however, is not the case; for‘the heat in the
condensation has increased the difference of elastic force between
the condensed stratum and the one in its front, and hence has in-
creased the velocity, while the cold in the rarefaction has caused an
equal difference between the rarefied stratum and the one in rear,
and has thus added an equal increment of velocity to this portion
of the wave. This is true for each stratum affected by the sound
wave. Hence the disturbance passes each stratum of the condensed
and rarefied portions with the same velocity, and this may be re-
garded as the velocity of the wave.
116 ELEMENTS OF WAVE MOTION.
189. Since the vibrational number of sound waves varies be-
tween 20 and 40000, for the extreme limits, the alternate condensa-
tions and rarefactions occur with sufficient rapidity to necessitate
the application of the formula of Poisson for the determination of
the velocity of sound in air and in other gaseous media.
190. Pressure of a Standard Atmosphere. Let p
be the pressure of the atmosphere when the barometric column cor-
responds to 76cm., the mercurial density being 13.5962, and g
981 dynes; we then have
p = 981 x 13.5962 x76 = 1.01368 x 108 dynes, (218)
as the corresponding pressure of the atmosphere upon a square cen-
timetre. But since the density of mercury, referred to the stand-
ard at the same locality, is independent of the locality, and hence
independent of g, we may assume as the standard atmosphere that
whose pressure on the square centimetre at all localities is equal to
106 dynes. Hence,
p=gxd,X h = 108 dynes. (219)
By substituting in this equation the value of g for the latitude
of the place, and solving with reference to 4, we will determine the
barometric height corresponding to the standard atmosphere at that
locality; g varies from 978.1 dynes at the equator to 983.11 dynes
at the pole.
191. Height of the Homogeneous Atmosphere.
If the atmosphere be supposed replaced by an atmosphere of uni-
form density D, as that of standard dry air at 0° C., and height Z,
exerting the same pressure, H may be obtained from the equation
p=y- D-H = 10° dynes; (220)
from which we have
__ 108 = 108 =
H= 7-D = Wix.0012750 = 7.9894 cm. x 10 (221)
= 7989.40 m. = 26212.18 ft.,
which is constant at the same locality, for the same temperature
RELATING TO SOUND AND LIGHT. 117
and barometric height. If the temperature become 6° C., we have,
by the law of Charles or Gay-Lussac,
H' = (1+0)H = Har, (222)
in which rt is the absolute temperature, and « the coefficient of
expansion.
192, Replacing the elastic force H by its equal, in terms of the
homogeneous atmosphere, in Eq. (214), we have
E Hat-g-D ae
VeA/>= \/ i = /Haryg, (223)
which is Newton’s formula for the velocity of sound in air. Mak-
ing tT = 273°, corresponding to zero Centigrade, and g = 981 dynes,
we have
V = 77.9894 x 10% x 981 = 2.8 x10! = 280.0 metres. (224)
For any other temperature, we have
V = V7.9894 x 105 x 981 xar = 28071 + a6. (225)
193. These values of the velocity of sound in air are about one-
sixth less than those determined by experiment, the discrepancy
being due to the supposition that Mariotte’s law expresses the rela-
tion of pressure and density. The law of Poisson is, however, appli-
cable; hence we have
,_ (DV,
= (5) :
f —1 t
differentiating, dp' = yp (ZY —
dp’ pp (DV _ ow.
Whence we see that when a sound wave is passing through air,
the ratio of the increment of the elastic force to that of the density
is equal to the ratio of the elastic force to the density, multiplied by
the constant y. The value of y can be determined from a direct
observation, by accurately measuring V, «, and 9, and substituting
in the equation
118 ELEMENTS OF WAVE MOTION.
v= 7% = 10!x2.80Vy (1 + 6), (227)
and solving with respect to y. Its value has been found to be,
approximately, 1.41 for all simple gases not near their points of
liquefaction. The final formula, therefore, is
V = 332.64 m. x V1 + a
at Ee ae, (228)
= 1091.35 ft. x 1 + .003666°,
for the velocity of sound in air at the locality where g = 981 dynes,
barometric height 76 cm., and temperature 6° Centigrade.
194, At West Point, assuming the barometric height to be
76 em., and g = 980.3 dynes, we have, for the velocity of sound
in air at any temperature,
V = V980.3 x 7.9894 x 141 x 10° x ar
= 332.3 m. x V1 + « (229)
= 1090.23 ft. x V1 + «6. J
Since the value of «= 54,, we see that the velocity increases
nearly 2 feet for each degree Centigrade, and hence is greater in
warm than in cold weather, all other things being equal. At
60° F., we may take the velocity of sound in air to be approxi-
mately 1123 feet per second.
195. The value of the velocity of sound in any gas can, in like
manner, be obtained theoretically by substituting in the equation
VS y £ ’ (230)
for D’ the density of the gas referred to that of air as unity, and
for p’ the value of the pressure in terms of the barometric height,
y being taken as 1.41; or it may be obtained more simply by di-
viding
V = 332.3 m. x V1 + «6 (231)
by the square root of the density of the gas referred to air as unity.
RELATING TO SOUND AND LIGHT. 119
At zero degrees Centigrade, we have for the theoretical value of the
velocity of sound in the following gases:
ATP. cc. 5 332] Carbon dioxide, . . . . 262
Hydrogen, . . - . . 1269/Carbon monoxide, . . . 337
Oxygen, - - .- . . - . 317) Olefiantgas,. . 2. . 2 . 3814
196. Velocity of Sound in Air and other Gases, as
affected by their not being Perfect Gases, The for-
mule of Mariotte, Charles, and Poisson are only applicable to
perfect gases. This condition requires the elasticities to be perfect,
and the excess of the elastic force which gives rise to wave propaga-
tion to be indefinitely small when compared with the elasticity of
the gas in its quiescent state.
A series of experiments made by Regnault, the results of which
are given in the Comptes Rendus, Vol. 66, page 209, show that
these conditions are not fulfilled, and that the theoretical velocity
therefore differs from the actual. The sounds were made in tubes’
of different cross-section, by discharging a pistol with different
charges of powder. The results are grouped in the following table:
Diameter of Tube, 0.108 m. Diameter of Tube, 0.3 m. Diam. of Tube,
Length, 566.74 m. Length, 1905 m. 1.10 m.
Charge, 0.3 gr. | Charge, 0.4 gr. |/Charge, 0.3 gr. oe Charge, 1.5 gr. || Charge, 1.00 gr.
is. | Mean is. | Mean || pig. | Mean) Mean | yj, | Mean is. | Mean
sree wee te aoe tances. bene ve. tances. pie eee co
566.74 | 330.99 | 1351.95 | 329.95 1905 | 831.91 | 332.37 | 3810.3 | 332.18 749.1 | 334 16
1138.48 | 328.77 | 2708.09] 328.20 || 3810 | 328.72 | 380.34 | 7620.6 | 330.43 9201] 333.20
1700.22 | 328.21 | 4055.85 | 326.77 11430.0 | 329.64 1417.9 | 332.50
2266.96 | 327.04 | 5407.80 | *323.34 15240.0 | 328.96 || 2835.8] 831.72
2838.70 | 327.52 5671.8 | 331.24
8507.7 | 330.87
11848.6 | 330.68
ee 1419.5 | 330.56 |
certain. 19. |
17015.4 | 330.50 |
19351.8 330 52 f
}
197. From these results we see: 1°, that the mean velocity of
the same wave decreases from the origin; 2°, that it is less: for the
same charge and route in tubes of smaller diameter; 3°, that it
120 ELEMENTS OF WAVE MOTION.
decrease less rapidly in tubes of larger diameter. Regnault also, by
means of sensitive diaphragms, followed the course of the waves
after they became inaudible, and obtained similar results with re-
spect to these. He found that a sound produced by a pistol dis-
charge, of one gramme of powder, became inaudible at distances of
1150, 3810, and 9540 metres, in tubes of 0.108 m., 0.30 m., and
1.10 m. diameter, respectively, and that the waves became insensible
after traveling distances of 4056, 11430, and 19851 metres respec-
tively. In the tube of 1.1 m. diameter, with a charge of 2.4 grains,
the wave ceased to be audible at 58641 metres, and ultimately ceased
at 97735 metres. These distances of audibility are, approximately,
directly proportional to the diameters of the tube.
198. The mathematical theory discusses the case of a perfect
gas, and assumes that the propagation in an indefinite tube is con-
tinuous. The above experiments show that this is not really the
case. The assumptions made by implication in a perfect gas are:
1°, That the laws of Mariotte, Charles, and Poisson are true,
but it is well known that no gas obeys exactly these laws.
2°. That its elasticity is unaffected by admixture with other
gases.
3°. That the gas offers no opposition by its inertia to wave
transmission; but experiment shows that an intense disturbance
always produces a real motion of the surrounding particles, which
increases the velocity, especially within sensible distances from the
origin. Such is the case, no doubt, in cannon discharges, violent
lightning-flashes, and other like instances.
4°. Theory supposes the excess of pressure due to a vibrating
body small, in comparison with the quiescent barometric pressure ;
but in the cases cited above, the excess of pressure at the origin
may be large, and hence cause an increase in the value of V near
the origin. Therefore, the correction of Art. 193, called that of
La Place, in such cases is not exact.
199. Regnault ascribes as the principal cause of the diminution
of the intensity, the loss of kinetic energy by the reaction of the
sides and ends of the tube, and confirms this by the fact that the
sounds are quite audible outside the tube during their first passage,
and in a less degree at each succeeding passage. As a secondary
cause, he ascribes the influence of the walls of the tube in dimin-
ishing the elasticity without affecting the density. This is con-
RELATING TO SOUND AND LIGHT. 121
firmed by the fact that in the above experiments, where the waves
have been produced by the same charge, and hence have the same
sensibility at the origin, they have not the same intensity after
traveling over equal routes. The mean limiting velocity ought,
therefore, to be the same, if the weakening is due to the loss of mv?
on account of the sides. The experiments show that this is not the
case; hence, the sides exercise another effect on air different from
that indicated as the principal cause of the diminution of the in-
tensity, an action affecting the elasticity and not the density. In
free air this effect would be null, and in the tube of 1.1 m. it is
taken as approximately so. The mean velocity of propagation, in
dry air at 0° C., of a wave produced by the discharge of a pistol,
and estimated from the origin to the point at which its sensibility
can no longer be appreciated by the ear is, according to Regnault’s
experiments,
V = 330.6 m.
The mean limiting velocity, considered from the origin to the
point at which its existence can no longer be detected upon a sen-
sitive diaphragm, is
V = 330.3 m.,
which differs from the mean limiting velocity in the 1.1 m. tube by
only 0.32 m.
200. Velocity of Sound in Gases independent of
the Barometric Pressure. Since an increase in the baro-
metric pressure increases the elasticity and density in the same
proportion, theory indicates that no change, due to this cause alone,
will take place in the velocity. The experiments of Stampfer and
Myrbach in the Tyrol, in 1822, between two stations whose differ-
ence in altitude was 1364 m., and of Bravais and Martins in Swit-
zerland, in 1844, between two stations whose difference of level was
2079 m., indicated no variation in the velocity, due to the change
in the barometric pressure. Regnault’s experiments upon air in
the tube 0.108 m. in diameter, over a distance of 567.4 m., with
pressures varying from 0.557 m. to 0.838 m., and over a distance of
70.5 m., with pressures varying from 0.247 m. to 1.267 m. found
no variation in the velocity, due to this cause.
The theoretical ratio of the velocities of sound in gases, given by
122 ELEMENTS OF WAVE MOTION.
y= as (232)
was experimentally confirmed to a near degree of approximation in
the cases of hydrogen, carbon dioxide, and air. The tube 0.108 m,,
filled for a length of 567.4 m., gave for hydrogen 3.801 m., for car-
bon dioxide 0.7848 m., which differ but little from the theoretical
values 3.682 m. and 0.8087 m., the velocity in air being taken as
unity. Hence the formula may be taken as an expression for the
limiting Jaw. The determination of the velocity of sound in free
air was made by means of reciprocal cannon discharges. There
were two series of these experiments. For the first, consisting of 18
discharges, the membrane being 1280 metres distant, the mean
velocity, referred to dry air at 0° C., was found to be
V = 331.37 m.
For the second series, of 149 discharges, over a distance of
2445 m., during 11 days of trial, with the temperature of the air
varying from 1.5° to 21.8° C., and with great variations in the
wind, the mean velocity, referred to dry air at 0° C., was
V = 330.7 m,
a senstble dirninution of the velocity, due to the increased distance.
201. Velocity of Sound in Liquids. The value of the
yelocity of sound in liquids is likewise given by the general formula
GH _ , |e 1
BAT D
= A D
Decatily E
=4/8x5= D’
in which A is the arbitrary barometric height, d, the density of
mercury, and g the acceleration due to gravity. The numerator is
then the pressure due to the height of the barometer, and when
divided by A, which is the diminution of the volume due to the
increase of pressure, gd,,H gives the ratio of the pressure to the
corresponding compression, and is therefore the measure of the elas-
tic force of the medium. The square root of this quantity, divided
(233)
RELATING TO SOUND AND LIGHT. 123
by the square root of the density, will be the value of the velocity
of sound in the liquid.
202. Colladon and Sturm made a series of experiments to de-
termine the actual value of the velocity of sound in water, in Lake
Geneva, in the year 1826. The sound was caused by the strokes of
a hammer upon a bell submerged one metre below the surface, and
so arranged that the epoch of the stroke could be determined by a
flash of powder. The instant of hearing the sound was indicated
by a stop-watch to within one-quarter of a second. The distance
traveled by the sound was found to be 13487 m. to within 20 m.,
and the time of this travel, from a mean of many experiments, was
found to be 9.4s. The temperature of the water was 8.1° C., its
density at that temperature, referred to that of water at the stand-
ard temperature, was unity plus a negligible fraction, its compressi-
bility was taken at .0000495, and the barometric height at 76 cm.
The density of mercury referred to the same temperature is 13.544,
and g = 9.8088.
Making these substitutions in the preceding formula, we find
9.8088 x 13.544 x. 16
v= VV 0000495 a dee mi
The actual velocity found was eat = 1435 m., differing
from the theoretical value but 7m. The latter may itself vary
within wider limits, on account of the inexactness of the value of
the compressibility of water, whose most probably correct value,
from the experiments of Regnault, is assumed to be .00004685.
203. The principal facts derived from these experiments of
Colladon are (Tome XXXVI, Annales de Chimie) that at distances
beyond 200 metres the quality of the sound is changed, and the
sensation is similar to the quick, brief noise produced by the strik-
ing together of two knife-blades in air. The diminution of inten-
sity with the distance is noticed, and at short distances, greater
than 200 metres, it is not possible to tell whether the sound origi-
nates at a near origin of weak intensity, or at a distant origin with
increased intensity. The duration is less than in air; as it should
A being greater and V being smaller in air
be from its value cm
124 ELEMENTS OF WAVE MOTION.
than in water. When the vibrations proceeding from the sounding
body reach the surface of the water at great angles of incidence, the
sound does not pass into the air. At distances greater than 400 to
500 metres, the ear in air does not hear the sound originating in
the water. At 200 metres the sound is readily heard. In these
experiments, the bell being placed 2 metres below the surface, the
angle of incidence at 400 metres is approximately 89° 43’; at 200
metres, 89° 26’.
Finally, the existence of a sharper acoustic shadow shows that
the wave lengths are proportionally shortened in water compared
with the waves made in air by the same sounding body.
204. Velocity of Sound in Solids, The ordinary solids
upon which experiments have been made for the determination of
the velocity of sound are glass, the various metals, and wood. In
the latter, from the manner of its growth in the tree, the three di-
rections, along the axis, in the direction of the radius, and normal
to the plane of these two, possess necessarily different elasticities.
The coefficients of elasticity also differ in different species, and in
the same species, when grown in different localities, under different
circumstances of soil, temperature, and moisture. Reasonably
exact determinations belong then only to the particular specimen
experimented upon, and mean values are usually taken for any one
kind of wood in a given direction. In metals and glass, variations
of the coefficients arise from the methods of their manufacture, and
modifications result from every circumstance which affects their
density and other physical properties. None of the solids can be
said to be perfectly homogeneous; but on the assumption that they
are approximately so, different experimenters have obtained values
for their coefficients which do not vary between very wide limits.
205. In solids, the sound may result either from transversal or
from longitudinal vibrations. In the cases here considered, the
vibrations are understood to be longitudinal, that is, the molecular
displacements are in the direction of the propagation.
When a solid bar, taken as homogeneous, transmits a longitudi-
nal vibration, the velocity of the propagation has been found to be
given by the equation _
V= / Z (234)
RELATING TO SOUND AND LIGHT. 125
in which 4 is the elongation due to the weight of the bar. Substi-
tuting for 4 its value in terms of Young’s modulus,
and making s equal to one square centimetre, 7 equal to one metre,
and P the weight of the bar, we have
V= 5. (236)
the same in form as has been found for gases and liquids.
206. Different methods have been employed to find Z, viz., by
the direct method of elongations or compressions, by flexure, by
transversal and by torsional vibrations of the bar. The values
given for the different metals, in Art. 23, have been obtained by
Wertheim, by the method of elongations. Could we accurately
determine the velocity of sound in solids by direct experiment, the
value of # could be readily found by the solution of the above
equation. But this velocity being very great compared with that in
air, and because of the impracticability of finding sufficiently long
homogeneous lengths, an accurate determination of # by this means
is impossible. Biot, by a direct experiment on 951 metres in length
of cast-iron pipe, found that the velocity was 10.5 times that in air;
but the want of homogeneity, due to the numerous leaded joints,
without doubt influenced this result appreciably. Wertheim found
about the same value in wrought iron, by experimenting upon
4067.2 metres of telegraph wire.
207. Assuming the experimental values for # given in Art. 23,
and taking g to be 981 dynes, the velocities of sound are, by the
above formula, found to be as follows :
E. Dz V IN CENTIMETRES. Bee
Lead, 177 x 981x108 | 11.4 1.23 x 108 3.7
Gold, 813 x 981 x 108 | 19.0 1.74 x 108 5.3
Silver, 736 x 981x108 | 10.5 2.61 x 10° 8.0
Copper,. - | 1245x981x10°! 8.6 3.56 x 105 10.7
| Tron, - | 1861 x 981 x 108 7.0 5.13 x 105 15.5
Steel, . 1955 x 981 x 108 7.8 4.99 x 105 15.0
126 ELEMENTS OF WAVE MOTION.
For glass, with density of 2.94, V has been found to be, by the
same method, 4.53 cm. x 105; and for brass, of density of 8.47,
T = 3.56 cm. x 105; or 13.6 and 10.8 times the velocity in air,
respectively.
208. The following velocities of sound in wood, deduced from
the observations of Wertheim and Chevandier (Comptes Rendus,
1846), are taken from “ Everett’s Physical Constants,” page 65,
from which also several of the above numbers have been obtained:
ALONG FIBRE. RADIAL. TANGENTIAL. |
Pie} nat ee Soe Get 52 3.32 x 108 2.83 x 105 1.59 x 105
Beech, . .. . 3.34 x 10° 3.67 x 108 2.83 x 105
Birch, te cae aut ae 4.42 x 105 2.14 x 105 3.03 x 105
| Fir, . de itch oad 4.64 x 105 2.67 x 105 1.57 x 105
209. The preceding values of the velocities of sound in solids
are true only when the medium is in the form of a bar of small
cross-section. Wertheim has shown by his investigations, based on
the theory of Cauchy, that the corresponding velocities in extended
homogeneous solids are greater than the above results in the ratio of
3
a: 1.
210. Reflection and Refraction of Sound. The
laws deduced in Art. 77 for the reflection and refraction of wave
motion are applicable to the undulations of sound. From the
equation
sind = pw sin ¢’, (237)
the direction of any deviated ray or that of any deviated plane wave
by a plane surface, can be found when Vi is substituted for p. If
V'> VV, then ¢' > @, and the refracted ray is thrown from the
normal; conversely, if V' < V, then ¢’ < ¢, and the refracted
ray is bent towards the normal. A ray of sound in air, incident on
the surface of water, will be refracted, provided the angle of inci-
dence be less than 13° 26’; for since V’ in water is about 1428 m.,,
and V in air about 332, we have
RELATING TO SOUND AND LIGHT. 127
_ 832
Y= TRB
and sin @ = .2325, or @ = 18° 26’.
= .2325,
For greater incidences the ray is totally reflected, and does not
enter the water.
211. Consequences of the Laws of Reflection.
1°. Ifa sound originate at one of the foci of an ellipsoid, it will
be reflected to the other focus.
2°. If at the focus of a paraboloid, the rays of sound will be
reflected in lines parallel to the axis, and can be again collected at
the focus of another similar paraboloid, with sensibly undiminished
intensity. The slightest sound, as the ticking of a watch, may be
employed to illustrate this case of reflection.
3°. The speaking-trumpet and speaking-tube are employed to
-prevent the too rapid dissipation of sound. The former, partly by
reflection from its sides and largely by resonance, concentrates the
sound within the volume of the cone whose apex is the mouth-
piece and whose section is that of the other end of the trumpet.
The speaking-tube confines the energy in the narrow compass of
the tube, the loss being insignificant in the ordinary lengths em-
ployed.
4°, When a sound is reflected by any
obstacle which prevents its direct trans-
mission, and the observer is at such a dis-
tance that the direct and reflected sounds
are not confounded, the reflected sound is
called an echo. Thus, if A be the position
of the observer, S the origin from which a Figure 28,
sound of short duration emanates, and W
the obstacle, such as a wall, then the direct sound will reach the
s.
Q
observer in the time eeu and the reflected sound in the time
SW + WA s . Sw + WA—SA
33017 the temperature being 0° C. If = aa be
sufficiently great, so-that the reflected sound arrives after the ces-
sation of the. direct sound, then the echo will be heard, provided
-the intensity be of sufficient value. If the two sounds commingle,
128 ELEMENTS OF WAVE MOTION.
the resultant sound will be prolonged, and partial resonance will
ensue. The number of distinct impressions distinguished by the
ear will determine the shortest difference of route necessary to es-
tablish the echo. Thus, if we take nine per second, or 37 m.
is the shortest difference of route at 0° C.
5°. The conditions of interference of sound are the same as
those discussed in Arts. 65-68. Hence, it is theoretically possible
that two sounds affecting the ear simultaneously will result in
silence, and practically it will be shown, in the lectures on this part
of the course, that such an experiment is also possible. Other illus-
trations of interference are also reserved for the lectures.
212. Refraction of Sound. In order that the rays of
sound shall converge after deviation by refraction, we see from the
formula that p = = must be greater than unity. Then the
deviated wave will, in general, become converging, and the energy
accumulate on an ever decreasing surface. Examining the table,
Art. 195, we see that V’ in carbon dioxide is 262 m., and hence,
when the incident medium is air,
332
L — 262 = 1.25,
and sin @ = 1.25 sin ¢’. (238)
The sound lens devised by Sondhaus is a double convex lens of
collodion filled with carbon dioxide, which collects the sound rays
proceeding from any sonorous body and concentrates them appre-
ciably at another point on the opposite side of the lens. By means
of a concave lens of the same material, filled with hydrogen,
V’ = 1269 m., it will be evident, after the study of the properties
of lenses, as explained in optics, that a similar result would be
effected. The slight noise produced by the ticking of a watch may
be collected by this means at a point so that the noise is audible,
when without this assistance it would be inappreciable at the same
poy
218. General Equations for the Vibratory Mo-
tion of a Stretched String. The bodies usually employed
to produce musical sounds by their vibrations are strings, rods air-
RELATING TO SOUND AND LIGHT. 129
columns, plates, bells, ete. When the vibrations of the particles
are perpendicular to the direction of wave propagation, they are
called transversal, and when in the same direction, longitudinai.
We will first consider the vibrations of a perfectly elastic and
flexible string, supposed to be stretched between two points whose
distance apart is 7, by a force which produces a tension 7. Let the
elongation be that given by
( T
U—l= Bo (239)
in which 7 is the natural length, 7’ the length after the tension 7’ is
applied, and # is the longitudinal modulus. If the displacements
of the string from its position of rest be due to the incessant action
of forces whose rectangular accelerations are X, Y, Z, these with
the tension 7' will be the only extraneous forces considered.
Let m be the mass of any element; x, y, z, + dz, y+ dy,
z+ dz, the co-ordinates of its extremities and its length ds; « the
area of its cross-section, and p its density; then
m = pads.
Let the components of 7 at 2, y, 2, be
da dy dz.
Tay Tay? Ta.
and at «+ dz, y + dy, 2+ dz, be
dz da pe = dz dz
T5,t+arz + dT 7 Tatas,
The general equations of motion will then be
2 1
pa ds (x —*) + ar& =0,
a= 240
pads (v— 54) + art! =0, (240)
pa ds(z —S4) + art =0.
214, These equations are simplified when we suppose that the
string is arbitrarily displaced from its position of equilibrium, and
9
130 ELEMENTS OF WAVE MOTION.
abandoned to itself, without the action of the forces _Y, Y, Z It
will then oscillate about its position of rest, and the only extraneous
force that acts will be the tension 7, whose intensity will vary be-
tween known limits. Let the axis of x coincide with the string in
its position of rest, and the co-ordinates of the clement m, at the
time /, be x + & 9, 6 If the displacement be supposed small, é,
n, and ¢ are functions of x and ¢, and « is independent of ¢, and the
above equations reduce to
Pe d(a+é&)
pa dx —- dP — dT a 0,
weit : a 7 dn = 241)
ede —aT = 0, (241)
pe dx = are = 0.
Let 7’ be the tension when the string is straight, and 7 when
the string is displaced; the length of the element is in the first
case dz, and in the second ds ; these are connected by the equations
T—T'
ds = dx (1 + ==>}, (242)
ds? = (dz + dé)? 4+ dr + d?; (243)
from which, when dy and dé are very small, we have
ds = dx + dé, (244)
Pa T' 4. B a. (245)
Substituting in Eqs. (241), we have
rs oa ae
oie = lt a
ca Pq 246
“ae ae, dx’ )
2
PS = Fi ae
Pen * ae
RELATING TO SOUND AND LIGHT. 131
i
Replacing S and tn by w? and v? respectively, we have
PF _ Ge
at? da?
adn an
a= a (247)
Po 4 FS,
de — ” dae
The integration of these three partial differential equations give
(Analytical Mechanics, Appendix IV),
1 = F(e+0t) +f (e—2),
6¢=F («+ wt) +f(«— vA).
&= F(a -+ ut) +f (# —ut),
(248)
215. The first equation determines the longitudinal vibrations,
or those along the axis of the string, and the other two give the
transversal vibrations along y and z respectively. Because of the
independence of the differential equations, the three vibrations in
general coexist and are wholly independent of each other, and since
the differential equations are of the same form, we see that the two
kinds of vibrations are subjected to the same laws. They may each
be discussed separately. Each is due to a progressive motion for-
ward and backward along the string. These motions may be of
the most varied character, but the particular form of the motion
depends on the form of the functions whose symbols are /’ and /.
The only conditions imposed so far are that for = 0 and x = J,
&,, and ¢ are zero for all values of ¢. These, together with any
assumed initial conditions, will enable us to determine the form of
the functions # and f, and thus complete the solution of the
problem.
216. Since the vibrations parallel to y and z are exactly alike in
every particular, the discussion of one will do for the other, and we
will consider that of y, given by the equation
n= Fat vt) +f (e—). (249)
182 ELEMENTS OF WAVE MOTION.
Assume the conditions that at the epoch, or when ¢ = 0,
ad t
n = $(#) and 7 = vy’ (c), (250)
in which the functions @ and y are supposed known, and that ~’ is
the derived function of w. If ¢ = 0, we have
1 = (2) = F(z) + f(a), (251)
1ST = Wi (0) = Fe) —f' (a); (252)
v(t) = F(x) —f (2); (253)
and hence, F(z) = —— (254)
f(s) = a (255)
Therefore (x) and f(z) are known for all values of « from
0 to 7, when, as is supposed, ¢ (x) and w(x) are known between
the same limits.
For the extremities, we have, by placing x = 0 and z = J,
F (vt) + f(— ot) = 0, (256)
F(i+ vt) +f(l— vt) = 0; (257)
whence, F' (vt) and f(— vt) are equal, with contrary signs, and
thus become known for all values from ¢ = 0 tof = w.
217. The value of 7 can be expressed by means of a single func-
tion by substituting vf + 7 —« for vt in Eq. (257) ; whence,
—FQ—a2+4+ rt) = f(c«—vI); (258)
which in Kq. (249) gives
7=P(a+vt) —-F Q—2 + vX). (259)
Again, for vt, in Eq. (257), substitute 7 + vt; then
F(Ql+4+ tt) = —f(—vt) = F (vt); (260)
whence we conclude that the function /' takes the same value when
the variable vt is increased by 27; and therefore by 41, 61, 81, ....
RELATING TO SOUND AND LIGHT. 133
or 2n1, m being a positive whole number. Therefore, if 2 (vf) is
known from vt = 0 to vt = 21, its value is known for all values
from =0tot=o.
Replace vt by / — vi, in Eq. (257), vt being less than 7; then
F (21 — vt) = —f (vt); (261)
but f (vf) is known for all values of vt between 0 and J; therefore
F (vt) is known for all values of vt between J and 2I.
Hence, the value of #' (a + vt) is known for all values of 2+ vt
from 0 to «©; and, similarly, the value of f(# — vt) can be found
for all values between 0 and — o; and therefore the problem is
completely solved.
Figure 29,
218. The function whose symbol is Fis subject to the following
conditions, derived from Eqs. (256), (257), (260), (261),
F («) = —F(—2), (262)
F(l+2) = —F(l—®), (263)
F =FQ+2)=F(4l+2)= Sieatekisity . ‘
(2) Se ee. } (264)
F(e) = —F(Q—2) = —F(4l—2z)=.... (265)
From Eq. (262) we see that the curve represented by 7 = F (2)
is continued in similar forms on each side of O in the figure; from
Eq. (263), that the forms are similar on each side of A; from Eq.
(264), that the form is repeated from O' to O” exactly as from O to
0’; and from Eq. (265), that the form of the curve inverted is the
same from O’ to A as from O to A.
The motion of any particle is that of oscillation about its place
of rest, and of which the period is a This vibratory motion is
gradually diminished, while the period remains unchanged, because
134 ELEMENTS ‘OF WAVE MOTION.
of the energy communicated to the air, and through the points of
attachment to other bodies. The time of one complete oscillation is
t= " = 2l JE (266)
and the number of oscillations in the unit of time is
Ot FEY
Therefore in the transversal vibrations of a string, the resulting
pitch is inversely proportional to its length, directly as the square
root of the tension when straight, and inversely as the square root
of the density by the area of cross-section.
219. The number of longitudinal oscillations in the unit of
time is
, u L. JP ..
ye = a = aI pa’
whence the pitch depends only upon the Jength of the string and
the material of which it is made, and is independent of the tension,
unless the latter should be so considerable as to change the value
of #. Experiment appears to indicate that the longitudinal pitch
increases slightly with the tension; but this may be accounted for
in the elongation experienced, which is always accompanied with a
slight diminution of density p, and should this occur, the formula
indicates that the pitch should rise.
The ratio of the numbers for the same string is given by
We hg
PaVlpaVa ee)
M. Cagniard Latour experimented on a cord of 148 m. in
length, and found
2 188 and) = 0.05 m.
n 7
Substituting in the formula, we have
198 _ 4. /8,
Eo Al’
whence, AJ = 0.052 m., a sufficiently near approximation.
(268)
RELATING TO SOUND AND LIGHT. 135
220. The preceding values of ” and 7’ are the least numbers of
transversal and longitudinal vibrations of the string, and therefore
correspond to its fundamental tones; but we know that each of the
vibrations is decomposed into any number of vibrations of equal
periodicity, when the string is divided into a like number of sym-
metrical parts. This can be shown more readily when the integral
equation is expressed in a series which is a function of sines and
cosines. Thus, it is evident that a possible solution of the differen-
tial equation
dn dn
ae a? Gi)
is given by
iTut . ort) . 4
7= (4; cos > + B, sin ) sin * (271)
when the conditions with respect to the extreme points are un-
changed. In this equation, ¢ is any entire positive number which
marks the order of the term, and .4;, B;, are constant coefficients
depending on 7 and on the initial state of the string. If then this
state is such that 7 is constant only for the terms for which 7 is a
multiple of another entire number %, the string will return to the
same state at the end of each interval of time = » which is the
duration of its similar and isochronous vibrations. Under this sup-
position, the 2 —1 points of the curve corresponding to distances
1 27 31
=-, =-, =-—, = eit,
n n n
will be nodes, that is, will remain at rest during the whole period
of the motion.
Since the value of 7 is linear, every value corresponding to 7 =
1, 2, 3, 4, ete., will be a solution, and the sum of all the values of 7
will also be a solution of the differential equation; hence we will
have for the general integral equation
i sizr( 4s cos = + B; sin ner sin (272)
221. The values of 4,B,, A,Bz, etc., are in general arbitrary,
and we may suppose all to vanish up to any order », while the rest
remain arbitrary. If A,B, are not zero, there are no actual nodes
136 ELEMENTS OF WAVE MOTION.
except the fixed ends, and the first simple tone is that whose period
is.r and whose wave length is 27. If there is one node, the period
is 5, and the gravest simple tone is that of wave length 7; and,
generally, if there are —41 nodes, the period is = , and the
gravest tone is the (x — 1)” harmonic of the fundamental tone.
When the string vibrates without nodes, the series of harmonic
tones is in general complete, and a practised ear can distinguish ten
or more. It is also possible to make a string vibrate in such a man-
ner that for any proposed value of 2 the coefficients 4nBn, Aon Bon,
etc., shall disappear, so that the component harmonic vibrations.
: 7. fF Pee a
whose periods are ye oe ete., are extinguished. When this is
done, the ear does not distinguish these tones, and we may therefore
conclude, from what precedes, that each component tone actually
heard is produced by the corresponding harmonic vibration of the
string.
222. The same general method may be applied to the longitudi-
nal vibration of a rod, and the differential equation will be, as in
the case of the longitudinal vibration of a string, of the form
aE ae
p= ve ie? (273)
of which the integral equation is
f= P(x +Vi) +f(e—Vo), (274) °
and which may be put under the form of
&é=2+ = cos = (4 cos ae + Bi sin ah (275)
in which £ is the distance from the fixed origin at any time ¢ to the
particles in a plane section of the rod, of which the natural distance
from the end of the rod is 2 The value of z therefore depends
only on the particular section considered, and is independent of the
origin of €; but if the vibrations cease, the periodic part of Eq.
(275) would vanish, and we would have ¢ = @ for all points of the
rod, and therefore the periodic part gives the displacement (£— 2)
at the time ¢ of the section determined by the value of a.
RELATING TO SOUND AND LIGHT. 187
The periodic part does not in general vanish for any value of z,
so that there are in general no nodes. But there will be nodes at
sections for which z is any odd multiple of a » provided .A;, B;
vanish for all values of 7 except odd multiples of x. Thus the rod
may have any number of nodes, of which those next the ends are
distant from the ends by half the distance between any two consec-
utive nodes.
223. Differentiating Eq. (275), we have
dé wo... inal in Vt ~ one).
a 1 —7%t sin — (4; cos 7 + B; sin 7 ) (276)
which, when z = 0 and 2 = J, becomes
dé
Za (277)
dep
But wa p?
in which p’ is the natural density, and p is the changed density.
We see, therefore, that there is no change of density at the free
ends. If A;, B;, vanish, except where 7 is a multiple of , the
variable part of = vanishes when % is a multiple of . Hence,
where there are nodes, the sections in which there is no variation of
density are those which bisect the nodal intervals in the state of
equilibrium, and these sections of no variation of density are also
sections of greatest displacement, since, Eq. (275), cos > is equal
to +1 for values of « which make sin = = 0.
224, The vibration represented by Eq. (275) consists of an infi-
nite number of simple harmonic vibrations, each of which might
subsist by itself; the n” component would have » nodes, and its
eo Rb
period would be eae the period of the fundamental tone being re
therefore the wave length is twice the length of the rod. For the
general case in which there is a node at the middle of the rod,
cos > vanishes for all values of 7 when z = i Then A;, Bi,
138 ELEMENTS OF WAVE MOTION.
must vanish for all even values of 7. The gravest tone is then the
fundamental tone of the rod, and the higher tones of even orders
disappear. The first upper tone will be a twelfth above the funda-
mental. In this case, the middle section might become absolutely
fixed, and either half be taken away without disturbing the motion,
so as to leave arod of half the length, with one free and one fixed
end. Therefore the fundamental tone of a rod with one end fixed
is the same as that of a free rod of twice the length. The wave
length is then four times the length of the rod, and the even orders
of the harmonics are wanting.
225, The vibrations of air columns are theoretically the same as
that of a free rod or one fixed at an end, and the same conclusions,
modified by the elasticity of the air and its velocity of wave propa-
gation, will theoretically apply. We will, however, determine the
positions of the nodes and ventral segments of vibrating air columns
in a simpler manner.
226. Vibrations of Air Columns. We will first sup-
pose a single sonorous pulse moving in an air column, and consider,
1°, the column closed at one end and open at the other. Hach
stratum passes through all changes of density during the periodic
time 7, while the pulse moves a distance 4; the air particles de-
scribe longitudinal vibrations, whose amplitudes depend on the
intensity of the sound. When the condensation, which we suppose
is in advance, reaches the closed end, the air stratum at that place,
not having freedom of motion, undergoes changes of density alone.
These changes are each immediately reflected in succession, and the
condensation moves from the closed end with the same velocity with
which it would have proceeded beyond had there been no obstruc-
tion to its progress. Hence we see that at the instant the rarefac-
tion first reaches the closed end the reflected condensation affects
the same strata as the incident rarefaction, and disregarding the
loss due to incidence, the air strata will, at this instant, in the
a
length 5 from the closed end, have their normal density through-
out. The velocities of the air particles, at the same instant, will
likewise be that compounded algebraically of those belonging to the
reflected condensation and incident rarefaction. Now when a
sonorous body is vibrating, the sound undulations follow each other
RELATING TO SOUND AND LIGHT. 1389
periodically, and therefore the reflected and incident pulses will be
distributed throughout the column. The densities and motions of
the strata will therefore result from the combination of the same
elements in the incident and reflected pulses.
227. Let the curve rm'mA represent the direct wave at any
instant, and its ordinates the corresponding compressions and dila-
tations of the air on the line »’d’ due to this wave ; the curve
m''m'bA and its ordinates will, in like manner, represent the re-
flected wave from the stopped end AA’.
Figure 30,
We see that at points such as », v’, etc., at $A, 24, ete., from
AA’, the condensations or dilatations due to the direct wave will
always be contrary and equal to the dilatations or condensations due
to the reflected wave; hence, at these points, the normal density of
the air will ever exist. But at points such as 7, n’, etc., at distances
of $A, A, 34, ete, from AA’, the condensations or dilatations of eacli
are of equal value, and of the same kind, and exist simultaneously ;
therefore the resultant condensation or dilatation is double that due
to either. At these points then the air undergoes all variations of
density during the period 7, The density at all points from x to v
and to ov’, undergoes decreasing variations from the maximum at n
to zero at v and v’.
228. With regard to the velocities of the air particles at differ-
ent distances from AA’, since the motions of the particles change
direction abruptly at reflection, the ordinates of the curve A’dmo
will represent the velocities due to the reflected wave, and those of
Amm"r may now represent those of the direct wave. Then at vr, 2’,
etc., the velocities are zero only at instants separated by 5 and at
all other times have values that vary from zero to that represented
by double the maximum ordinate; at n, n’, etc., the velocities are
140 ELEMENTS OF WAVE MOTION.
always zero, and therefore the air at these points is quiescent, while
undergoing changes of density. At intermediate points, both
changes in velocities and density occur.
Hence, we conclude that xodes will be developed in a column of
air closed at one end, when it is traversed by a sonorous wave, at
24 40 6A
re 7 a? etc.
The vibrating parts between the nodes are called ventral seg-
: j A BA 5A
ments, and their middle points are at distances of 2 - = etc.,
from the stopped end.
distances from the stopped end of 0,
229. 2°. Open Air Columms. Let the two tubes AM
and MB, of unequal diameter, be united at M, and admit that there
is no abrupt change of density of the air at M. The consequence
of a contrary supposition is that the opposite sides of the infinitely
thin stratum M would be subjected to unequal pressures, whose
finite difference wonld generate in M an infinite velocity in a finite
time. Hence, the density has the property of continuity in its
variation throughout AB. It is not essential that the variation of
the velocities of the particles of air should be continuous, nor is it
incompatible with this condition.
Let s and s’ be the areas of sections M B
in AM and MB, indefinitely near M; 4————
v and v’ the velocities of the air parti- ———
cles in s and s’, at the time ¢; then
vs dé and v's' dt will be the volume of Figure 31,
air passing s and s’ during dt, and
(vs — v's') dé will be the increment of the quantity of air in the
volume ss’ in the time dé, which will be proportional to the increase
of density in ss’. But in order that the increment of density may
be compatible with the supposed continuity of the pressure, it is
evident that (vs — v's’) dt must be an infinitesimal of the second
order, and equal to zero when s and s’ are coincident. Hence, at
the limit we have
vs = v's'
therefore, there will be a wave propagated in MB, whose intensity,
determined by the value of v’, will become more and more inappre-
clable as s’ becomes greater and greater than s. Let MB be in-
RELATING TO SOUND AND LIGHT. 141
creased indefinitely in area, as when the tube AM opens into the
external air, then v’ becomes very small, and the transmitted wave
becomes negligible, as is the case in open pipes. There will then
be a reflected wave in AM, composed of a rarefaction followed by
a condensation, when the direct wave is a condensation followed by
a rarefaction. The velocities of the air particles will then be theo-
retically equal in value, and the same in direction in the two waves.
The curves of Fig. 30 will illustrate the case of open pipes, if A’bof
represent the densities and Adm" the velocities of the air particles
in the reflected wave. The nodes and middle points of the ventral
segments will then be at distances of
A 384 5A WA
4 ’ 4 2 A 2 4 ? etc. ?
24 4A GA
and eg 8 ae ee etc.,
from M, the open end of the tube, respectively.
230. These laws, which determine the positions of the nodes
and ventral segments of vibrating air columns, are known as Ber-
nouilli’s laws. From them we see that the harmonics of open pipes
are in the order of the natural numbers, and that those of closed
Pipes are as the odd numbers. Thus, the open pipe can give, by
an increased pressure, the octave, the twelfth, the fifteenth, etc.,
while the closed pipe gives the twelfth, the seventeenth, etc. Ex-
periments with organ-pipes verify the laws of Bernouilli only approx-
imately ; that is, that the nodes are not exactly at the positions
defined above, nor are the nodes exactly places of rest. Organ-
pipes are usually made to speak by forcing a current of air through
a narrow slit, and causing it to impinge against a thin lip. Of the
many vibratory motions produced in this manner, there is always
one whose periodicity is such that, by the resonance of the pipe, its
intensity will be raised to such a degree as to produce a marked and
determinate musical sound, called the fundamental tone of the
pipe. Other vibratory motions, which undoubtedly exist, are either
destroyed by the interference of the reflected waves, or have so feeble
an intensity as to be negligible. The wave length of the funda-
mental tone is, as we have seen above, double the length of the
open pipe, or four times the length of the closed pipe, approxi-
142 ELEMENTS OF WAVE MOTION.
mately. The discrepancy between experiment and theory arises
from the fact that the hypothesis is not in accord with what actu-
ally occurs in the pipe. Without considering these minutely, it is
sufficient to note the perturbations at the embouchure by the air
current, the modifications in the pipes by the moving air, and the
induced vibrations of the material of the pipe at the sides and
closed end, to account for the greater discrepancies.
231. Relative Velocities of Sound in Different
Material. Since in any medium, we have A = Vr = ve, in
which 7 is the vibrational number for a note of definite pitch, a the
corresponding wave length in the same medium, and V the velocity
of sound, it is readily seen that if free rods of different material be
taken, of such lengths as to give the same note when put into
* longitudinal vibration, we will have
i=", Hert, vers, ae
n n n
whence
AMA" 22 ViVi V",
x I yw
But as 3°93? 9? etc., are the lengths of free rods that give the
fundamental tone, we see that such lengths are directly propor-
tional to the velocities of sound in the several media, when their
lengths are great compared to their cross-sections. Knowing then
the velocity of sound in any material, we can by experiment find
that in others by this method. Then having the velocities, we can
by substitution in the formula
BE
v=4/%,
find the value for the longitudinal modulus Z.
232. Applying the same principle to any gas and comparing the
velocity in it with that im air, by the formula
hA ace
V= v/ “= [1 + eb] y, (278)
RELATING TO SOUND AND LIGHT. 1438
the values of y, or the ratio of its specific heats, can be readily
obtained. By this means Dulong found the following results :
DENSITY. VELOCITY. eS - *
Air BY OE he een se Me 1. 333. 1.421
Oxygen. . . Se CR 1.1026 317.7 1.415
Hydrogen. . . oe 0.0688 1269.5 1.407
Carbon Dioxide. an od 1.524 261.6 1.338
Carbon Monoxide, - ae 0.974 337.4 1.427
Under the assumption that the gas is perfect, simple, and far
from its point of liquefaction, y is assumed to have the constant
value of 1.41. The above results show that this value should be
considered as the limit to which y approximates and only reaches
under the particular suppositions made.
233. Transversal Vibration of Elastic Rods. An
elastic rod is a rigid body whose cross-section, considered uniform
throughout, is taken as very small compared with its length. The
rod or bar may be arranged in six different ways, depending on the
method by which its ends are sustained, viz. :
1°. The rod may be free at both ends.
2°. It may be firmly fixed at both ends.
3°. It may be fixed at one end and free at the other.
4°, It may be supported at one end and free at the other.
5°. It may be fixed at one end and supported at the other.
6°. It may be supported at both ends.
It may yield its fundamental tone by vibrating as a whole, or
give tones of higher pitch by dividing itself into vibrating parts
separated by nodes. The formula
nt “E
N= ma / 9% (279)
gives the number of vibrations in all cases, as has been verified by
experiment. In this formula, V is the number of vibrations per
second; 2 a constant depending on the manner in which the rod is
arranged at the ends and on the number of nodes formed ; ¢ is the
144 ELEMENTS OF WAVE MOTION.
thickness, measured in the plane of vibration; 7 is the length, #
the rigidity, and D the density of the rod.
234, This formula shows that the vibrational number is inde-
pendent of the width, provided it be small as at first supposed ;
that it is directly proportional to the thickness, inversely as the
square of the length, and directly as the square root of the rigidity
divided by the density.
1°. The rod is free at both ends. Lissajous has determined
by careful experiments that the following formule apply, viz. :
21 bl __ 0.66087
ay te
Sea Shoe? oo aa
in which / is the length, 7 the number of nodes formed, d the dis-
tance between two consecutive nodes, s the distance from the free
ends to the nearest nodes, and s’ the distance from the free ends to
the second nodes. Hence from these formule, we see that the
intermediate nodes are equidistant; that the distance from the
extreme nodes to the next adjacent is nearly 0.92 of the distance
between two consecutive intermediate nodes; that s:s' :: 0.2643: 1,
and s:d::0.383:1. Experiment confirms these results whatever be
the number of the nodes. The positions of the nodes are made visi-
ble by sprinkling sand on the bar, and noticing the lines on which
it accumulates when the bar or rod is put in vibration.
2°. Both ends are fixed. When the ends are so fixed as not to
modify its elasticity at these points, it can vibrate freely, and the
nodes are found to be located at the same places as in a free rod of
the same length, except that the extreme nodes are at the fixed
ends. The first two of formule (280) are then applicable to this
case.
3°. The rod is fixed at one end and free at the other. There
will then be 0, 1, 2, 3, .... nodes depending upon the manner by
which it is put in vibration. If the fixed end be regarded as a
node, the first of the above formule is applicable, and the other
two apply to the free end only. Therefore these first three cases
are all reducible with the modifications mentioned to that of a free
rod at both ends.
4° and 5°. In these cases the supported end may be considered
as an intermediate node, and we can consider the rod as half of a
rod of double the length, free or fixed at both ends in which the
RELATING TO SOUND AND LIGHT. 145
number of nodes is 2x—1. Replacing 7 by 27 and nm by 22 —1,
we then have
as 4l ‘ 54 _ 1.82162
nag? * ~gBoe 2~= a as
of which the last two apply only to the case where one of the ends
is free.
6°. If the supported ends be regarded as intermediate nodes
we have
os (282)
235. Harmonic Vibrations of Elastic Rods. When
the vibrating parts are known, the harmonies of the rod are easily
determined, and considering the fixed extremities as nodes, the
formule of Lissajous above given become general for the six cases.
In the first three cases the sounds resulting are the same for the
same number of nodes, whatever be the condition of the extremity,
whether fixed or free. The numbers of vibrations are as 8%, 5%, 72,
.... (2a — 1)*, when there are 2, 3, 4, ....2 nodes. In the 4° and
5° cases where one of the extremities is supported, the vibrational
numbers are as 5%, 9%, 13%, ....(42 — 3)?; and in the 6° case the:
numbers are 1%, 27, 32, ....(7 —1)?, » being the number of nodes.
Comparing in all these cases the vibrational numbers for two
nodes, we have
25
9 25
9:74 or Zz: agi4s
and therefore generally 4
(2n—1)? , (4n—3) aya
For d we have
al Al i
Qn —1’ 4n — 3’ 1
Substituting the value of m taken from the latter in the former,
we have
P
Zz
If d = 1, which corresponds to a rod supported at both ends ana
yielding its fundamental sound, we have V = 1. We therefore con-
10
N= (283)
146 ELEMENTS OF WAVE MOTION.
clude that when a rod gives a harmonic, the parts comprised between
the nodes vibrate as rods whose extremities are supported and whose
length is the distance between the nodes, and that the vibrational
number is inversely as the square of this length. This conclusion
is inapplicable to the first nodes, because they are more or less
influenced by the extremities.
236. Tuning Forks. A tuning fork may be regarded as a
rod or bar free at both ends. Experiment shows that in proportion
as a bar free at both ends is bent or curved the extreme nodes
approach each other. Thus, in the
figure the bar ad if supported at the . e
points 1, 2, one fourth the length m \ / 3"
of the bar from the extremes, will 7 % \ /
when vibrated transversely develop \ \ | a ae
nodes at these points. In the forms Me le
ay’. a’b"", ab" the length remain- Fi Peek
ing unchanged, the nodes approach
each other as indicated in the figure.
The laws which govern the yibration
of a fork whose section is rectangular have been experimentally
found to be; 1°, that the vibrational number is independent of the
width ; 2°, proportional to the thickness; 3°, inversely proportional
to the square of the length increased slightly. The length 1s taken
as equal to the projection of the prongs on the medial line of the
fork. For a fork of rectangular cross-section we have from the
experiments of Mercadier.
N
Figure 32,
t
=k 2R(.012)2
in which WV is the vibrational number, ¢ the
thickness, and 7 the length; « is a constant
which for steel is found to be 818270. When
the fork yields its fundamental note its method
of division is shown in the figure. The over-
tones of a fork correspond to vibrational
numbers which are to each other, beginning
with the first; as, 32:5?:'7:cte. The vibra-
ticaal number of the first overtone is about Bo
4 Figure 33.
RELATING TO SOUND AND LIGHT. 147
that of the fundamental. Helmholtz found by experimenting on
many forks, that it varied from 5.8 to 6.6 that of the funda-
mental. These overtones are so high, that they are generally of
short duration, and they are also inharmonic with the prime.
Tuning forks are generally mounted on their resonant boxes, by
which arrangement the prime tone of the fork is greatly rein-
forced to the disadvantage of the overtones. The duration of
the vibration of a fork although theoretically constant, is found
to increase slightly with an increase of amplitude and tempera-
ture, thus slightly lowering the pitch. This is, however, not
appreciable to the hearing, but can be detected by any of the graph-
ical methods for determining the number of vibrations in a given
period. It is a matter of importance in determining the initial
velocity of projectiles, by means of the Schultz chronoscope or other
devices, where the vibrations of a tuning fork enter into the calcula-
tion, to limit the amplitude and to take note of the temperature, in
order to obtain uniform and reliable results. When the amplitude
does not surpass 3 or 4mm. and the temperature varies but little
beyond the ordinary atmospheric temperature, the vibrational
number may be taken as constant within .0001 of its value.
237. Vibration of Plates. Plates are rigid bodies, gener-
ally of metal or glass, whose length and breadth are very great
compared with their thickness. To put them in vibration, one or
more points are fixed and a violin bow is drawn across an edge.
The circumstances of vibration are exhibited by sprinkling fine
sand over the surface and examining the nodal lines formed by the
sand which seeks that part of the plate which is at rest. The parts
of the plate separated by a nodal line, evidently vibrate in opposite
directions, and therefore for permanent figures the number of
vibrating parts must be even. When the plate yields its funda-
mental tone the resulting figure is the simplest that can be formed,
and as the plate separates into a greater number of vibrating parts,
the figures become more complex. Chladni has given to these
figures the name of Acoustic figures. As yet, from the inherent
difficulties of the problem, the mathematical laws have not been
deduced, but experiment has assigned the following as the laws of
vibrating plates, viz.; 1°, the vibrational numbers of plates of the
same form and of the same material are inversely as the squares of
148 ELEMENTS OF WAVE MOTION.
the homologous dimensions ; 2°, and are proportional to the thick-
ness. Hence we have
by Ob
Mim! it yap
If a rectangular plate be so constructed that a system of nodal
lines parallel to the length be formed by a sound, which gives
another system of nodal lines parallel to the breadth, when it is
vibrated in these two ways, then if at any of the middle points
of the ventral segments it be vibrated so as to produce the same
sound, these two systems will simultaneously exist and the acoustic
figure will result from the combination of these two systems. The
figure illustrates five such plates where the numbers of the nodal
lines are in the ratios of 2:3, 2:4, 3:4, 3:5, 4:5. Other combi-
nations illustrative of the vibrations of plates are reserved for the
lectures.
2:3 2:4 3:4 3:5 435
Figure 34,
238. Vibration of Membranes. When a stretched mem-
brane is near a sounding body, the air transmits to it the vibratory
motion. It can respond, however, only to certain sounds depend-
ing on its tension, and thus enter into synchronous vibration.
This fact is made evident by the acoustic pendulum, or by the
nodal lines formed by sand sprinkled upon it, as in the case of the
px
RELATING TO SOUND AND LIGHT. 149
vibration of plates. The frames upon which the membranes are
stretched are generally square or circular. Experiment has con-
firmed the following deductions of Poisson and Lamé, with respect
to the vibrations of square membranes, viz. :
1°. Membranes respond only to certain sounds, separated by
determinate intervals.
2°. To each sound a system of nodal lines corresponds, parallel
to the sides of the membrane, and whose numbers are represented
by 2 and 7’.
3°. The nodal lines which correspond to the same sound form a
system of figures, such that we can pass from one to the other by
continuous changes in varying the mode of disturbance, without
changing the sound; but we can never pass in a continuous man-
ner from the lines of one sound to those of another.
Circular membranes can only give nodal lines along the diam-
eters or circumferences, either separate or combined, depending on
the method of vibration and on the point or points of enforced rest.
239. Because of the limited time allotted to this part of the
course, many subjects of mportance are necessarily omitted in the
text. ‘Among these are,
1°. The theory of beats, and resultant sounds.
2°. The phenomena of interference. whose consequences, how-
ever, are readily derived from the discussion in Arts. 65-68.
3°. The graphical and optical methods of the study of sonorous
vibrations, and that by sensitive and manometric flames.
4°, The phenomena of vibrations of air columns in organ-pipes,
of elastic rods, of plates and membranes, with the applications of
the latter in the phonograph, phonautograph, and telephone.
By means, however, of a very complete acoustical apparatus,
mainly from the workshop of Koenig, the celebrated physicist of
Paris, the omitted parts, as well as those treated of above, are illus-
trated in the lectures, which largely supplement and complete the
study of the text.
240. The nature and essential principles of undulatory motion,
as illustrated by sonorous vibrations, have received sufficient atten-
tion to enable the student to prosecute understandingly the study
of similar principles connected with light in the analogous subject
of optics.
PART III.
OPT tes
241. Light is the agent by which the existence of bodies is
made known to us through the sense of sight.
That branch of physical science which treats of the properties
of light and the laws of its transmission is called Optics.
242. It is divided into two parts:
1°. Geometrical Optics, which embraces all the phenom-
ena relating to the propagation of rays, based on certain experi-
mental laws, and which is entirely independent of any theory as to
the nature of the luminous agent.
Experiments in Geometrical Optics, however carefully made,
can never accurately prove the laws of light propagation, but serve
merely to establish a certain degree of probability of their truth,
and which, when applied to other phenomena of the same nature,
strengthen this probability in proportion as the application is more
extended.
2°. Physical Optics, which is based on the theory of un-
dulations, and seeks to explain by this theory the nature of light,
and of all the phenomena arising from the action of rays on each
other.
243. That light is not a material substance, but is merely a
process going on in some medium, is proved by the phenomena of
interference, in which results of various magnitudes occur, from
less to greater, or the reverse, depending upon the manner in which
the interference takes place, even when the combining magnitudes
are themselves constant in value.
244. The undulatory theory asserts that light is due to the
transmission of energy from luminous bodies to the finely-divided
parts of the optic nervc, spread over the interior concave surface of
RELATING TO SOUND AND LIGHT. 151
the eye. This energy is conveyed by the optic nerve to the brain,
and there transformed into the sensation of sight.
The transmission of the energy is accomplished by undulatory
motion in a medium called the luminiferous ether. There is no
direct proof of the actual existence of the ether, and its assumption
can only be regarded as an extremely probable hypothesis, supported
by nearly all the known phenomena of light, and directly contra-
dicted by none.
Within the present century, its reality has been almost uni-
versally accepted, and as a consequence the undulatory theory has
entirely supplanted the rival hypothesis of the materiality of light
molecules, known as the emission theory, which had, however, held
its ground for many years.
245. The accepted properties of the luminiferous ether have
resulted from theoretical considerations, modified from time to time
by deductions from experimental observations, and while there are
several imperfections yet to be removed, nevertheless the strong
array of unquestioned facts, both observed and predicted, has estab-
lished these properties as a satisfactory foundation upon which
modern physical optics is now constructed.
The luminiferous ether is considered to be a material substance
of a more rare and subtile nature than the ordinary matter affecting
the senses, ard to exist not only within these bodies, but through-
out space. It has great elasticity, and is capable therefore of trans-
mitting its particular energy over vast distances, with great velocity
and with inappreciable loss. That this energy is not transmitted
instantaneously has been proved by direct experiment, and con-
cluded from several astronomical observations.
246. That light is propagated in right lines from the source is
a fact of observation and experiment. This statement, however,
while absolutely true, is subject to modification when taken in the
ordinary sense of the language. Thus, we have seen that while
sound is propagated in right lines from its source, it is capable of
spreading around an obstacle, so that sound can be heard out of the
direct line of the source ; so, in a less degree, we can see around an
obstacle, as will be shown in the discussion of the diffraction of
light.
The acoustic shadow, however, is as much less marked than the
optical shadow as the wave lengths of sound are greater than the
152 ELEMENTS OF WAVE MOTION,
wave lengths of light. But for the explanation of the principles of
geometrical optics it is unnecessary to consider this refinement.
247. Bodies are called self-lwiminous when they are themselves
the sources of light, and rays proceed directly from them. They
are visible because of their emanating rays. Other bodies are called
non-luminous, and become visible because rays from luminous
bodies are reflected from their surfaces.
A luminous point, or origin of light, is a very small portion of a
luminous surface. When light emanates from a luminous point,
we consider it made up of rays of light, each of which is the small-
est portion of light which can be transmitted. The ray is the right
line along which the undulation is propagated, and is practically a
mere conception, indicating direction.
A collection of parallel, diverging, or converging luminous rays
is called a beam of light, and sometimes a pencil of light, the latter
name being generally applied to the last two cases.
The ais of a beam is the geometrical axis of the cylinder or
cone of rays; when the axis is normal to the deviating surface, the
beam is direct, and when inclined to it, odlique.
248, When a beam of light is incident upon any surface, it is
generally separated into three portions, viz.,a part is scattered or
diffused over the surface, by which the surface becomes visible, a
second part is reflected, and the remainder is refracted.
The proportion of the several parts depends on the polish of the
surface, the angle of incidence, and the nature of the medium. A
perfectly polished surface would be invisible, and the incident beam
would be separated into a reflected and refracted beam alone; of
course, such a polish is not practicable. Light regularly reflected
has its intensity increased with the degree of polish, while the in-
tensity of irregularly reflected light is similarly diminished. The
intensity of regularly reflected light from the surface of water is, at
the incidences of
0°, 40°, 60°, 80°, 894°,
about 1.8%, Raves 6.5%, 33%, 72%.
At normal incidence, water, glass, and mercury reflect 1.8%,
2.5%, and 6637, respectively. The differences at small angles of
incidence are more marked than at greater angles, since while both
RELATING TO SOUND AND LIGHT. 153
water and mercury reflect the same at 894°, the former reflects but
gs as much as the latter at normal incidence.
249. A mediwm is any substance which permits the passage of
light through it.
Since the luminiferous ether is supposed to pervade all matter,
it might be inferred that all bodies could be classed under the head
of media for light. Gold, although one of the most dense of sub-
stances, does permit the passage of light, when beaten into a very
thin leaf; and no doubt if other opaque bodies possessed an equal
malleability, the same property would belong to them.
But owing to internal reflection and consequent interference, it
is assumed that an inappreciable quantity of light, if any, passes
through very small thicknesses of opaque bodies. Glass, air, water,
and all other matter which permit the passage of light freely, are
said to be transparent. Transluceucy is a term applied to such
bodies as permit the passage of diffused light; thus, ground glass
and flint are translucent, while clear glass and quartz crystal are
transparent.
250. Since light is assumed to result from undulatory motion
in the luminiferous ether, all the consequences deduced in the dis-
cussion of the properties of this kind of motion in Part I are at
once applicable to the phenomena of light.
251. Shadows and Shade. From each luminous point
considered as an origin of disturbance, undulations proceed along
right lines in all directions from this origin. Therefore, whenever
they meet an opaque body, this undulation will be deviated from
its original direction, and the effect of light will be wanting along
this direction prolonged.
The absence of this effect is called the shadow of the point of the
opaque body.
The line of the surface of the opaque body, along which rays
drawn from the luminous point are tangent, is called the line of
shade. Since each point of the luminous surface is an origin of
light, we see that in all actual cases the shadow of an opaque body
must be indistinct near its boundary, and gradualiy merge into the
illuminated surface surrounding the shadow, whenever the lumi-
nous source is of an appreciable area. This modified portion of the
shadow is due to the overlapping of the cones of rays proceeding
154 ELEMENTS OF WAVE MOTION.
from each ]uminous point, and 1s called the penumbra. It is lim-
ited by the space between the two cones, whose elements are tan-
gent to the luminous surface and the opaque body, one having its
vertex between the two, and the other its vertex on the further side
of either one of the surfaces. The softness of shadows in general is
due to the finite extent of luminous surfaces.
252. Every point of the luminous source emitting rays in all
directions, each will carry an image of its Inminous point.
Figure 35.
Thus, if a lighted candle be placed in front of a small aperture
of a darkened chamber, the aperture will permit the passage of a
limited number of the ravs from every point of the candle, each ray,
however, carrying an image of its radiant. The image, as shown in
Figure 35, will be inverted.
If another aperture be made near the first, a second image of
the candle will be formed, overlapping the first, and, while the
luminosity will be increased, the image will lose distinctness, be-
cause of this overlapping. The diffused light of a room during the
day is due to the overlapping images of external objects, caused by
rays proceeding from each of them, thus making their individual
images indistinct. A small aperture in a darkened room will per-
mit the formation of an inverted image of the external scenery upon
a screen placed within the room near the aperture.
253. Photometry. The eye possesses the property of dis-
tinguishing color and intensity.
In determining variations of intensity, the judgment is only
approximate when the colors are the same, and the difficulty of this
appreciation is increased when the colors differ. Equality of in-
RELATING TO SOUND AND LIGHT. 155
tensity can readily be determined by the eye, while it is not possible
to ascertain the numerical ratio of different intensities by direer
observation. :
Photometry has for its object the measurement and comparison
of the intensities of different lights.
254. The principle of all photometric methods is to arrive at
this comparison, by the appreciation of the equality of iumination
of two near surfaces, physically identical. In assuming the dis-
tance of the luminous source from the illuminated surface to be
great in comparison with the dimensions of the surface. and remem-
bering that the intensity of the light is due to the molecular kinetic
energy, we readily see, if there be no absorption of this energy dur-
ing transmission through the intervening media,
1°, That the intensity of the illumination on the unit area of
any surface, taken normal to the direction of propagation, at a
2 : : ak
distance d from the luminous source, varies as me
az
2°, That if J represent the intensity of any given light. and if
it be supposed to illuminate uniformly any area 4. the intensity on
: z 1
a unit of area varies as —"
3°. That the quantity of light emanating from any luminous
element, and hence the intensity of illumination on the unit area.
is proportional to the cosine of the angle made by the normal to the
element with the direction considered, and hence varies as the co-
sine of the inclination, or cos ¢.
4°, That if the area on which the light falls is inclined to the
direct line of propagation, the illumination on the unit area is pro-
portional to the cosine of the angle made by this line and the nor-
mal to the surface, or to cos ’”.
5°, That the illumination on the unit area will vary with the
intrinsic brightness of the source. The intensity of the illumina-
tion on the unit area, parallel to the source, at the distance unity,
may be taken as the measure of the intrinsic brightness.
255. Let S and S’ be the projections of the luminous and the
illuminated surfaces. respectively, on a plane normal to the direc-
tion of the luminous rays; B the intrinsic brightness of the
156 ELEMENTS OF WAVE MOTION.
source ; @ the distance apart of the two surfaces, and J the intensity
of the illumination; then, from the above principles, we have
r= 8-5 (285)
Making S’ = 1, and calling J, the total brilliancy of the source
at the distance d, we have
S
L= BR (286)
Zz is the apparent area of the source seen from the illuminated
surface, and making this equal to unity, we have
EB (287)
Therefore the intrinsic brightness of the source is the total
brilliancy of the apparent unit of area of the luminous surface at
the distance 1.
The general method of comparison of the intrinsic brightness of
two sources consists in permitting the rays from each source to fall,
nearly normal, upon adjacent portions of the same surface; then
to increase the distance of the stronger light, until] the eye judges
the illumination to be equal. We then have
BS _ BS,
CG ~ @e-
(288)
from which, by substituting the known values of d, d,, S and 8,,
the ratio of B, to B ean be determined.
256. The apparent intrinsic brightness of an object is equal to
the quantity of light received from it by the eye, divided by the
area of the picture on the retina. Therefore, since the apparent
y
illumination of the object is B = and the area of the retinal pic-
“ heey coos calar a!
ture is 7B the apparent intrinsic brightness will vary with the real
intrinsic brightness B, and the object will appear equally bright at
all distances.
This result is deduced under the supposition that no light from
the object is absorbed by the medium through which it passes, and
is therefore only an approximation.
RELATING TO SOUND AND LIGHT. 157
257. Velocity of Light. In 1675, the Danish astronomer,
Reemer, noticed certain discrepancies with regard to the observed
times of the eclipses of Jupiter’s satellites, which he correctly at-
tributed to the finite velocity of light.
To show this, let 8 be the sun, EE’ the earth’s orbit, JJ’ the
orbit of Jupiter, and ss’ the orbit of Jupiter’s inner satellite,
Figure 36,
~~
The planets and satellites shine by the reflected light of the sun,
and therefore cast shadows, whose axes are on the right lines join-
ing their centres with the centre of the sun. Because of the posi-
tion of the orbit of the satellite with respect to the plane of Jupiter’s
orbit, the satellite enters Jupiter’s shadow at every revolution, and
is eclipsed. If light traversed space instantaneously, its entrance
into and exit out of the shadow might be noted at the exact in-
stants at which these phenomena occurred, independently of the
relative positions of the earth and Jupiter.
But when Jupiter is near opposition, as at J, the interval be-
tween two, successive disappearances of the satellite in entering, or
between two successive reappearances on emerging from the shadow
is found to be about 42 hr. 30 min. The periodic time of Jupiter
being about 11 yr. 10 mo., he advances but a short distance, as to
J', while the earth moves to E’ near conjunction.
Their distance is now increased by very nearly that of the diam-
eter of the earth’s orbit, and the times of apparent immersion of the
satellite are delayed beyond the computed times by about 16 min.
26 sec. Since the periodic time of the satellite is constant, Roemer
158 ELEMENTS OF WAVE MOTION.
therefore concluded that light required 16 min. 26 sec. to traverse
this diameter.
Tf this diameter were accurately known, and the exact instant
of the eclipse could be noted, a very nearly exact measure for the
velocity of light could be computed. The reduction of more than
a thousand eclipses of Jupiter’s satellites, by Delambre, gave 473.2
mean solar seconds for the time of travel, which corresponds to a
solar parallax of 8.878”, and to a velocity of 298,793 kilometres per
second.
258. By the Aberration of Light. Bradley, in 1728,
accounted for the aberration of the fixed stars by assuming that the
velocity of the earth’s orbital motion had an appreciable ratio to the
velocity of light. By assuming an ideal star at the pole of the
ecliptic, the value of the constant of aberration, according to his
determination, is 20.25", which corresponds to a solar parallax of
8.881". According to W. Struve, this constant should be 20.445”,
decreasing the parallax to 8.797”, and corresponding to a velocity
of 296,067 kilometres per second.
The principle on which this method is based is given in the text
on Astronomy.
259. By Actual Measurement. Owing to the great
velocity of light, it is not possible to measure directly the very small
interval of time required for light to traverse any terrestrial dis-
tance. But Fizeau, Foucault, Wheatstone, Cornu, and more
recently Michelson, have succeeded in obtaining its value within
very near limits. The essential principle of the experiment by
Fizeau consists in causing a toothed wheel to revolve with great,
but uniform velocity, in a plane perpendicular to the track of a
small parallel beam of light. The toothed wheel in its rotation
alternately permits and obstructs the passage of the beam, accord-
ing as an interval or a tooth is interposed in its track. The beam
of light, after traversing the distance determined upon, is reflected
by a small mirror, and may or may not be intercepted on its return,
depending on the ratio of the velocity of rotation of the wheel and
the velocity of light. Should the velocity of rotation be such that
the returning beam passes through the next interval, the circum-
ference of the wheel would have moved through an angle equal to
RELATING TO SOUND AND LIGHT. 159
that subtended by a tooth and an interval, while the light has tra-
versed double the distance from the wheel to the reflector.
When the angular velocity of the wheel is doubled, the light
passes through the second interval, and soon. The value for the
velocity of light determined by this method is 315,364 kilometres.
Cornu has recently made use of the same method, but with a very
much improved apparatus, and has found, as the mean of 504 ex-
periments, the value of 300,400 kilometres for the velocity of light
in vacuo, with a probable error of less than .001.
Figure 37
260. Foucault’s method is a modification of the preceding. Let
A, in Figure 37, be a luminous line, BC a lens whose focal length
for the position A is Bb + 2D, b0c a revolving mirror, D and E
circular mirrors whose centre is at O, MM’ a glass plate, Ra reticle,
and I an eye lens to view the image of A. Now if the mirror O is
at rest, the path of a ray from A, passing through the lens BC and
reflected from O, is ABJD; returning by reflection from D, its path
is DeCA. A part of its light is reflected from the first surface of
MM’, and the image of A is seen coincident with its object at «
If now the mirror O is put into sufficiently rapid rotation, the re-
turning ray meets it at 5'Oc’, and the ray is reflected along cO'A’,
and its image is seen at e’. The angle G00’ is known from the
velocity of rotation, the distance OD is given, and the displacement
aa’ ig measured by a micrometer.
These data serve to measure the velocity of light in terms of the
160 ELEMENTS OF WAVE MOTION.
angular velocity of O. By the addition of a tube filled with water
at FF’, the velocity of light in water was found and shown to be
less than that in air.
In the diagram annexed to the figure, ad is the position of the
image when O is at rest, c’ when O has a determinate velocity, and
a'b' the corresponding position of the image after the ray has tra-
versed the water. The result of this determination is 298,187
kilometres for the velocity of light, corresponding to a solar parallax
of 8.86". Michelson, by an ingenious modification of the method
of Foucault, by which he separated his mirrors 2000 feet, and
caused one of them to revolve 257 times per second, obtained a de-
flection of his image exceeding 133 millimetres, and thus obtained
results which are claimed to be exact to within one ten-thousandth,
due to this element of deflection.
As the mean of 1000 observations, he has determined 299,930
kilometres per second for the velocity of light in vacuo.
A new investigation of this important constant, under the di-
rection of Prof. Newcomb, is now in progress, and which, when
completed, will undoubtedly be as close an approximation to the
true value as the present state of experimental science can furnish.
261. Assuming that light is due to the transversal vibrations of
the luminiferous ether, we see, Eq. (119), that in isotropic media
the velocity of light depends on the coefficients «, 3, ¢, ete., which
are functions of the elasticity and density of the medium.
In homogeneous light, or that in which 4 is constant, V will
therefore vary when light passes from one medium into another.
The conclusions derived by supposing a variation in A, the medium
remaining the same, will be considered under the dispersion of light.
:
St GEOMETRICAL OPTICS.
262. In geometrical optics it is only necessary to take account
of the variation of the velocity due to a change in the elasticity and
density of the ether, in passing from one isotropic medium into
another. Hence, we consider homogeneous light alone in the dis-
cussions which follow. These changes are given by the formula
ce a y ain ¢. (289)
RELATING TO SOUND AND LIGHT. 161
The ratio » is called the index of refraction ; it is the ratio of
the velocity of light propagation in the two media, and is called the
absolute index when the medium from which it passes is the ether.
When any two other velocities are compared, the ratio is called the
relative index; the relative index is then only the ratio of the two
absolute indices. When reflection is considered as a particular case
of refraction, » is always taken as — 1.
263. A radiant is a point from which the rays proceed ; it is
said to be real when the beam is parallel cr diverging, and virtual
when converging. A focus is the point in which the rays meet
after deviation, or in which they would meet if prolonged in either
direction ; in the former case the focus is real, and in the latter, if
the point of meeting is found by prolonging the rays backward, it is
virtual. A radiant and its focus are the centres of curvature of the
undeviated and deviated pencils, respectively. In the following
discussions, distances estimated in the direction of wave propaga-
tion, from any origin whatever, are taken as negative, and in the
contrary direction as positive.
264. Deviation of Light by Plane Surfaces. Let us
suppose the incident medium to be any whatever, as air, and that.
the ray enters any other medium, as glass, whose surface is plane,
Then, Figure 38, we have, for the first refraction,
sin @ = pw sin @¢’, (290)
in which p is the relative index of air referred to glass; and for the
first reflection we have
sin @ = — sin ¢. (291)
The angle ¢’ is less than @, because
— 7 is greater than uuity, since the
,
velocity of wave propagation of light in
air is found by experiment to be greater
than that in glass. Should the velocity
in the medium of intromittance be
greater than that in the medium of in-
cidence, » would be less than unity and
¢' would be greater than # The re- Figure 38.
162 ELEMENTS OF WAVE MOTION,
fracted ray will meet the second bounding surface of the medium,
which we suppose plane, and its direction on emergence into air is
obtained from the equation
sin @, = 7 sin @”’ = ; sin @”. (292)
Tf the second plane surface is parallel to the first, then @, in the
last equation will be equal to ¢' in Eq. (290), and we will have
o” = ; whence we see that the ray is not ultimately deviated in
passing through a medium bounded by parallel plane surfaces, and
emerging into the incident medium.
The refracted ray is, as shown by the figure, not coincident with
the incident ray, but is displaced, parallel to it, by a distance de-
pending upon the thickness of the plate and the angle of incidence ¢.
From Eq. (292) we also see that the refractive index of one medium
referred to another is the reciprocal of the refractive index of the
latter referred to the former.
265, The parallelism of emergent and incident rays is likewise
true when light traverses any number of parallel plates of different
refractive media. Thus, let 4, B, C, be consecutive parallel plates
of different media, and p, pv’, j2"’, their respective absolute refractive
indices; then, in supposing incident light to proceed from and to
emerge finally into a vacuum through these plates, we will have
: aa ; ‘
sind = > sin ¢ =peSsIin®’,
V,
: gaa tame ae ae
sin ¢’ = v, sing” = ae sin @” = : sin 6”,
V
v, L (293)
E Vos Fy ee
sin 6” = T sin ¢/” = ae sin o> i sin 6”,
V
F Veggies 1.
sin d’” = Tv sin d” = 77 sin gs J
RELATING TO SOUND AND LIGHT. 163
whence we have,
sin @ = pv’ sin @” = p" sin 6’ = sin o". (294)
Experiment shows that the relative index for solids and liquids,
referred to air, differs but little from their absolute indices, and in
geometrical optics, these relative indices may be assumed for the
absolute indices, when air is the incident medium.
266. Refraction by Optical Prisms. An optical
prism is a refractive medium bounded by two plane surfaces enclos-
ing a diedral angle; the faces of this
angle form the deviating surface for
luminous rays. In the optical prism
-BAC take the plane of incidence nor-
mal to the axis of the prism.
Let « be the refracting angle of
the prism, ¢, ¢’, and ¢,, %,,, the first
angles of incidence and refraction,
and second angles of incidence and Figure 39.
refraction, respectively; let uw be the
relative index, and 6 the total deviation of the ray after passing
through the prism. Then we have
sin dé = psin ¢’,
gid = lain, ey
Be
g' +4, = 4,
e+, =at+, (296)
J = (¢—¢') + (6, —%)-
é is therefore a function of but one independent variable. Let
this be ’; then we have, from Eqs. (296), by differentiation,
db, 297
agi = 4 oe
dé _ db , db, db, _pwoose’ pcos >, (298)
do! — de’ * dd, dd’ — cos ~ C08,”
164 ELEMENTS OF WAVE MOTION.
which, when placed equal to zero, gives ¢' = ¢, and ¢ = ¢,,.
Taking the second differential coefficients, we have
OO Gg Tey aor
dp? — de?" do? de”
= (rere neon (296)
u cos? @, sin ¢,, — cos? ¢,, sin oy,
= cos? @,, /
Substituting the above values, we get
aS ue (uw cos? ¢' sin @ — cos? ¢ sin 9’)
do? cos? @
, (300)
which is positive when # >1, and negative when » <1. The
former, therefore, corresponds to a minimum and the latter to a
maximum. ‘Therefore we conclude, that when the angle of inci-
dence is equal to the angle of emergence, the deviation is a mini-
mum when the medium of the prism has a refractive index greater
than unity, and the deviation is a maximum when the refractive
index is less than unity.
The former is the more usual case, and will hereafter be as-
sumed.
Substituting this value of @,, in Eq. (296), we have
é
o=- =S andalso ¢’ = 7
This furnishes a simple method of determining experimentally
the relative index of any medium referred to air. Thus, if the
medium be a solid, a prism of it is formed whose refracting angle
and the position of a small parallel beam of homogeneous light at
its minimum deviation are experimentally determined. The values
of « and , substituted in the equation
sin (° t *) = sin a (301)
enables us readily to find p.
If the substance be a liquid, a hollow prism whose sides are
plates of glass, each with parallel faces, is filled with the liquid, and
the same measurements made as in the case of the solid prism.
RELATING TO SOUND AND LIGHT. 165
267. The refractive indices for the same medium vary with the
wave length, since, as shown by Kq. (119), the velocity varies with
this length. Therefore, the color of the ray must be stated for
which the refractive index is taken.
TABLE OF REFRACTIVE INDICES FOR YELLOW RAYS.
Solids. Liquids.
Chromate of Lead, 2.50 to 2.97 Carbon Bisulphide, . 1.678
Diamond, . 2.47 to 2.75 | Olive Oil, .- . - 1.470
Carbonate of Lead, 1.81 to 2.08 Sulphuric Acid, . 1.429
Flint Glass, . 1.57 to 1.58 Hydrochloric Acid, 1.410
Crown Glass, - 1.525 to 1.534 | Alcohol, . : - L372
Plate Glass, . 1.514 to 1.542 | Water, : 1.336
Ice, . 1.310
268. Deviation of Light by Spherical Surfaces.
Let O be any radiant, BAD any spherical reflecting surface, and C
its centre of curvature; then the direction of any ray deviated by
reflection can be readily determined when ¢ is known. Let OI
be any incident ray and If’ be the corresponding reflected ray;
then all incident rays which proceed from the same radiant, and
Figure 40,
make the same angle with the axis OA, will after reflection meet in
f'. As this point of incidence I is taken nearer to A, the corre-
sponding point /’ will move to the right and approach some limit-
ing position on the axis, as f,, and the pencil may be taken so
small as to make the distance /'7, less than any assignable value.
The position f, is called the geometrical focus of the reflected pencil.
We can approximate in the same way to the geometrical focus of a
166 ELEMENTS OF WAVE MOTION,
pencil refracted by a medium bounded by a spherical surface, when
@ and pw are known.
The principal focus of a spherical reflecting or refracting sur-
face is the geometrical focus of a small parallel beam of light,
whose axis coincides
with the axis of the
surface,
Let F be the virtual
focus of the rays which
proceeding from O, Fig.
41, meet the surface at
the angular distance 0
from the axis, after de-
viation by refraction; then we have, in the triangles CIO and CIF,
Figure 41;
sng f—r sinO6 su
sn@” 5 ” sing’ f’—9r’
(302)
in which f = AO, f' = AF, r = AC, wu = FI, and s = OL.
Eliminating 0, and combining the resulting equations with
sing — psin ¢’,
we have po = i (303)
From the triangles we have
v= (f—r)y +r 4+ ar (f—r) cos 6
= f?—2r(f—7) versin 6,
w= (f' —ryP + 7 + 2r (f' — 7) cos 8
= f? —ar(f' — 1) versin 6.
(304)
Substituting « and s from these equations in (303), and re-
ducing, we have
(f —7r) Vf? — 2r (fF — Pn) versin 0 (305)
The relation existing between f and f’ by this equation shows
RELATING TO SOUND AND LIGHT. 167
that for every constant value of f the values of f' will vary with @,
and therefore rays proceeding from a radiant on the axis will not in
general meet the axis, after deviation, in a single point; and for
particular values of 6, every value of f will give a real value for f’.
The two distances f and /', being thus related, are called conju-
gate focal distances.
269. The surface of accurate refraction, which is independent
of 6 for a given radiant O, on the axis, is formed by revolving about
the axis OA a curve such that, for any incident ray OI, OI’, ete,
we have OI + vpIF = Ol’ 4+ wl'F = aconstant. Among the par-
ticular cases, we have
__ 1°. When, if the incident rays are parallel, then the incident
surface must be convex and a portion of a prolate spheroid, whose
eee 1
eccentricity Is ¢@ = 7?
2°. When the incident rays diverge from the further focus of an
hyperboloid of revolution and the eccentricity of the generating
curve is e = p, the deviated rays will then be parallel.
3°. The refraction at a spherical surface is accurate, for a diverg-
ing pencil at a concave surface, and for a converging pencil at a
convex surface, when the radius is a mean proportional between the
conjugate focal distances, estimated from the centre of curvature.
270. If versin @ be zero, we will have
poe | a
Pi B? 06
* PG iper
whence, 7 =" F e + - (307)
an equation from which the distance of the geometrical focus /’,
corresponding to any radiant f, can be found.
For all pencils for which @ is so small as to make its versed-sine
approximately zero, the above equation can be employed to find the
conjugate focal distances.
Such pencils are called small direct pencils.
168 ELEMENTS OF WAVE MOTION.
271. Application of Eq. (305) to Reflection at
Plane Surfaces. Wet BAD be any plane reflecting surface ;
then, in Eq. (305), becomes zero for all positions of the radiant O
on the axis O’AO, » = —1, and Eq. (305) reduces to
f=-f'3 (308)
therefore the conjugate focus is as far behind the reflector as the
radiant is in front, and the vergency of rays is not affected by re-
flection from plane sur-
faces. The image of
any object seen by re-
flection appears to oc- petted
cupy a corresponding Og :
position behind the re- Ses
flector, no matter where =
the eye be placed to view
it; the only change be- :
ing that the apparent Figure 42.
right-hand side of the
image corresponds to the left-hand side of the object.
Hence the image of a printed or written word presents the letters
in an inverse order, from left to right, with the letters also reversed ;
the clear impression of a word on a blotter, when reflected by a
plane mirror will, however, exhibit the letters and word in their
proper form and arrangement. It is also evident that a second
reflection of an object would reverse
the first image and thus make the
second image similar in all respects
Cad
to the appearance of the object. ae
Figure 43 shows the a of a oe
rays apparently proceeding from ee
the virtual image formed by a plane ee a
reflector for two positions of the 99 sy
SA
eye, and itis evident that wherever
the eye be located in front of the
reflector, it will receive a reflected
pencil of the same vergency as if it
were viewing a real object located at the position of the virtual
image,
Figure 43,
RELATING TO SOUND AND LIGHT. 169
272. Multiple Reflection. When a radiant P, Figure 44,
is seen by reflection in two parallel mirrors, the distance of the
image P’ from P,, each being due
to a single reflection, is double the Pao
distance separating the mirrors. =
Each image, P’ and P,, considered 7
as a new radiant, will determine a =
new virtual focus at a distance be-
hind the mirror which it faces equal
to its distance in front, and so on;
therefore the images of the point P
will be theoretically unlimited in
number, and will all be located on the normal to the murrors
through P. The loss of light by reflection will diminish the inten-
sity of each successive image, and ultimately cause the luminosity
of the succeeding ones to be inappreciable.
Figure 44,
273. Multiple Reflection by Inclined Mirrors. Let
OM and OM’, Figure 45, be two plane mirrors inclined to each
other at the angle 7, and P be any luminous point in the included
angle. With O as a centre, describe the
circumference MPM’; then the two
series of images of P formed by the mir-
rors M and M’ will be P’, P”, P’”, etc,
and P,, P,, P,,,, ete. Regarding P as
an image, the number of images formed
will depend on the angle of the mirrors,
the number increasing as the angle 7 di-
minishes. If the angle 7 be an aliquot
part of 180°, the number is limited San Pk
wherever P be located in the arc MPM’. Figure 45.
If 7 be 60°, 90°, or 120°, the number of
images will be six, four, and three, respectively. When MM’ is
contained an entire number of times » in the circumference, there
will be # + 1 images of P, counting P as one image, except when
n is even, or when 7” is uneven and P is at the middle of MM’. In
the latter exception, two images coincide and are considered as but
one. The kaleidoscope is based on this principle of multiple re-
flection.
170 ELEMENTS OF WAVE MOTION,
274. Angular Velocity of the Reflected Ray. When
a ray is incident on a plane mirror, and the latter is turned about
an axis in its plane, the angular displacement of the ray is double
that of the mirror; for the incident ray being constant in direction
if the angular displacement of the mirror is «, the normal to the
mirror has also turned through the angle «, and the angle of inci-
dence has been either increased or diminished by the same angle;
and since the angles of reflection and incidence are equal, the angle
of reflection has been increased or diminished by the same value ;
therefore the angle that the reflected ray makes
with the incident ray has been augmented or di-
minished by 2«, due to the change of direction of
the mirror «,
275. When a ray is deviated by reflection by
two mirrors inclined at an angle 7 to each other,
the angle included between the original direction
and the direction after the second reflection is
double the angle included between the mirrors.
Thus, in Figure 46, let @ and ¢’ be the first and
second angles of incidence and « the required an-
gle; then we have
20 = 2¢' + 2, and or
v= 2 — 29' = 21.
Figure 46,
(309)
The sextant is constructed on this principle ; M is the index-
glass, and M’ the horizon-glass, and the angle 2 is read off on the
circular arc, the units of which, for this purpose, are marked double
their true value.
276. Deviation of Small Direct Pencils by Spher-
(cal Surfaces. Resuming Eq. (307), and limiting the discus-
sion to small direct pencils deviated by any number of spherical
deviating surfaces, we have
1 p—l *¢ 1
fae aE
If now we regard /’ as a new radiant, its conjugate focus f”,
after a second refraction by another medium bounded bya spherical
surface, will be given by
RELATING TO SOUND AND LIGHT. 171
1 "1
7 oe tay oo
in which f’, f”, are the first and second focal distances from the
origin, 7’ the radius of curvature of the second spherical surface,
d the distance of the second surface from the first, measured on the
axis of the pencil, and wv’ the relative index of refraction of the sec-
ond medium. In the same way, for any number of media, we will
have for the focal distance,
1 | 1
fun = pry + pe (f” + dnt (311)
By the substitution of these values in succession, the final focal
distance can be found. In ordinary cases, the thickness d can be
neglected in comparison with the radiant distance f, and hence the
equation can be written
1 pr — 1 1
fra = pny + pr fr (312)
If the pencil emerge from the first refracting medium into the
incident medium, p’ in Eq. (310) will be equal to 7 and we will
have
1 [pb p—1. 1
an = +e oa
S z on (313)
=o 08-244 |
Considering this focus f” a new radiant, its conjugate focus,
f", after deviation by another medium, bounded by two other
spherical surfaces, whose radii are r” and r'”, will be given by
p= w" (4-4) + -—DE-7) +3 (314)
and so on.
172 ELEMENTS OF WAVE MOTION.
277. Representing principal focal distances by /, properly ac-
‘ 1
cented, we will have, since f = 0 and F = 0,
1 p—l
F, oe ur 2 (315)
1 ae!
q=uHD (: —>), (316)
1 ' 1 1 —
A=W (msm) +e —0 (5-5)
; (317)
fe igs
By BY
and, generally,
1 1). a 1 1
Se EE a . oe Sa
Pr, Rt yt Rp te ee)
Substituting z in Eq. (313) for its value, Eq. (316), we have
2
Si cafliy Zt
f" & ft’
an equation suitable for the discussion of the deviation of a small
direct pencil of light, by refraction by a medium whose bounding
surfaces are spherical, and whose thickness can be neglected.
This equation being of the first degree, and containing two varia-
bles, f and f", there are an infinite number of sets of values of the
conjugate focal distances which will satisfy it. It is usual to con-
sider the radiant distance f as the independent variable, and the
focal distance f” as the function.
(319)
278. A lens is any medium bounded by curved surfaces, used to
deviate light by refraction. The lenses ordinarily used have spher-
ical surfaces, because of the readiness with which these surfaces can
be ground and polished. Spherical lenses are either concave or
COMED,
The convex lenses are three in number, viz., the double conver,
the plano-convex, and concavo-convex or meniscus, and are designated
RELATING TO SOUND AND LIGHT. 173
in Figure 47 as 1, 2, and 3, respectively. There are also three con-
cave lenses, viz., the double concave, the plano-concave, and the
convexo-concave, designated as 4, 5, and 6 in the figure.
1 ( ( 4 5 6
Convex Concave
Figure 47,
279. The principal focal distance of a spherical lens being
given by
1 1 1
R= e-D(--7)
we see that it is constant for the same medium and the same curv-
ature. Taking » > 1, as is the case when light enters from air
into the several media of which the usual lenses are composed, and
remembering the rule with respect to the sign of measured dis-
tances, we see that, from any origin,
z= o-0G-3)
is negative for all convex, and positive for all concave spherical
lenses. The numerical value of /, will, however, depend on the
particular values of », 7, and 7’.
280. Discussion of the Properties of a Lens. By
this is meant the deterinination of the focus corresponding to any
position of the radiant taken on the axis of the lens.
1°. Convex lens. The equation is
1 1 1
ee ea oo 320
fee Ff, + f (3 )
By assuming values of f equal to +o, +2, #,, 0, and
—o, we have, from the solution of Eq. (320), for the correspond-
ing values of /”,
—Ff,, — 2F,, —O, 0, and ee oe
174 ELEMENTS OF WAVE MOTION.
and since the conjugate focal distances are connected by the law of
continuity through Eq. (320), we see that as the radiant moves
along the axis
from +x to + 2#;, the focus will move from — F, to —2F,;
ay +2F,to Fy, 6“ ce 6c —2F,to —x;
* t F, tothe lens, “ ee “ .t eo to the lens;
“* thelensto—«, es “the lens to — F..
Again, since positive values of f correspond to real radiants and
negative values to virtual radiants, and negative values of f” to
real foci and positive values to virtual foci, the above results may
be grouped in the diagram of Figure 48.
Incident Light.
Virtual Radiants, Real Radiants,
¢ SiC 5 We >
ie give 1
between these limits, virtual foci. \ give real foci.
give real foci located between <\ between. 5 aan
O anda Fy: eo and O between-Fz,and-0
ss =O, =F ) +h; 42F, +0)
Figure 48,
2°. Concave lens. Its equation is
1 1 1
df a Of
The special values of the radiant distances necessary to be con-
sidered are
+o, 0, — Ff, — 2F,, —,
for which the corresponding focal distances are
+f, 0, —o,; + 2h, + #3
Incident Light. aes
1
Virtual Radiants, Real Radiants;
‘
'
'
between these limits,
real “¥~
foct. give virtual foci located between
between +F, and O
4
‘
1
|
I
1
1
: i :
a) -2F, -F, \ + Fp +2 Fy +00
Figure 49,
give virtual foet.
between+oand +F,
RELATING TO SOUND AND LIGHT. 175
and in the diagram of Figure 49 for this lens, it will be noticed
that there is a complete analogy, if the words virtwal and real be
interchanged in the convex and concave lenses.
281. Relative Velocities of the Radiant and Fo-
cus. Differentiating the equation for lenses,
iL 1 1
portent -P
and dividing both members by dt, we have
df" 1 df 1. a
dé f™ = dp FP (322)
or replacing the differential coefficients by their equals V" and T,
we have
2 ee
oe
Since the ratio of the velocities is positive, whatever be the
values of f and f”, we see that the motion of the conjugate foci is
in the same direction, and that their velocities are directly propor-
tiqnal to the squares of their distances from the lens.
(323)
V/ 282. Discussion of Spherical Reflectors. Eq. (307)
gives the circumstances of the deviation of light at a single surface,
and which in case of reflection becomes, since » = — 1,
1 2 1
a f (324)
and since 7 is positive, the reflector is concave. Making f= o,
we have, for the principal focal distance, /, = a3 therefore, the
equation for a concave reflector is
1 1 1
ah Fae ae 825
PR” fF )
For a convex reflector, r being negative, the equation takes the
form of
es
ft
S| %~
(326)
SIH
176 ELEMENTS OF WAVE MOTION.
1 1 1 ;
or pe aay ee
In reflectors, when the origin is taken at the reflector, it is
easily seen that real radiants and real foci have the positive sign,
and virtual radiants and virtual foci have the negative sign.
Making the discussion as in lenses, we readily obtain the follow-
ing conclusions, viz. :
Concave Reflector. Beginning with a real radiant at + 0,
the focus will be real and at /,; as the radiant moves toward the
reflector, the focus moves in the opposite direction, these points
passing each other at the centre of curvature; real radiants from
this point to + /, correspond to real foci, located between the
centre and + 0; as the radiant approaches the reflector, the focus.
changes from real to virtual, and approaches the reflector from
—o, both radiant and focus meeting at the reflector; finally, the
radiant being virtual and proceéding to —o, gives real foci be-
tween the reflector and + F,.
These relations are shown in the diagram of Figure 50.
Incident Light.
ee
Virtual Radiants, Real Radiants,
¢ SIC 2 ~
give real foci. give give real foct.
“a Of virtual |F, C ral
foci.
Figure 50.
Convex Reflector. A similar discussion of the equation
ee ee
eo
will show that all real radiants from + o to the reflector give vir-
tual foci, located between — F, and the reflector; that all virtual
radiants between the reflector and — F, give real foci, located be-
tween the reflector and + ; and that all virtual radiants between
— F, and — ~ give virtual foci between — oo and — F,. These
results are grouped in Figure 51.
RELATING TO SOUND AND LIGHT. 177
Incident Light.
KK
Virtual Radiants, Real Radiants,
1
dE
. '
give virtual foct. real give virtwal foci.
Ft
Fy
~o c
Figure Sl.
283. Differentiating the equation for reflectors,
1 Ti hook
Fi =+ Rf’
dividing by d#, and replacing the differential coefficients by the ve-
locities, we have
(328)
ee ae
poe
from which we see that the velocities of the conjugate foci are di-
rectly proportional to the squares of their distances from the
reflector; and that the motion of these points is in opposite
directions.
284, From the preceding discussions it is evident that the con-
vex lens and concave reflector increase the convergency of incident
light, and therefore either change incident diverging pencils into
converging ones, or make them less diverging; also, that the con-
cave lens and convex reflector increase the divergency and diminish
the convergency of incident light. Therefore, the office of tiie con-
vex lens and concave reflector is to collect rays, and that of the
concave lens and convex reflector to separate them.
(329)
285. Power of a Lens. The expressions
: and z
f pe
may be taken to measure the vergency of the incident and deviated
pencils, respectively ; for, since the pencils are small and direct, f
and f” may be considered as the radii of the same arc, subtended
by the incident and refracted pencils at the distance of the lens
from these points ; hence their reciprocals will have the same ratio
as the angles of the pencils themselves. By the power of a lens is
12
178 ELEMENTS OF WAVE MOTION.
meant its capacity to deviate that ray of the small pencil which
meets the lens at the unit’s distance from the axis, and since
1 1 1
et 336
Fi F ae (336)
we see that the vergency after, diminished by that before deviation,
and which is evidently the vergency increment, is for the same lens
‘constant and equal to the reciprocal of the principal focal distance.
Hence, to ascertain the power of a lens, we have the following rule:
Draw that ray of the pencil which meets the lens at the unit’s dis-
tance from the axis, and find the corresponding conjugate focus of
the radiant; the angle between the incident ray prolonged and the
refracted ray is the power of the lens. Or, find the principal focal
distance of the lens, and take its reciprocal. Since the deviation of
each ray of the pencil varies with its inclination to the axis, it is
better to find the devia-
tion of the ray at a 2 il a
unit’s distance, al- = ~~~ /\
though any other ray
would answer, not too wl ; f
near the axis; thus the ry
axial ray is undeviated, Figure.32.
the next is but slightly
deviated, and so on to the extreme ray, which is deviated the most.
It is also evident that two lenses, constructed of the same material
and with faces having the same curvature, are of the same power.
Thus, the two lenses, A and B, Figure 52, will bring the divergent
pencils P and P’ to the same focus f”.
286. To find the Principal Focal Distance of a
Lens. Since we have
a
it 1s evident that the power of the lens depends upon the refractive
index and the curvature of its faces. When these are known, the
power, and hence the principal focal distance, can be readily found.
RELATING TO SOUND AND LIGHT. 179
1 ;
Thus, the value of Fr for a meniscus crown glass lens whose
2
radii are 6 and 3 feet, respectively, and refractive index 1.525, is
1 1 1
625 (5 —5) = — ap
therefore the principal focal distance is negative, and 11.424 ft.
from the lens. Practically the principal focal distance of a negative
lens can be found by measuring the distance of the focus of a beam
of sunlight from the lens, since these rays may be taken as parallel
before deviation, and the radiant as at an infinite distance; or, since
when the radiant is at a distance of + 2/, the conjugate focus is
formed at a distance of —2F,, these positions can be found exper-
imentally, and one-fourth of their distance apart will be the value
of F#,. The principal focal distance of positive lenses can be found
by associating them with negative lenses of greater power, finding
the principal focal distance of the combination by either of the
above methods, and subtracting from the power of the combination
the power of the negative lens; the remainder will be the power of
the positive lens, and its reciprocal the principal focal distance.
This method depends on the equation
1 1
m= F (832)
287. For reflectors, we have
FtF +R!
whence, the sum of the vergencies before and after deviation is
equal to the power of the reflector. The principal focal distance of
a spherical reflector being equal to half its radius of curvature,
there is no difficulty in finding its power. When the reflector is
plane, z ‘becomes zero, and we have, as in Art. 271,
1
ji=-f
Hence, plane reflectors have no power to change the verge of
the incident pencil.
180 ELEMENTS OF WAVE MOTION.
288. Deviation of Oblique Pencils by Reflection
or Refraction at Spherical Surfaces, When the axis
of the pencil is not coincident with the axis of the deviating surface,
the pencil is said to be oblique. When the axis of the oblique pen-
cil meets the surface at a point with reference to which the area
covered by the pencil is symmetrical, the incidence is called centri-
cal, and when otherwise, excentrical. We have seen, Art. 268, that
the rays from a single radiant on the axis of the deviating surface
are not in general brought to a single focus, but that by Eq. (805)
we can find the particular direction of any deviated ray. The prin-
ciples enunciated in Art. 85 show that the consecutive deviated rays
will be tangent to some surface, which is the locus of their consec-
utive intersections, and which may therefore be regarded as their
envelope. This surface is called the caustic surface. The general
problem of caustic surfaces, as outlined in Art. 85, is one of con-
siderable intricacy, and it is sufficient to consider only those of
deviating surfaces of revolution, since such surfaces are usually
employed in optical instruments. A surface of revolution has in
general two generatrices ; one being the meridian curve, and the
other a circle in a normal plane to the meridian curve.
If we consider any indefinitely small area of the deviating sur-
face, we see that the deviated rays will form two systems of develop-
able surfaces at right angles to each other, when the surface is
conceived to be generated by the curves above referred to, Each
deviated ray will belong at the same time to both systems, and the
consecutive intersections of the rays in a plane of each system will
form a curve of regression, which will be the caustic curve cut out
of the caustic surface by the plane in question. To illustrate, let
HAK, Figure 53, be a portion of a spherical deviating surface,
whose axis CQ contains the radiant Q. We may regard the inci-
dent pencil as being made up of a series of conical surfaces of rays,
whose angular spread 6 on the surface varies from the extreme value
HOC by continuity to zero. Each cone of rays will have a single
focus, such ag 7, g, 8, since, by Eq. (305), f’ is a function of
(f, +, 9) only, and the consecutive foci of these cones will be spread
over the axis from that point corresponding to the extreme cone to
the geometrical focus of the small direct pencil. The intersections
of the consecutive deviated cones will form consecutive curves of
the caustic surface, whose planes are normal to the axis of the de-
RELATING TO SOUND AND LIGHT. 181
viating surface. Consider now the annular area of the deviating
surface formed by revolving the indefinitely small area HAK around
the axis CQ. The corresponding annulus of the caustic surface
will be at the limit the are of a circle, and when HK is very small
it may be taken as such, the diameter of which in the figure is qf.
If we consider the small beam of light incident on the area HAK
alone, it may be taken as a small oblique pencil centrically incident,
whose axis after deviation will be Ag.g,. Every deviated ray will
Figure 53.
now pass through a point of the small circular arc at g,, which
may be regarded as coincident with its tangent and normal to the
plane QCA. This line and the point g, are called the primary
focal line and the primary focus, because the points of the line are
the first intersections of the consecutive deviated rays. Again, con-
sidering the second generatrix H’AK’ of the area, every deviated
ray will pass through the corresponding part of the intersection of
the caustic surface made by the plane H’K’g, at the point g., and
this intersection and point g, are called the secondary focal line
and secondary focus of the oblique pencil. The secondary focus is
then the intersection of the axis of the oblique pencil and the axis
of the deviating surface. The secondary focal line, although like
an elongated figure 8, may be taken approximately as a right line.
The plane containing the axis of the incident pencil and the axis of
the deviated pencil, or the plane QCA, is called the primary plane ;
and the plane containing Ag,g,, normal to the primary plane, is
the secondary plane. Hence, we may conclude that a small oblique
182 ELEMENTS OF WAVE MOTION.
pencil, after deviation, converges to or diverges from two right lines
called focal lines, the primary and secondary focal lines being nor-
mal to the primary and secondary planes, respectively. When the
deviation is by reflection at a single surface the focal lines are real,
and by refraction, when « > 1, virtual. That property of a de-
viating surface by which the rays of a pencil are caused to pass
through the two focal lines after deviation is called astigmatisin.
289. Circle of Least Confusion. If the deviated pencil
be cut by a plane parallel to the tangent plane to the surface at A, we
see that at g, the section will be a right line, normal to the primary
plane ; and as the cutting plane is moved towards g,, the breadth
of the section increases in the primary plane and decreases in the
secondary, until at g, the section becomes a right line in the pri-
mary plane. The section, which at some point between ¢, and q is
nearly circular, is called the etrcle of least confusion, and may be
taken as the approximate focus of the oblique pencil.
_ 280. The Positions of the Foci. In Figure 54, let Q be
the radiant from which the small oblique pencil, whose axis is QA,
is incident centrically upon the refracting surface at A; O the cen-
tre of the surface, and the
relative index. The inter-
section g, of the ray through
O with QA, after deviation,
is the secondary focus; “4,
where any ray of the primary
plane, after deviation, inter- |.
sects the axis Ag, of the de- Figure $4,
viated pencil, is the primary
focus. Draw Cs and Cr perpendicular to AQ and Aq,, respectively,
and let K and T indicate the angles made by the intersections of
OC and AQ, and OC and Agq,, respectively ; let AQ=f, Ai =fy
Ag = f,, AO =r, QOA = 6, o and ¢ + do the angles of inci-
dence, and ¢' and 9’ + dq’ the angles of refraction of QA and QC,
respectively. Then we have
AC cos
oi 333
COA = % nearly =
RELATING TO SOUND AND LIGHT. 183
Og = AC cos @
1
; (334)
dod = QCO — QAO
= angle K — CQA — (angle K — AOC)
_ _ AC AC cos
do’ = q,CO — g,AO
= angle T — Cq,A — (angle T — AOC) j (336)
(335)
= AOC — 0g,A = AL _ AC ose
ji
Differentiating sin @ = p» sin ¢’, we have
cos @ dp = p cos @' dd’. (837)
Substituting the values of d@ and d¢', we have
1 cos¢ _ {1 cos@' j
(;- =") cos ¢ = (= 7 ) w cos ?', (338)
Le cos? ¢’ __ cos? cos o' — cos 339
hi f ie r 2 ( )
from which the distance of the primary focus f, from A can be
found.
For the secondary focus we have
r AO _ sin (¢’ + 6)
= ‘4. sin ¢' cot 6 340
tr Ag sin @ ee (ee)
r _ AO _ sin (@+ 9)
f° AQ” sind
=cos¢@?+sing cot @, (341)
- Eliminating cot 6, and reducing by sin ¢ = p sin $', we have
; } lt (342)
1s4 ELEMENTS OF WAVE MOTION,
For refraction at plane surfaces, 7 = «, and the second mem-
bers of Equations (339) and (842) reduce to zero. For reflection,
p = —1, and these equations become
1 1 2
i + fF ~ reos@? (848)
1 1 _ 2cos¢ (344)
iF 8
291. The position of the circle of least confusion and the value
of its diameter can readily be found from the properties of similar
triangles; the resulting values will be given in terms of the length
and breadth of the beam on the deviating surface, as determined
by the intersections of the primary and secondary planes, and in
terms of the focal distances measured on the axis of the deviated
pencil from the surface. The limiting position for the centre of
the circle is midway between f, and f,, at which point it may be
assumed for all pencils of small obliquity.
eo t Figure 55.
292, Particular Cases of Caustic Curves. Let trt,
Figure 55, be a concave spherical reflector, and O any radiant on
its axis ; O/ and Ov’ any two consecutive rays, and m their point of
intersection after reflection; then, from Eq. (343), we have at once,
by representing ia by 4c,
1 1
of ees 345
pe oo (345)
and if the reflector be convex, f, and ¢ are negative, and we have
1 1 1
RELATING TO SOUND AND LIGHT. 185
The caustic curves may, from any assumed position of the
radiant, be constructed by points, and when revolved about the
axis they will generate the caustic sur-
face for the spherical reflector. For : Pee
S RP. Ne SON
parallel rays, f is o and fi=e In f A I ss
: . cd . ‘ * Lee has. §
this case, Figure 56, the curve is an epi- ae EN
a * . . ee SN x /
cycloid generated by a point whose ini- , {7 vi Nic V4. 1
. eye . * . ‘ t
tial position is v, in the circumference of Ne ANd ce }
the small circle Av, with a radius equal Naber | f
fy
r : é Af
to 7 rolling upon the circumference vp, oe
bo : : :
whose radius is 5, being the radius of Figure 56.
the spherical surface. To show this, let m be any point of the
caustic; then —
- r
wm = 5 cos o = fy
Again, the arc pm is equal to the arc pv, since the angles pim and
pev are equal. The virtual caustic arising from reflection on the
convex surface indicated by the dotted curve is symmetrical with
and equal to the real caustic. As the
radiant moves from o along the axis,
the cusp v approaches the centre OC, the
virtual caustic diminishes until the ra-
diant reaches B, when it is zero, and
the real caustic springs from B in both
directions to v. o this case, the radius
of both circles is 7 ‘as shown in Fig-
ure 57%; for we have
Figure 57.
Ti Sse
ff arcos
when the radiant is at B, which substituted in Eq. (343) gives
1 2 1 3
Fares Weose Wweosg?
(347)
a , cos @. J
186 ELEMENTS OF WAVE MOTION.
As the radiant moves within the spherical surface, the cusp
approaches the centre, uniting with the radiant at this point, and
the caustic reduces to zero. For radiants on the axis within the
surface, the caustic has a point of regression on each side of the
axis near the reflector, from which asymptotic branches proceed,
and a virtual caustic of two branches is formed exterior to the re-
flector on the convex side. Figures 58, 59, and 60 represent these
{ ax on KE | \
ee, OS 7 ey
Figure 58. Figure 59. Figure 60.
f
/
¥
‘
‘
i
1
4
\
At
A
caustics for positions of the radiant within the spherical surface at
distances from the centre > 5? = a and < 5 respectively. All
these, as well as other special cases, can be obtained from the gen-
eral principles enunciated in Art. 86.
293. Critical Angle. From the equation
sind = p sin ¢’, .
since is greater than unity when the incident ray is in air, sin @!
is always less than unity for all angles of incidence, and therefore
o < 90°. Then at all angles of incidence, when light passes from
a rarer to a denser medium, there will be refraction; but when the
light passes from a denser to a rarer medium, there can be no re-
fraction when the angle of incidence surpasses the value
d 1
¢' = sin“! -.
22
Rays making greater angles of incidence than this are totally re-
flected. This angle is called the limiting or critical angle of inci-
dence for refraction. Its value for water is
1
sin-1 1.336 = 48° a7! 40” ;
RELATING TO SOUND AND LIGHT. 187
for crown glass about 40° 30’, and for chromate of lead about
19° 28’ 20". Therefore, to an eye placed in water, Figure 61, all
external objects would appear crowded within a conical space NA,
of about 48° 28’ on either side of the normal to the surface.
Figure 61 Figure 62.
The remaining angle of 41° 32’ would be filled with rays, pro-
ceeding from objects in the water, which have been totally reflected
at the surface.
Let us now consider rays refracted from a denser into a rarer
medium. Let f, Figure 62, be the radiant, in water, for exam-
ple, and consider any ray whose angle of incidence is less than
48° 28’, incident at I. Draw fAs normal to the surface, and make
As = Af. Describe a circle through f, I, and s, and construct the
lines as in the figure. We have, geometrically,
Os Wns
OF xf"
Ea
and gS ae
sf _ Of _ sinfrl _ sing _
rs+rf rf sinfOr” sing’
°
?
rs + rf = e = a constant.
The locus of r is therefore an ellipse, whose foci are s and /;
rI ig normal to the ellipse at every point 7, since it bisects the angle
included between the foci.
The caustic curve is therefore the evolute of the ellipse whose
. 18
foci are s and f, and transverse axis e
15 ELEMENTS OF WAVE MOTION.
LH
By constructing the caustic for any radiant, as in Figure 63, we
see that the rays emerging into a rarer medium are tangent to the
caustic curve, and hence the
radiant Q in the denser me-
dium would appear to an
eye in the rarer medium \\| i,
lifted up to Q’, nearer to ce
the separating surfaces, and ah Hi Wiy/
along the successive tan- VS yy
\ 4)
gents to the caustic curve, \
depending on the position \\\ y
of theeye. If PQ, Fig. 64, i
be any object, as a straight Figure 63
rod, parallel to the surface
of water, points of the image P’Q’ may be constructed as in the
figure, showing that, to an eye in the air, the right line appears
curved, and that the portion directly under the place of the eye is
apparently deeper in the water than the more remote portions.
at
Figure 64.
The horizontal lower surface below clear water presents, for the
same reason, a basin-shaped appearance.
/ 294. Spherical Aberration. In Figure 65, let F, be the
principal focus of the spherical reflector whose centre is CO, and z
the point in which the extreme ray cuts the axis after deviation;
then, since the triangle CIz is isosceles, the perpendicular from x
to CI will bisect the radius at s. The triangles TIA and zsF, are
RELATING TO SOUND AND LIGHT. 189
similar, and hence F,# is equal to $TA, or approximately to $AN.
a and therefore the distance of the
focus « from the limiting position F, varies as the square of the
semi-apertausey 4 as
twice the r of the sur- '
face inversely.
This distance is called the
longitudinal, and Fz the lat- i
eral spherical aberration. 2
These departures of large /
pencils after deviation from a
single focus F, is due to the
form of the deviating surfaces
alone, and hence is called
spherical aberration.
As the incident ray ap-
proaches the axis, the focus Figure 65.
approaches F,, asaty. The
point , where that one of the intermediate rays which intersects
the extreme ray at the furthest distance from the axis AC, limits
the extent of the circular area through which all the deviated rays
pass. This circle is called the least circle of aberration. ‘To find
the value of its radius, let «, «’, represent the arcs AI and AI’,
and @ the aberration F,2. hen, if the ray Iz be invariable and I'y
vary, mn will vary directly with wm, and we have only to find when
zm will be a maximum. i
But AN is nearly equal to a
Zlsaasasoasesee eS
The aberration a and xy give
of — ¢'2
ay = a-—s (848)
and we have, from the figure,
a
mn = xm z- (849)
my = mnt = zm! . =: (350)
Taking Ay = Aa, we have approximately,
190 ELEMENTS OF WAVE MOTION.
ry = am + am a =e + “am. (351)
a
Equating these values of «y, we have
2g
e < = il + © om,
a (352)
or am =u (a—«'),
a
a
‘ a é a .
which is a maximum for « = 3? and for this value we have
a AI Fe _ Fz .
am =a and =omn = i a (353)
Hence, the least circle of aberration is at three-fourths of Fz
from F,, and its radius is one-fourth the lateral aberration.
The longitudinal aberration being from Hq. (353) we see
iB a
that the lateral aberration will be approximately on
295. In the discussion for lenses, we have omitted all considera-
tion of the thickness, and limited the question to the deviation of
small pencils, and hence the results are only approximately true.
The more complete investigation shows that the aberration not only
depends on the thickness of the lens, but upon the particular sur-
face which is first presented to incident light in the same lens. On
this account it is sometimes usual, in naming lenses, to designate
the first incident surface; thus, plano-concave indicates that light is
incident on the plane surface, while concavo-plane is the same lens
with light incident on the concave surface. This designation has
not been followed in the text, because our limits have not permit-
ted more than a passing allusion to the subject of aberration. When
in a single lens, or in a combination of lenses, the aberration is de-
stroyed, the lens or combination is said to be aplanatic. In the
former case, the lens is usually a meniscus of a particular form for
a beam proceeding from a determinate point as a radiant; in the
latter, the aberration of one lens counteracts or destroys the aberra-
tion of the others. The ocular of Huyghen’s, referred to hereafter,
RELATING TO SOUND AND LIGHT. 191
is an example of an approximate aplanatic combination. Surfaces
such as the ellipsoid, paruboloid, and hyperboloid are of course free
from spherical aberration, so that the rays of incident pencils, hav-
ing their radiants at certain points, meet after deviation accurately
in another point. From the geometrical properties of these sur-
faces, and the laws of refraction and reflection, the circumstances of
the deviation of light can readily be determined. The difficulty of
accurately forming these surfaces gives them only a theoretical
interest, and the discussion of their properties is for this reason
generally omitted. In grinding and polishing the surfaces of lenses
and reflectors, it is not always possible for the practical optician to
attain the best mathematical form of the surfaces, nor is it always
desirable, since the lack of complete homogeneity assumed in theory
is rarely attained in practice. A great part of the ultimate value of
a lens depends on the skill of the optician in removing the defects
which crop out in the course of preparation, and for which no
merely theoretical discussion can provide.
296. Optical Centre. The point in which the line joining
the extremities of two parallel radii intersects the axis of a lens is
called its optical centre. LetOAA’,
Figure 66, be the axis of a lens, O
and O’ the centres of the first and
second surfaces, I and I’ the ex-
tremities of any two parallel radii.
Join I and I', and let C be the
point in which II’ meets the axis;
this will be the optical centre. To
find its distance from A, let r and.
r be the radii of the surfaces, x the
distance AC, and ¢ the thickness of the lens. By similar triangles,
we have
co 2
r—2
rf —t—2
rt
t= .
rf —P
CO’ r
r
or =753
7’?
(354)
A similar result can be obtained from Figure 67, and we see that
the distance of the centre from the lens depends only on the radii
/
192 ELEMENTS OF WAVE MOTION.
of its surfaces and the thickness of the lens, and that it is independ-
ent of the refractive index and therefore of the material of which
the lens is constructed. By
substituting the values of 7, 7’,
and ¢, in the value of 2, and
paying attention to their signs,
we readily find: ms
1°. That, if the curvatures
of the surfaces are in opposite
directions, the centre lies within
the lens.
2°. If one surface be plane, the centre is on the curved surface.
3°. If the curvatures are in the same direction, the centre is
outside of the lens, and on the side of the greater curvature.
4°, If the thickness ¢ be negligible, the centre may be assumed
at any point of the lens on the axis.
The optical centre of a spherical reflecting surface is by defini-
tion at the centre of curvature of the surface. The equation of the
small pencil, when the focal distances are measured from this point
as an origin, can be obtained readily from Eq. (307), and is
1 2 1
ao ea 7? (855)
for concave and convex reflectors.
When a ray is refracted through a lens in such a manner that
its direction between the two refractions passes through the optical
centre, the ultimate direction of the ray is parallel to its primitive
direction; for it is in the same condition as if it had passed through
a portion of the medium bounded by parallel planes ; and when the
lens is thin and the obliquity small, the lateral displacement is neg-
ligible. Therefore, if we draw a ray from any radiant through the
optical centre of a lens or reflector, we may consider it as the axis
of a small oblique pencil on which the other rays of the pencil met
by the lens or reflector are brought to a focus. This property is
made use of in constructing the images of objects formed by a lens
or reflector.
Figure 67
_ 29%. Focal Centres of a Lens. The limiting positions
on the axis of the lens, where the directions of the incident and
RELATING TO SOUND AND LIGHT. 193
emergent ray cut the axis, when the direction of the ray while pass-
ing through the lens contains the optical centre, are called the
focal centres of the lens. Supposing, in Figure 66, the incident
ray through I’ be prolonged and its point of intersection with the
axis be designated by m, and that the point where the emergent ray
cuts the axis be designated by 7; then passing to the limit, that is,
when these rays coincide with the axis, Eq. (307) is applicanle, and
we have
pb 1 p—1
Mae = oe? (ae
RO- wa =F (857
also, AC = a as ~ (858)
and by combination,
Am = GF SPLI TA er (359)
An = es (560)
ate —e—h te
hence, these distances depend on the refractive index, thickness
and radii of curvature of the lens.
298. The Eye. The human eye, Figure 68, is very nearly
spherical in form, the front part being slightly protuberant. The
globe of the eye consists of the sclerotic coat, known generally as
the white of the eye, which covers all but a small part of the ante-
rior surface, and the cornea, a thin transparent membrane covering
the front, through which the iris and pupil are seen. The iris
gives color to the eye, and is a membranous plate crossing the axis
of the eye transversely from the junction of the sclerotic coat and
‘he cornea. The dark circular aperture in the iris or the pupil
permits the passage of light, and is capable of involuntary adjust-
ment, becoming larger or smaller as the intensity of light is little
or great.
The interior of the eye behind the iris is lined with two thin
membranes. That next the sclerotic coat is called the choroi/, and
has its cells filled with a dark pigment, giving to ‘t a brownish
13
104 ELEMENTS OF WAVE MOTION.
lack velvety appearance. The iris is a part of the anterior portion
-of the choroid coat, drawn together like a curtain, and is thickened
by muscular tissue. The other part of the choroid coat surrounds
the outer edge of the crystalline lens in a series of folds, about
seventy in number, and is called the
ciliary processes.
The membrane lying within the
choroid coat, and resting upon it, is ¢p- 4
called the retina. It is really a
nerve tissue completely lining the
interior of the eye, as far forward as
the ciliary processes, where it forms
an irregular border. Near the point
where the axis of the eye meets the
retina is the yellow spot, which hasa
diameter of about 3 of an inch.
One-sixth of an inch from this point
towards the nose is the centre of the Cj, conjunctiva; Co, cornea; Ag,
optic nerve; this portion of the adueons humor; I, iris; Om, ciliary
a fi muscle; Cp, ciliary processes ; Z, zon-
retina where the optic nerve enters — we of Zinn; L, crystalline lens; V, vit-
is called the punctum cecum, not res ies cig ac ae
being sensitive to light. sclerotic; O, optic nerve.
By making two dark spots on a
piece of paper a few inches apart, and directing the axis of either
eye to the nearer spot, the other eye being closed, the outer spot
will disappear when the paper is at the proper distance from the
eye; the rays from this spot then meet the retina at the punctum
cecum of the open eye. A displacement of the paper in either
direction brings the spot again into view.
The retina consists of nine layers of nerve fibres, cells, tissues,
and blood vessels, forming a very complex structure.
The layer nearest the centre of the eye, called the daciliary
layer, consists of numerous elongated elements, arranged side by
side, normal to the surface of the retina. Some of these elements
‘are cylindrical, called rods, and others are conical and shorter than
the rods. At the yellow spot there is a slight depression, and the
rods are replaced by cones, closer in arrangement, longer but more
slender in form than the other cones; this part of the eye is by far
the most sensitive to light.
Figure 68.
SECTION OF THE EYE.
RELATING TO SOUND AND LIGHT. 195
The humors of the eye are three in number. Their object is to
refract. the light, and bring it to a focus on the retina. The first
in order, beginning at the front of the eye, is the aqueous, which
is a watery fluid, having a little sodium chloride in solution.
It occupies the space between the cornea and the front of the
rystalline lens. The latter is contained in a membranous double-
convex capsule, the greater curvature being to the rear; its thick-
ness is about one-third its transverse diameter. The crystalline
lens consists of several nearly concentric layers of varying density,
decreasing from the centre outwards; the transparency of this lens
is destroyed by the disease known as cataract. The vitreous humor
fills the larger volume of the eye, from the crystalline lens in front
to the retina; it is clearer than the most transparent glass, and its
consistency is like that of jelly.
The eye, as an optical instrument, is similar in its function to
the camera obscura. The sclerotic coat forms the wall of the cham-
ber; the choroid coat, the dark lining for the absorption of surplus
light ; the cornea and humors, the system of lenses and deviating
surfaces for the formation of the image; the pupil, the aperture in
the diaphragm ; and the retina takes the place of the sensitive plate
on which the image of the external object is thrown.
299. Vision. By vision is meant the sensation of sight aris-
ing from the action on the retina of the molecular encrgy of the
rays of light. Since the image of any point, or the focus corre-
sponding to any radiant, is on the opposite side of the axis of the
eye from the radiant, it is evident that the sensation is referred by
the mind along the ray in the direction of the radiant. We there-
fore do not look at the image on the retina, which is in reality in-
verted, but at the object which is considered erect, although the
inverted image is formed on the retina, and can be seen under
proper conditions by an observer.
The range of distinct vision for the normal eye is from five
inches to infinity. The normal eye is capable of so adjusting the
lenses which compose it as to make a perfect image on the retina
for any position of the object between these limits. The near-
sighted or myopic eye brings parallel rays to a focus, in front of the
retina, and, according to the magnitude of this defect, the range of
-the eye may vary from any distance less than infinity'to a distance
196 ELEMENTS OF WAVE MOTION.
less than five inches, say from four feet to four inches. The lenses
of the eye in this case possess greater refractive power than those of
the normal eye. The defect may be artificially remedied by the use
of positive or concave eye-glasses. The far-sighted or presbyopic
eye differs from the normal eye in a lack of power of adjustment
for near objects. It brings parallel rays to a focus on the retina
equally well with the normal eye, and fails alone in doing so for the
near objects, whose images would be formed behind the retina.
This defect can be remedied by the use of convex lenses, to assist
the lenses of the eye in refracting the rays when near objects are to
be viewed.
When the eye has a curvature of the cornea, or of its crystalline
lens, which differs in the horizontal and vertical planes, the radiants
will be brought to a focus on the focal lines instead of to a point,
and this defect is called astigmatism. The vptical centre of the eye
may be taken in the plane of the iris at the centre of the pupil, for
all practical purposes relating to the discussion of optical images.
300. Optical Images. An optical image is an assemblage
of foci, which are conjugate to the consecutive radiants on the sur-
face of the ohject whose image is to be formed. The optical image
may, or may not, have a general resemblance to the object. When
the deviating surfaces are irregular in form, these images are unlike,
and when regular, the image is a picture more or less similar to the
object in appearance. Every ray proceeding from a radiant carries
with it an image of the radiant, and if a small pencil of these rays
be deviated by media bounded by regularly curved surfaces, the
focus will present a visible image of the radiant.
For the ordinary optical instruments, it is sufficient to consider
only small direct or oblique pencils, and to construct the foci of the
salient points of the object, considered as radiants, and thus get the
salient points of the image.
301. In Figure 69, let the radiant PQ bea right line perpendic-
ular to the axis of the lens, whose optical centre is at O. The only
rays proceeding from P which are met by the lens are those of the
cone /Pl’. The axial ray of this cone, taken as a small oblique pen-
cil, passing through O is undeviated, and the focus of all rays of
this pencil may, without material error, be taken on this line. To
find the focus, draw the extreme ray P/; its deviation by the lens is
RELATING TO SOUND AND LIGHT. 197
represented by the general symbol a in which @ is the distance
2
from the axis to the point in which the ray meets the lens, and
is the principal focal distance, expressed in the same unit of meas-
ure. This represents the power of the lens to deviate any ray, and
its value for any ray depends therefore on the angle the ray makes
with the axial ray before deviation, the curvature of the surfaces
P
Figure 69.
and index of refraction. The intersection of the rays jp and PpO
is the image of P, and all rays from P passing through the lens,
produced backward meet at y. This is then the image of P, and
similarly g is the image of Q, and pg of PQ. Let 0 be the angle
made by PO with the axis of the lens; then taking O as an origin,
we have
0Q j
eS cos 6 cos 0”
in which / represents the distance of the point of the object on the
axis of the lens from the optical centre. Substituting for f its
value in
1 1 1
fo an (861)
F,
we have, finally, f'=— Fr (362)
F, ’
1 = cos 0
v3
in which f” is the distance of the focus, measured on the axis of
the pencil, and 7, the principal focal distance of the lens. This
equation is similar in form to the polar equation of a conic section,
and therefore f”, the distance of the focus of each radiant, meas-
‘ured on the axis of each pencil of rays, is a radius-vector of a conic
section, referred to the optical centre as a pole. The excentricity of
198 ELEMENTS OF WAVE MOTION.
the conic section depends on the sign of —, and its value com-
pared with unity. It is, however, sufficiently exact for the practi-
cal purposes of illustration to regard the image of PQ similar in all
respects to PQ.
802. The equation for discussing optical images for lenses will
then be
foe, (363)
in which /, is negative for convex, and positive for concave lenses,
By transferring the origin for reflectors to the centre of curva-
ture, Eq. (307) readily becomes
1 22
and the corresponding equation for images for reflectors becomes
fata. (365)
1 + = cos 6
f
in which /, is negative for concave, and positive for convex re-
flectors.
303. It is necessary now simply to determine whether the image
is real or virtual, erect or inverted, and larger or smailer than the
object.
1°. Image Real or Virtwal. Since 9, in the usual cases where
this discussion ‘applies, differs but little from 0°, cos 6 is positive
and nearly unity, and since in lenses positive values of f correspond
to real objects, and negative values of f to virtual objects, and posi-
tive values of /” to virtual, and negative values of f” to real
images, we at once draw the following conclusions, viz.,
With a concave lens or a convex reflector:
All real objects give virtual images.
All virtual objects between the lens and — F, give real
images.
All other virtual objects give virtual images.
RELATING TO SOUND AND LIGHT. 199
With a convex lens or a concave reflector:
All real objects beyond + F, give real images.
All real objects within + /, give virtual images.
All virtual objects give real images.
2°. Image Erect or Inverted. If the image and object are on
the same side of the optical centre, the image will be erect; if on op-
posite sides, inverted, since the axes of all the pencils pass through
the optical centre and the object is always considered erect. But,
in the former case, f and f” cos 6 will have like signs, and in the
latter contrary signs; hence, if the ratio of the distances of the
object and image, measured on the axis of the lens, is positive, the
image will be erect, and if negative, it will be inverted.
8°. Relative Size of the Image and Object. The linear di-
mensions of the image and object are the distances of the conjugate
foci of the extreme points from the axis of the lens, and these dis-
tances are directly proportional to their distances from the optical
centre, measured on the axis of the lens. The ratio of f and
jf cos 6 will therefore determine the relative size of the image and
object.
304. Concave Lens. The equation is
fee’,
—. (366)
1+ £ cos 8
f
1°. Real objects.
/ is positive, and hence f” is positive, but f" < f:
the image is virtual, because f” is + 5
; ‘ F : :
the image is erect, because FT cos 6 is +3
the image is smaller, because f” cos 6 < f.
2°, Virtual objects.
(a.) Between the lens and — F,, f is negative and f" > f:
the image is real, because f” is — 5
‘he i is erect, because —,2—— is + 5
the image is erect, =F cae +5
the image is larger, because f” cos 0 > f.
200 ELEMENTS OF WAVE MOTION.
(2.) Between — F, and —o, f is negative, f” is positive
and until near —2F,, f" > f, after which
f" f;
the image is smaller, when within 2/,, because
f' cos6 J Aste, uw = 2f""
e — fo Ee (370)
A= 0, < a ?
that « will vary, and its corresponding values will be
, ¢
@€ Se; Sa, = 2a, > 2a, =O, a
The images in these cases are real, and being on the opposite
side of the lens from the object, will appear inverted. Finally,
when the object is at the principal focal distance, the emergent rays
will be parallel, and the visual angle will be independent of the
position of the eye, and the same as if it coincided with the optical
centre of the lens.
307. Optical Instruments. These embrace the various
combinations of reflecting and refracting surfaces which form
images of objects, in such positions and of such dimensions as are
needed for special observations. Dispersion, which generally ac-
companies refraction, is not considered in the following discussion ;
its influence in modifying the construction of particular instru-
ments will be referred to subsequently.
308. The Camera Lucida enables an observer to trace
readily a reduced image of an external object. It consists of a
quadrangular prism of glass MKNH, Figure 70, whose diedral
angles at K and H are 90° and 135°. respectively. The rays pro-
ceeding from an external object A are deviated by the positive lens
Z, making @ the virtual image. from which rays enter the prism
202 ELEMENTS OF WAVE MOTION.
nearly normal to KN; they are then reflected at NH and HM, to
enter the pupil of the eye at P with direct rays from a’; the image
may be traced by the pencil
of the observer at a”, since
the rays from the pencil
point and image have the
same vergeney when they
enter the eye. The con-
struction and course of the
rays are shown in the figure.
309. The Camera
Obscura, Figure 71, is
used to forma real image
of an external object on a
screen or upon a sensitive
photographic plate, placed
at the back of a darkened
chamber. Since the image
is real, the lens is negative, and the relative distances of the object
and image from the optical centre are given hy Eq. (320). The
magic lantern is essentially the same in principle. The object in
this case being 2 transparent picture, painted or photographed on
Figure 70.
Figure 71.
glass, and strongly illuminated by artificial light, which is concen-
trated on the object by a condensing lens, and the picture is placed
near the principal focus of the projecting lens, in order that the
image may be as large as the illumination will permit.
310. The Solar Microscope, Figure 2, the same in prin-
ciple as the magic lantern, consists essentially of a plane mirror
MM’, to reflect the solar rays upon a condensing lens OC, by means
RELATING TO SOUND AND LIGHT, 2038
e
of which the transparent object PQ is strongly illuminated, and of
a short-focus lens L to form an image of PQ. Since the solar light
is intense, the Image of PQ can be made very large in comparison
with it, and yet be very bright. The object PQ is placed bevond
Figure 72,
the focus F, of the lens C. to diminish the injurious effect of the
heat rays which are collected in or near the foeus F,, and alse to
interseet the diverging cone of rays. so that all the object will be
illuminated.
811. Microscopes. A microscope is an optical instrument
used to form a magnified image of a very minute and near object,
and to view the image. If the eve possessed the property of dis-
tinet vision at all distances. a clear and magnified image would be
formed by it as the object approached the optical centre of the eve.
But whai is gained in magnitude after the object passes the inferior
limit of distinct vision is more than lost by the indistinemess aris-
ing from the inability of the lenses of the eve to bring the rays toa
focus on the retina. The simple micrascane is a convex lens. whase
principal foeal distance is less than the inferior limit of distinet
vision. Such a lens, whose principal focal distance is five inches,
would be a migroscope for an eye Whose inferior limit is six inches.
but would not be for one whose limit is four inches. When the
object, as in Figure 73. is placed nearer the lens than F,. the image
is virtual. erect, and larger than the object. When the object is
placed at F,. the emerging rays of each pencil are parallel to the
undeviated ray. and the apparent size of the image is Just the same
as if the optical centre of the eye were placed at the optical centre
of the lens
2OF ELEMENTS OF WAVE MOTION.
312. The Magnifying Power is the ratio of the apparent
diameter of the image to that of the object, the object and image
being supposed properly placed for distinct vision.
But since objects viewed by the microscope can be placed at any
convenient distance from the eye, it is usual to compare their
apparent diameters with those of their images when they are at the
inferior limit of distinct vision, because their details can best be
observed at this location. If we suppose the object to be so placed
that the image formed by the simple microscope is at the limit of
distinct vision from the eye, then the direct ratio of the diameters
of the image and object, since they are compared at the same dis-
Figure 73. me
tance, will be the measure of the maynifying power of the micro-
scope. Representing the magnifying power by M, we have then,
from Figure 73,
OX’ = ie
M=oy => (371)
Letting @ represent the limit of distinct vision, and A the dis-
tance of the eye from the Jens, we have
f" = d—aA,
1 il a
and ey eg
poe ff,’
d—a
whence, Ma=1+ a (372)
When the object is at the principal focus, the image appears of
°
RELATING TO SOUND AND LIGHT. 205
the same size to the eye as if the latter were at the optical centre of
the lens, and we have
1
rats x (373)
2
The same value results when 1 is taken so small compared with
d as to be negligible.
When F’ is so small compared with d that unity may be omitted
with respect to = we have, approximately,
2
d P
1 x (874)
For a particular observer, the utility of the microscope is deter-
mined by the greatest visual angle under which it enables him to
see clearly any definite length, whatever may be the angle under
which he sees it without the lens.
Thus, if the millimetre be the unit of length, the visual angle
without the microscope would be - and with it would be = :
therefore, the advantage would be, from Eq. (372),
A
La
en + = (375)
da FF ‘
or approximately as before, calling P the useful power of the mi-
croscope,
Pe ake 378)
era 2 i
which shows that a myopic eye has an advantage over a presbyopic
or normal eye, unless 7; is so small as to make this difference
negligible.
313. The Field of View of the microscope is the angular
space embraced by the eye within which spherical aberration does
not appreciably affect the clearness of vision; it is experimentally
found to be within 9° or 10° around the principal axis of the mi-
croscope. But although the magnifying power of the microscope
206 ELEMENTS OF WAVE MOTION.
increases when the radii of curvature of the surfaces diminish, the
spherical aberration increases also.
To diminish this effect, the field of view must be correspond-
ingly contracted, by means of a diaphragm which stops out the
marginal rays.
314. Oculars. Simple microscopes formed of two lenses are
generally used to view images formed by optical instruments, and
of these it is sufficient to describe two, viz., the positive, in which
the image is formed in front of the lenses of the ocular, and the
negative, in which it is formed between the lenses.
ssn ye
a Ber ee >_>
_-—— ee op
wet wee ger
oes
ee Zoe- yet
--257 OP ge
A
ae ee
4
ee, I
i
I
|
|
See I
Maa as
~ aha
Ss,
ES
i =. lo’
Figure 74 PPE
1°. Positive Ocular. Let the two plano-convex lenses L and
L’ (Figure 74) be placed with their convex surfaces turned towards
each other, and let PQ be the object to be viewed. The image of
PQ formed by Lis pg, and the image of pq formed by L’ is p’y’, as
is readily seen from the figure.
We have from L, with the origin at O,
i Ae
FOF AF
and from L’, with O' as an origin, representing the distance OO’ by
D, and the limit of distinct vision by d,
(377)
1 1 1
PPR (378)
RELATING TO SOUND AND LIGHT. 207
The magnifying power of the ocular is
PY vd. wy aa
Ey ag Sg Hage ee
but from Eqs. (378) and (379) we have, since f,"" = d
aa
DF" = ae Fy’
AR
ig D
Pe oeg ge
and —
2 2 2
_ fl _D
= 1 1-D a ae approx. |
= + F, a Fy = PF,’ P] ie J
When F, = #' and D = 2F,, the ocular is called Ramsden’s,
and we have for the magnifying power,
re a (381)
and when /, is small compared to d, equal to
4 d
Maar (382)
2°. The negative or Huyghen’s ocular consists of two convexo-
plane lenses. The images formed by the two lenses L and Li’ of
this ocular are shown in Figure 75 ; thus the converging pencil of
rays would form the object PQ, were they not met by the lens L;
but being deviated by it, they form the real image pq, which the
lens L’ magnifies into p’q’.
A discussion similar to that for the positive ocular gives for the
magnifying power,
M= (1+ r(! + Fe AGLP A a (383)
208 ELEMENTS OF WAVE MOTION.
Figure 75
identical in all respects. Making 7,’ = 47; and D= 3h, we
have
6d + F,
6d ,
M= a 3F? approximately. (384)
315. The Compound Microscope consists essentially of
an objective O, which is a simple microscope formed of one or sev-
cral lenses, whose purpose is to form a real, magnified, and inverted
image of a minute object, and an ocular E, also a simple micro-
scope, to magnify and view this image. Figure 76 indicates the
ae
rete pg
Figure 76 —
size and location of the image corresponding to the object PQ. If
the distance of the ocular from the objective can be varied, the
ocular can be placed always in such a position that the image formed
by it will be at the inferior limit of distinct vision from the eye,
RELATING TO SOUND AND LIGHT. 209
wherever the image of the objective may be. In this case, however,
the magnifying power, which depends on the ratio of p'g’ to PQ,
will vary with the position of the ocular with respect to the
objective.
Therefore, in order to keep a constant magnifying power, this
distance should be invariable, and the tube containing them may be
moved towards or from the object. The magnifying power is evi-
dently ao os
= eV a 8 85
M= 5G = “ny * PO (385)
The first factor is the magnifying power of the ocular, and the
second is that of the objective.
316. Telescopes. A telescope is an optical instrument com-
posed of an object-glass or reflector to form a real image of a real
and distant object, and of an ocular to magnify and view the image.
Therefore, the object-glass must be essentially convex, if the tele-
scope is a refractor, and if a reflector, the object mirror must be
concave ; the ocular may be either concave or convex.
1°. The refracting telescopes are the astronomical, the Galilean,
and the terrestrial.
317. The Astronomical Refracting Telescope, Fig-
ure 77, consists essentially of a convex lens L, called the field lens
Figure 77
or objective, and a convex lens L’, called the ocular or eye lens. The
incident pencil of rays from the upper point of the object is brought
by the objective to a focus at p, and the rays of the pencil from the
lower point are united at g; pg is therefore a real image whose
14
210 ELEMENTS OF WAVE MOTION.
linear magnitude is to that of the object directly as their distances
from O. From pa pencil of rays proceeds which is met by the eye
lens L’, and so deviated in the figure as to have the virtual focus of
pat p'; p' is determined by drawing the ray through O' and pro-
‘ducing it backward; it is the axial ray of the small pencil and
passes through O’ undeviated ; any other ray from p will be deviated,
and when produced backward will meet pO’ in p', which is the vir-
tual image of py. The image of any point, as q, is found in the
same manner, and thus p’qg’ is determined. If the lens L’ were
drawn out until the image pq occupied its principal focus, the pen-
cils from pg after deviation would be composed of parallel rays, each
being parallel to the axial ray of its small pencil. For the normal
or myopic eye, O'X is always slightly less than the principal focal
distance of the ocular, but for an infinitely presbyopic eye it is
equal to it. In determining the magnifying power of telescopes,
the distance O'X is usually taken equal to #,', or the principal
focal distance of the ocular. When the object is so far off that the
rays of each pencil may be taken as parallel, the image pq is formed
in the principal focus of the objective, and the length of the tele-
scope is then equal to #,+ 4’. The objective is usually formed of
two lenses, the outer one a double convex and the inner a double
concave. The adjacent surfaces of the two lenses have exactly equal
and opposite curvatures, and generally are in actual contact, and
the curvature of the inner surface of the concave lens is much less
than that of the outer surface. The forms and refractive powers of
the lenses are such as to satisfy, as far as possible, the conditions of
correcting both spherical aberration and color.
318. The principal oculars used in the astronomical telescope
are Huyghen’s or the negative, and Ramsden’s or the positive ocu-
lar. The former is called the achromatic eye-piece, because the
correction for color is nearly fulfilled. It is suitable for examining
the physical features of celestial objects, and is not well adapted for
taking observations which require measures of time or space, since
the image is formed between its lenses. For such observations, the
observer must be able to see distinctly fine threads or spider-lines
stretched across a stop fixed in the telescope-tube, in coincidence
with the optical image of the celestial object. This condition is
fulfilled in Ramsden’s ocular, since the image formed by the object-
RELATING TO SOUND AND LIGHT. 211
glass is situated close to the inner lens of the ocular, on the side
towards the object-glass, as shown in Figure 74.
319. If lines be drawn from the opposite extremities of the
object and eye-lens, as shown in the figure, the pencils of light
oe from points of the image included between these lines
aréSjacident on the eye-lens. Hence these points will appear of
equal illumination, while the points exterior to these limits will
exhibit decreasing brightness. This exterior ring is known as the
ragged edge of the field of view. A diaphragm is inserted in the
eye-tube to cut off the partial pencils and leave the whole pencils
alone, to pass to the eye-lens. This area appears uniformly bright,
and is called the field of view. The diaphragm is placed in the
plane of the image, and to it are attached the wires or spider-lines
referred to above. In telescopes such as the transit, when it is de-
sirable to have a more extended field of view, the eye-piece has a
motion perpendicular to the axis of the instrument, so that the
whole field in this case may be considerably extended laterally, in
order to enable the observer to notice the passage of a celestial body
over several wires.
820. The Galilean Telescope, Figure 78, commonly
known as the opera or reconnoitring glass, has a convex lens for an
Figure 78
objective L, and a concave lens E for the eye-lens. The image pg
of a distant object PQ would be formed by the objective as in the
preceding telescope, were not the pencils intercepted by the eye-lens
E; pq is then a virtual object for E, and its image is determined as
indicated by the construction lines in the figure. The lens E is
212 ELEMENTS OF WAVE MOTION.
usually taken to be at its principal focal distance from pg, and when
the incident rays on L are parallel, the length of the telescope is
equal to #, — F;’. This instrument is not adapted to use in astro-
nomical measurements. The image of PQ, as seen in the telescope,
is erect.
wf?
[/ 321. The Terrestrial Telescope, Figure 79, is nothing.
but the astronomical telescope, L, E, with the addition of two con-
vex lenses /, /', called the erecting-piece. The first image pq is
formed at the principal focus of 7, the pencils of which after devia-
tion are parallel; these parallel beams are brought to a focus by
i’ at its principal focal distance, and form the second image p’q’.
The eye-lens E deviates the pencils proceeding from p'g’, making
them appear to come from the virtual image pg”. The construc-
Pp
Figure 79 (392)
and for astronomical telescopes,
wa 4,
é
(893)
328, In all telescopes except the Galilean, the size of the pen-
cils is determined by the aperture of the object-glass or spherical
mirror, and A will be the value of its diameter. If e be greater
than the diameter of the pupil of the eye, some of the light of each
pencil is lost, and the effect is the same for the same magnifying
power as if the diameter of the object-glass had been decreased.
The inferior limit of the magnifying power with a given aperture A
is therefore =, e' being the diameter of the pupil. If the diameter
of the pencil e entering the eye is less than that of the pupil, the
image loses distinctness from loss of light, and to remedy this it
would be necessary to increase the diameter of the object-glass.
The quantity ¢ is practically the diameter of the image of the
object-glass formed by the eye-glass at the position of the eye. If,
therefore, the telescope be adjusted to distinct vision, with the eye-
piece attached whose magnifying power is to be determined, and
the diameter of this image be meastired, the quotient obtained by
dividing the aperture of the object-glass by this diameter will be the
218 ELEMENTS OF WAVE MOTION,
magnifying power of the telescope, The instrument designed to
make this measurement is called the Dynameter.
329. Brightness of Images. Let B be the brightness of
an object, S its surface, and d its distance. Then the quantities of
light falling on the unit area, on the pupil of the eye, and on the
object-lens or mirror, are
S
@
Bap 5, and Bras
B Ee
respectively, p and @ being the radii of the pupil and aperture.
After passing the ocular, supposing no light lost by absorption, and
the radius of the ocular ring to be 7, the quantity of light received
2
by the pupil will be Brat. 4 which is less than the quantity
2
of light passing through the ocular, in the proportion of f. On
the other hand, the apparent surface of the object is increased in
2
‘. a which may be placed
then the light received by the pupil may be expressed
2
proportion to ) and hence becomes
Ul
equal to a
a
“i and the brightness B is unchanged. If, however, the
ocular ring is smaller than p?, the whole quantity of light Bra? z
a
enters the eye. This expression may be written . e
Br (S@\ (2 ae
=(E2) (80) = =) Ss
2
whence we see that the brightness is now - less than before in
as Bry?
2
the proportion of a Therefore, when the ocular ring becomes
P
less than the area of the pupil, the brightness decreases and the .
image loses distinctness. This does not apply to a star, because,
having no disk, it cannot be magnified, and therefore the larger the
\aperture of the telescope, the greater the brightness of the image.
To study the details of objects having appreciable disks, less power-
ful oculars should be used than those used for stars. In the best
RELATING TO SOUND AND LIGHT. 219
achromatic telescopes, but 85% of the light is transmitted, and hence
objects appear brighter to the naked eye than through the telescope.
330. Magnifying Power of Telescopes. Applying the
general expression, Eq. (390), for the magnifying power, we have,
for the
Astronomical refractor, = = ~F (394)
Galilean refractor, +4 = a ; (895)
Terrestrial refractor,
Th ee th Ai
RR TRA tH oF
When Ji: = fs, and in which fy = fA,
Herschelian, +4 =— a (39%
Newtonian, +4 =— a (398)
ie ee Be
Gregorian, —f 6 ae (399)
Cassegrainian, th tho _ Ff, (400)
+A -A KA
From these results we see that the Galilean, terrestrial, and
Gregorian give images erect, and the remainder give images inverted,
as indicated by the essential sign of the expression for the power.
PHYSICAL OPTICS.
331. Heretofore, we have considered the luminous ray as a geo-
metrical right line, and have discussed its changes of direction in
reflection and refraction without regarding its physical properties.
We now proceed to explain, by the principles of the undulatory
theory, the nature of the luminous ray as exhibited by the various
220 ELEMENTS OF WAVE MOTION.
phenomena arising from the action of rays on the organ of vision
and on each other, and in their deviation by different media.
332. Color. When a small parallel beam of sunlight is inter-
cepted by a prism, a separation of its elements is effected, and each
ray of the white light is found not to be homogeneous, but to con-
sist of an innumerable number of other rays, each of which has the
property of so affecting the retina as to produce in our minds the
sensation of color. And generally all natural or artificial light is
similarly but variously compounded. When a small beam of homo-
geneous light, as red, for example, falls on a prism placed in its
position of minimum deviation, the refractive index for this color
and particular prism is readily obtained from Eq. (801),
. (d+ta
sin ( =) ait
sin 3
t=
Should the beam consist of two kinds of homogeneous light, 6
and hence p will differ for each. In the solar beam, the deviation
of the constituent homogeneous elements varies by continuity from
the red to the violet, and therefore each simple ray will have a par-
ticular refractive index. When the beam is intercepted by a screen
after deviation, a colored image called the solar spectrum, whose
width is uniform for a given position of the screen, is pictured upon
it. Its length, however, depends on the refracting angle of the
prism, and on the material of which it is made, as well as on the
position of the screen. The colors pass by insensible transition
from red to violet, in the order of
Red, Orange, Yellow, Green, Blue, Indigo, and Violet,
red being deviated towards the base of the prism the least, and
violet the most. Each colored ray has a particular refrangibility,
and is simple, for it suffers no farther decomposition when deviated
again by another prism.
333, This spectrum is, in reality, made up of the overlapping
colored images of the aperture through which the solar beam passes,
but its colors are impure, since the rays which enter the eye from
any part between its extremities proceed from two or more of the
RELATING TO SOUND AND LIGHT. 221
colored images. To obtain a pure spectrum, the light is admitted
through a narrow aperture or slit, about 3 or 4 mm. in width, with
parallel edges, and allowed to fall on a prism placed in the position
of minimum deviation, having its refracting edge parallel to the
slit. To project this spectrum, an achromatic lens of considerable
focal length is placed behind the prism, at double its principal focal
distance from the slit. The screen is placed normal to the mean
deviated ray, at double the principal focal distance of the lens, and
the spectrum will then be rectangular in shape, sharply defined,
and will consist of the contiguous colored images of the aperture,
arranged according to their refrangibility.
334, Dispersion. This property, by which refractive media
separate white light into its homogeneous elements, is called dis-
persion. All refractive media do not possess this property in the
same degree, as is readily shown by using prisms of the same re-
fracting angle, but of different material. The coefficient of disper-
sion is the difference between the refractive indices of the extreme
violet and extreme red ray, or pw, —p,. The dispersive power of a
medium is measured by the ratio of the excess of the index of re-
fraction of the mean ray over unity to the coefficient of dispersion,
or
Hy — be
ae (402)
To definitely fix the value of the dispersive power, pu,, ,, and 1,
are taken to be the refractive indices of rays coincident with certain
characteristic lines of the spectrum, C, EH, and G, and which there-
fore now fix the wave lengths of the particular red, green, and
violet, considered in this measure.
335. If the aperture be made sufficiently narrow, the resulting
solar spectrum will be crossed transversely with numerous dark
lines, indicating the absence or partial obscuration of certain colored
rays of particular refrangibility. Frauenhofer, who in the early
part of this century noted nearly 600, and determined the position
of 354 of them, called them the fixed lines of the solar spectrum ;
but they are now generally known as Frauenhofer’s lines. He se-
lected eight of the principal lines as standard means of reference,
and designated them by the letters A, B, C, D, E, F, G, and H.
os) ELEMENTS OF WAVE MOTION.
Their position, with that of two intermediate lines a and 8, are
shown in the normal spectrum, Figure 83. Owing to their fixity
of position and sharpness of delineation, they may readily be ob-
served, and the refrangibility of any ray can be determined by their
assistance with the greatest accuracy.
AaB Cc oD Eo F G H
|
| |
De K
Red Orange Yellow Green Blue Indigo Violet
Figure 83.
336. As has been before stated, the dispersion produced by
yarious media is different, and therefore the widths of the color
spaces and distances separating the fixed lines would not be invaria-
ble in comparing spectra produced by different prisms. But in the
normal spectrum produced by diffraction, the position of the lines
and colors depends on the corresponding wave lengths, and hence
affords a reliable and fixed standard of reference.
337. The index of refraction », being equal to the ratio of the
velocities of the ray before and after refraction, and since it differs
for each color, we will now see what conclusion can be drawn from
the expression for the velocity of wave propagation in any medium,
as given by Eq. (119), of Art. 95,
b d
Viszatatatstete (403)
In this, a, 3, ¢, etc., were shown to be constants depending on
the constitution and nature of the medium; therefore, for a partic-
war medium, V must vary with 4. This, together with the facts
of dispersion, indicates that 4 differs for each ray of homogeneous
light and is the basis of the explanation offered by the undulatory
theory for the unequal refrangibility of the elements of white light.
If white light is not dispersed by any medium, we must conclude
either that the wave lengths of rays of different colors are equal, or
that they are transmitted through this medium with equal veloci-
ties; and in the latter case attribute to such a medium a constitu-
tion compatible with the observed facts, by properly modifying the
RELATING TO SOUND AND LIGHT. 293
hypotheses which influenced the deductions based upon them.
Many independent experiments and observations completely estab-
lish the fact of a variation in the wave length due to a variation in
color, and therefore the supposition of a constancy of wave length
must be abandoned. The most careful observations on the light of
the fixed stars and other heavenly bodies have failed to indicate the
slightest dispersive power of the intervening media which transmit
their light to the earth, and therefore we are forced to conclude that
the velocity of transmission is the same for all rays in the inter-
planetary ether. The inference drawn from this constancy of V is
that the ether molecules of space are separated from each other by
distances which are practically insignificant, compared to the lengths
of the shortest light waves. The analogous supposition made with
respect to the wave lengths of sound and the distance apart of the
adjacent air-particles, in Art. 183, resulted in the theoretical deduc-
tion of the constancy of V for all sound waves of greater or less
wave length in air, which deduction is found practically confirmed
by observation.
338. Again, since the absolute refractive index of air is only
1.000294, and those of other gases are not greatly in excess of this,
we commit no sensible error in assuming the dispersive power of
air and other gases to be negligible compared with ordinary solid
and liquid media, and can therefore practically assume the relative
refractive indices of these media, referred to air as their absolute
indices, making the correction in particular cases, when necessary.
339. If two prisms of different material and same refracting
angle be used to form the solar spectrum, the widths of spaces occu-
pied by the same color, and the distances apart of the characteristic
lines, will not be the same in each, even if the spectra be of the
same lenoth. This is called ¢rrationality of dispersion. Assuming
that the velocity of wave propagation in air is the same for all
luminous rays, and remembering that the time during which the
motion is propagated over the wave length is the periodic time, then
either the length of the wave undergoes a change when homo-
geneous light enters a dispersive medium, or the number of vibra-
tions per second corresponding to this color varies. Certain
observations indicate that the number of waves by which a particu-
lar color impression is made on the retina is invariable, and there-
DOF ELEMENTS OF WAVE MOTION.
fore we conclude that the wave lengths change on their entrance
into such a medium. The wave length of any color for any me-
dium will then be found by dividing the wave length in air by the
refractive index of this color for that medium. The method of
finding the wave length in air will be described in the subject of
diffraction, and when once determined, together with the velocity
of light in air or vacuo, we can find the number of vibrations cor-
responding to any color of the normal spectrum, which therefore
becomes the constant for this color.
340. Color may therefore be defined to be the sensation aris-
ing in the mind, due to the effect of a certain number cf waves of
light affecting the retina per second of time. We have in sound the
analogous property of pitch, and it is a matter of common observa-
tion that the pitch of a sound due to a definite number of vibrations
per second may vary through a considerable range, as heard by the
ea”, rising and suddenly falling, as the observer rapidly approaches
and recedes from the sounding body. So, likewise, it is theoreti-
cally possible to have a color rise and fall in the scale, provided a
sufficient change be made in the number of light vibrations which
enter the eye ina given interval, although the rays from a luminous
source may themselves have but a single wave length or periodicity.
The application of this physical fact is of use in the spectroscopic
study of the solar prominences.
341. Beyond the extremities of the luminous spectrum, uléra-
violet rays of greater, and ultra-red rays of less refrangibility exist,
which are however not visible. The existence of the former is
made known by their actinic or chemical effect, and of the latter by
their calorific action. They are essentially of the same nature as
the Iuminous rays, differing only in their refrangibility, and neces-
sarily therefore in their wave lengths. Observation shows in reality
that each ray of the visible spectrum possesses heat, light, and ac-
tinic properties, differing only in degree, and if we consider the
spectrum extended to include the ultra rays, the heating effect is a
maximum in the ultra red, the luminosity in the yellow, and the
actinic in the ultra violet. These properties are more or less modi-
fied by the absorptive powers of the media through which the
propagations pass.
RELATING TO SOUND AND LIGHT. 225
342, Absorption. The undulation conveys kinetic energy
from the luminous source to the ether of other matter. Each ra-
diation possesses certain properties, exhibited in the complex effects
of heat, light, and chemical action, and these properties are found
to vary with A. Since the intensity of the radiation depends om
mv’, all the effects which the radiation produces must vary with this
energy, and we will see that these effects are also proportional to
the periodicity of vibration. In sound, for example, the intensity
varies with mv*, and the property of pitch, which is appreciated by
sensation, depends on 2. Now a careful examination of each radia-
tion throughout the visible and invisible normal spectrum discloses
the fact that each radiation has one or more of the properties of
heat, light, or chemical action, depending on its wave length or
periodicity ; and since these effects are different in their nature, we
have no means of referring them to a common standard, and by
subsequent combination get at the true measure of the energy
proper to each radiation. If we refer them to sensation only, the
last is inappreciable ; if we disregard sensation, the second is omit--
ted; if we consider mechanical work only, since heat has its equiv--
alent in work, it would appear that this would be the true measure:
for the intensity of the radiation. A careful study of the normal
spectrum enables us to ascertain how much of each or all of these:
properties belong to each radiation.
O7 Os Oz AaB D E F G H Lt M NO T
Figure 84,
343, Let us suppose that the ordinates of the curve OGa, Fig-
ure 84, represent the relative intensities of the heat in each radia-
tion, whose abscissa marks its place in the spectrum, the curve
O'deH the relative intensity of the light, and the curve ghT the
15
226 ELEMENTS Of WAVE MOTION.
relative intensity of the chemical action. Then each radiation has,
as we see, a complex effect, and the individual effects of light, heat,
and chemical action of a radiation are absolutely inseparable. ‘They
cannot be dissociated by prismatic refraction, since the ray has but
one value for its refractive index, and it obeys the law of Snell or
Descartes; nor by absorption, for, as will be shown, all absorbing
media act alike on each. Hence, in all cases, each property of any
given ray is found with its same specific value as those of the other
two which accompany it in the spectrum. The philosophical inter-
pretation of this fact is that the sun or other luminous body sends
out rays of the same nature, distinguished only by their periodicity,
which, by the action of a prism or other means, may be separated
according to their wave lengths, and at any point of the spectrum
there is but ove ray and that it is simple.
344. All bodies absorb, transmit, and diffuse the radiations
which they receive. Whatever part of the energy is received and
not transmitted must perform work or store up energy in the mole-
cule. If « represent the coefficient of transmission, ¢ the thickness
of the absorbing body, and J the original intensity of the luminous,
heat or chemical effect of a radiation, then we have
Ll’ = Tut (404)
for the transmitted intensity J’ of either of these effects considered ;
the ratio Z' to J being aconstant for each ray for the transmitted heat,
light, or chemical effect. Experiment proves that the coefficient w
is the same for each of these effects, when we consider the trans-
mission of any particular radiation of wave length A, and it varies
in value for different values of A. Thus, red glass is quite trans-
parent for red rays, and it is equally transparent for the heat and
chemical properties of the red ray; but it is not transparent for the
yellow ray for either the luminosity, heat, or chemical effect of
this ray.
345. The principle of absorption may then be stated to be, that
when any radiation is transmitted through any medium, its three
properties of luminosity, heat, and chemical effect are transmitted
in equal proportion. When w= 1, the body is perfectly trans-
parent for all effects; it is then said to be diathermous, and chemi-
cally and luminously transparent. Rock salt is an example of a
body transparent for all radiations, and other bodies, as calcium
2
RELATING TO SOUND AND LIGHT. 227
fluoride, quartz, and glass are less transparent for the ultra red rays,
while alum, citric acid, ice, and water, stop these rays almost com-
pletely. Black bodies stop the luminous and ultra-violet, but let
the ultra-red pass; a solution of iodine in carbon bisulphide affords
a striking example of such a body. In some bodies, generally col-
orless and transparent, « diminishes as u increases, while in trans-
parent bodies of a blue or violet color, « increases with ». In the
same way, all bodies may be classed with respect to their powers of
absorption.
346. Emission. When bodies are heated, they emit radia-
tions, which however are not simple, but complex, being a mixture
of rays of varying refrangibilities, determined .by the temperature
of the body. Thus, for Centigrade temperatures below 100°, the
rays emitted are those which pass through rock-salt with difficulty ;
from 100° to 525°, the rays are still obscure; at 720°, the rays lie
between A and ©, and at 780° they extend to G, while at 1165° they
include all to H. The emissive power of a body is directly propor-
tional to its power of absorption, and this ratio is found to bea
constant for all bodies; hence the ratio
a = C= 1,
expresses the law for all bodies, in which Z is the emissive and A
the absorptive power of any body for a particular radiation, and C
is the emissive power of black bodies, whose absorption is taken
equal to unity; both # and A depend on the temperature of the
body and the refrangibility of the radiation. A striking analogy
with respect to sound exemplifies this law. Thus, while the mole-
cules of an incandescent gas can only emit radiations of a certain
periodicity, experiment shows that they absorb radiations of the
same periodicity. So in the case of a composite sound, a tuning-
fork mounted on its resonant box will, by sympathetic resonance,
enter into vibration when any component sound of equal wave
length exists in the composite sound. The fork then absorbs this
particular vibration, which is the only one it can emit, and thus
becomes itself a new origin of sound, sending out in all directions
waves of this single pitch; but if the tone corresponding to the
pitch of the fork be wanting in the original composite sound, the
228 ELEMENTS OF WAVE MOTION.
latter will not lose energy, but continue its way without sensible
diminution.
347. If the homogeneous light of a sodium flame be deviated by
a prism arranged with its lens to produce a pure spectrum, the lat-
ter will consist of a single bright line in the yellow, coincident with
Frauenhofer’s line D. If Uthium, thallium, and indium be substi-
tuted in succession for sodium, their spectra will also be bright lines,
but colored red, green, and blue, respectively ; and finally, if a mix-
ture of these four metals be vaporized by combustion, the spectrum
of their combined light will consist of these four bright lines,
arranged in the order of their refrangibility. Other substances,
when in a glowing state of vapor, are found to give.one or more
characteristic bright lines, having particular positions in the nor-
mal spectrum. These spectra are called discontinuous. When
solids are raised by heat to incandescence, their spectra are continu-
ous, and as their temperature increases and they become white-hot,
their spectra extend from the red continuously to include the violet.
Finally, the spectrum of solar light, whether examined directly or
by reflection, and that of the fixed stars, are found to be crossed by
numerous dark lines, in positions peculiar to each iuminous source.
Spectrum analysis in recent years has given a rational explanation
of these phenomena, and has placed within our reach means of re-
search which, at the same time, have largely increased our knowl-
edge of the physics of the sun and stars, enlarged our powers of
chemical analysis, and extended our list of the elements of matter.
So great in extent is this new domain of physics, that a mere pass-
ing allusion to its more important principles is all that can be
referred to here.
348, The instruments employed in this branch of physics are
spectroscopes and spectrometers. They consist essentially of an ac-
curate slit attached to a collimating telescope, through which the
light to be examined is brought in parallel rays to incidence on a
prism or train of prisms in their position of minimum deviation,
and of an observing telescope armed with a micrometer eve-piece to
view the rays after passing through the train ; the spectrometer has
in addition an accurately divided circle. As these instruments will
be constantly used in the lectures on this subject, an extended de-
scription is omitted.
RELATING TO SOUND AND LIGHT. 229
349. Kirchoff found that when the brilliant light produced by
incandescent lime passes through a sodiwm flame, the spectrum
produced by it was continuous, except in the line D. When it tra-
versed in succession the glowing vapors of potassium, strontium,
etc., the bright lines which they would have formed alone were
found wanting in the resulting spectra. These are called, for this
reason, absorption spectra. This property of the luminous flames
of substances in a state of vapor of absorbing the light which they
emit, is found by experiment to belong to all bodies to which its
application has been made, and the conclusion has therefore been
drawn that the dark lines in the solar spectrum result from the
absorption exercised by the luminous vapors of those substances
existing in the solar atmosphere, which would produce bright lines
of the same refrangibility in a state of glowing vapor. This infer-
ence, which has as a result that these dark lines would be reversed
and become bright, could their spectra be seen without the influ-
ence of the direct sunlight, was first confirmed by the authentic
observations of Professor Young during the solar eclipse of 1870.
350. It has been inferred also that gaseous bodies have the
property of absorption at a lower temperature than that of incan-
descence, by the fact that many of the dark lines in the solar spec-
trum are due to the absorption exercised by aqueous vapor in the
atmosphere. This fact has also been shown by direct experiment
to belong to other gases; thus, when nitrous oxide or iodine vapor
is interposed in the track of the solar beam, a number of dark lines
are found to be present which do not occurin the normal spectrum.
Then when this light is examined by the eye, its color would be
modified by the loss of the concurrent influence of the absorbed
rays. Liquid solutions and solids likewise absorb certain constitu-
ents of white light, and transmit others more freeiy, and the color
of the transmitted light would result from the combined effect of
those which are permitted to pass.
351. If we represent the original intensities of the elements of
white light by the capital first letters of the names of their colors,
and by the small letters the quantity represented by w for each color,
we will have, from Eq. (404), us an expression for the intensity of
the transmitted light,
Rrt + Oot + Vy + Got + Bot + Mt + Vor. (405)
250 ELEMENTS OF WAVE MOTION.
A discussion of this expression will explain why the transmitted
light may vary in color, intensity, or in hue. by varying the thick-
ness.
352. The color which bodies possess is due then to the con-
stituents of white light which are either transmitted or reilected
from their surfaces. Certain constituents are absorbed at the sur-
face. and if the remainder is reflected, the body is colored and
opaque: if the remainder is partially transmitted and reflected, it
will be transparent and colored due to the former, and the color by
transmission may in some cases be different from that by reflected
light.
353. The constants of color are purity, tnfensity, and hue or
tinf ; the first has reference to the relative amount of white ligbt
mixed with it. the second depends on the light sensation arising
from the energy of vibration, and the third is determined by its
refrangibility. referred to the normal spectrum.
Mire) ¢Jors arise from adding the pure colors of the spectrum
together. A new color which is not in the spectrum is purple, ob-
tained by a mixture of the red and violet. The general results
obtained from the mixture of colors may be briefly stated to be
these, viz.: A mixture of two kinds of pure colored light produces a
color in which the ere fails to detect the ingredienrs. in the same
sense that the ear can analyze a compound tone; that the same
mixed color can be produced in several ways; that many mixtures
of two colors can produce erhive ; in this case. the colors are said to
be complementary ; that a mixture of three or more colors gives no
new hues. other than those which can be formed by a mixture of
two colors: that all the hues between red and yellowish-green. and
between blue-green and violet, will give hues located between the
same limits. respectively, and that green when mixed with any
other color vives no new hue. but rather a color less saturated than
that with which it is mised. The study of mixed colors is very
much facilitated by the use of Maxwell's color-disks, by means of
which two or more colors mav be united in various proportions.
The eff-ets are somewhat modified from those produced by the pure
colors of the spectrum. becaus: of the impurity of the pigments
employed. and from the effects of absorption by the material of the
disk.
RELATING TO SOUND AND LIGHT, 231
354, Phosphorescence and Fluorescence. Certain
substances, such as the sulphur compounds of calcium, strontium,
and barium, possess the property of emitting luminous rays in the
dark for a shorter or longer time, provided they have been pre-
viously subjected to the action of light containing the more refran-
gible rays of the spectrum. This property is called phosphorescence.
Other substances possess this property in a much less degree, such
as uranium glass, quinine solutions, chlorophyll, esculin solution,
fluor-spar, and many others, in which the phenomenon lasts but for
a.very short time. This is called flworescence, from the fact that it
was first noticed in a species of fluor-spar. Both are essentially the
same in principle, differing only in degree, and both are merely the
effects of absorbed light. The radiations absorbed by these bodies
are those of greater refrangibility than the rays they emit while ex-
hibiting these properties, and the conclusion is that the absorbed
energy of their ether molecules is given up to the molecules of the
bodies, which in turn control the new vibrations of their surround-
ing ether, forcing them to take up vibrations of less periodicity.
From this stored work, the expenditure of which is not instanta-
neous, the vibrations are continued until all is exhausted, when the
phenomenon ceases. A new exposure to the more refrangible rays
will again cause the same effects to be produced.
355. Achromatisin. Since the refractive indices differ in
the same medium for different colors, the results obtained in geo-
metrical optics for the deviation of light by refraction are only true
for homogeneous light. After refraction by a lens, a pencil of
white light does not converge to, or diverge from a single point ;
first, because of the spherical aberration of the lens, and second,
because of the unequal refrangibility of the colored rays of which
the beam is composed. This latter departure from accurate con-
vergence to a single point is called chromatic aberration, and is
independent of the figure of the lens, but depends on the material
of which it is made. Thus, if a lens L, Figure 85, intercept a beam
of white light, the violet rays will first be brought to a focus at v%
and the red rays will be brought to a focus at 7; other colors will
have their foci at intermediate points, in the order of their refran-
gibility.
If a screen be moved along the axis, the section of the deviated
232 ELEMENTS OF WAVE MOTION.
cones between L and ¢ will be red on the exterior and violet within,
while beyond ¢ the sections will be violet without and red within.
The section ad at c, being the intersection of the cone of violet and
red rays, is the least through which all the colored rays pass, and is
therefore called the circle of least chromatic aberration.
/
1
a
\ os Ag
fe
ofc------SSh>
Figure 85.
Each pencil of rays proceeding from an object will likewise be
separated into numerous cones of colored rays, and the object will
have many colored images formed in different focal planes; when
these images are viewed by the eye-lens, the resulting image will be
confused, indistinct, and colored.
356. To reduce the dispersion of light in passing through lenses
is the object of an achromatic combination. The possibility of such
a combination arises from the fact that the dispersion of a ray of
white light and its mean deviation are not proportional for different
media. If dispersion were proportional to deviation by different
media, then any combination which would destroy dispersion would
also destroy deviation, and the combination would be as valueless
for a telescopic object-lens as a piece of plate-glass. There are
media, however, which produce the same dispersion, while the de-
viation differs. The achromatism is effected generally for two
colors, preferably for those which are complementary. Owing to
the irrationality of dispersion, the remaining colors are not accu-
rately united, and the resultant effect is still somewhat colored,
giving rise to a secondary spectrum, in which the fixed lines may be
inverted, depending upon the relative dispersive powers of the two
media.
357. Resuming Eqs. (295) and (296), Art. 266, and supposing
the incidence small upon a thin prism, we have, taking the angles
for the sines
o =), (406)
%,, = HD, (407)
RELATING TO SOUND AND LIGHT. 233
c=¢'+%,, (408)
S=$4+o,—% (409)
From the first two, we have
+o, =H (O' + 4%) = He; (410)
which, when substituted in the last, gives
d= (u—l)a. (411)
Therefore, the deviation by a thin prism, with nearly normal in-
cidence, is equal to the excess of the refractive index over unity,
multiplied by the refracting angle of the prism.
The deviation by two such prisms is
8 = (tl) + —1e, (412)
and forany number, 06, = =(u—1)e. (413)
If the two prisms be turned so as to have their angles in oppo-
site directions, we have
6 = (u—lhe—(v’ —1l)e@’; (414)
and if 6. = 0, we must have as a condition,
«pl ¢
eT Gis)
Considering » and ,’ to apply in succession to the violet ray or
line H, and then to the red ray or line B, we have
6, = (wu — lhe — (v',— le’, (416)
6. = (4 —1he—(v'. le’. (417)
The condition that the difference of deviation should be zero for
these rays is
Bea be (418
By > B a )
Therefore the two prisms will be achromatized for red and vio-
let when the ratio of the coefficients of dispersion of the media is
inversely as the ratio of the refracting angles of the prisms.
234 ELEMENTS OF WAVE MOTION.
358. The following table gives the values of the refractive
indices from which these coefficients can be obtained, for water,
crown and flint glass:
TABLE OF REFRACTIVE INDICES.
| Rerractine SuB- B. Cc. D. E. F. a. H.
STANCES. He | Ho | Hy | By | He ) i] My
Waters sasse-ccvgeirtgadiece 1.330935 | 1.381712 | 1.833577 | 1.335851 | 1.337818 | 1.341298 | 1.944177
Crown Glass, No. 13....| 1.524312 | 1.525299 | 1.527982 | 1.581372 | 1.534337 | 1.539508 | 1.544684
Flint Glass, No. 13...... 1.627749 | 1.629681 | 1.635036 | 1.642024 | 1.648260 | 1.660285 | 1.671062
359. Achromatism of Lenses. Fora given radiant dis-
tance f, we have for the focal distance of red rays, by Eq. (314),
Art. 276, deviated by two lenses.
ie 1 NegoyGat\as
ze = (i —. 1) (: — ) + (u ro 1) yt yl! TFs (419)
i =, C i ) : ae ) Ls aan
ge =D ro" + (u', — 1) pl ltt + #3 (420)
whence, when f,” = f,", we have after reduction,
2
rs oe roll oe
TT Sea aoa T (421)
ro
The first lens being supposed determined upon, r and 7’ are
known, and r” is usually taken to be equal to 7", as the two lenses
are generally in contact throughout ; r'” is then the only unknown
quantity in the above equation, and can readily be computed.
The first members of Eqs. (418) and (421) being the same, we
have
1 1
gaa
ao I TO (422)
ror
whence we see that the problems of achromatism for lenses become
RELATING TO SOUND AND LIGHT. 235
those of prisms of the same material. Dividing both members of
Eq. (421) by a , we have
g
a
iH, bes (5 on)
by nes iL = (v', 1) yl oe nl _ F, ie
Wamp (423)
w,—1 ae
Therefore an achromatic combination of two lenses for red and
violet can be formed, when the ratio of the dispersive powers of the
media is negative and directly as the principal focal distances of
the lenses. For an achromatic objective, the negative lens must
have the greater power in order that a real focus may exist. The
focal lengths of the lenses can readily be found from the preceding
table for an achromatic combination composed of crown and flint
glass lenses.
860. To find the chromatic aberration of a lens, and the diam-
eter of the least chromatic circle, we have for the red and violet
focal distances,
tr é i ) a
pe = (u. = 1) , — r + i’ (424)
Ae. 1 4) 1 =
x= (4 — 1) (= Te (425)
Subtracting the first from the second, we have
1 1 EM a ee 1 1
Kk Ff = Fir = He) —5)- 426)
Hy —
and substitute in the first
Multiply the second member by no
g
member f/f,” for f,f,", and there results after substituting D and
F, for the dispersive power and principal focal distance respec-
tively, pp
fl fl = (427)
2
361. In Figure (85) we have approximately,
cv vl a 498
ab AB’ ab ~ ABS on
We
oo
oe
ELEMENTS OF WAVE MOTION,
rv Ir + Lv BS
whence - An = fe See (429)
in which f,” is taken to be the distance of the circle from the lens,
and « is the semi-aperture ; whence we have
rv ad: f,"
a 3 (430)
ab = re =
hence the diameter of the circle of least chromatic aberration
yaries as the aperture and dispersive power directly. When the
rays are parallel, 7,’ is very nearly equal to #,, and the diameter
ab becomes equal to aD.
362. By Eq. (423) we see that the conditions of achromatism
depend only on the focal lengths of the component lenses, and are
independent of the forms and order in which they are placed.
The spherical aberration depending on the latter, it is therefore
possible by a suitable arrangement of the forms and order to secure
a combination which shall be both aplanatic and achromatic at
the same time ; such a combination is made in Huyghen’s ocular.
363, Rainbow. The rainbow is a phenomenon frequently
seen during a shower of rain when we turn our backs to the sun
and view the opposite portion of the sky. It consists generally of
two colored bands im each of which the succession of colors is the
same as in the spectrum. The most brilliant, called the primary
bow, has the red outside and the violet within; the less brilliant,
called the secondary, is above the primary and has the violet with-
out and the ved within. The common centre of the bows is on the
right le drawn from the sun through the eye of the observer.
The space between the bows is darker than the rest of the sky, and
in brilliant bows looks like a dark ribbon fringed with prismatic
colors. Below the primary and above the secondary other bows,
colored red and violet, are often seen, which are known as super-
numerary or spurious bows.
364. The complete theory of the rainbow which has been given
by Airy, is based upon the form of the emergent wave after refrac-
tion by the rain-drops, and is a consequence of the theory of
caustics. The explanation ordinarily given of this phenomenon,
that of Descartes, is inexact and merely approximate, fails to
RELATING TO SOUND AND LIGHT. 237
account for the supernumerary bows, and is not confirmed by exact
measurement.
Let the spherical drop of {
water I, I’, I’, Fig. 86, in- Z
tercept the parallel rays of + :
homogeneous light coming
in the direction SI, and let
a and 7 be the angle of inci-
dence and refraction. The
ray SI will be refracted to I’,
reflected to I, and emerge Ny
Figure 86. Ne
in the direction IR. Let
6 be the deviation of any ray
from its direction before incidence to its direction after emergence.
Then the deviation produced by the first refraction at I is ¢—7,
by the first reflection at I’ is — 2r, and by the second refraction
at I is ¢—r, since the angle of incidence is here r and angle of
refraction is 7. Then the total deviation is
6 =2(i—r) + (7 — 2r), (431)
and for any number of internal reflections will be
d= 2(i—1r)+n(r—2r). (432)
To find the value of 7 corresponding to a maximum or mini-
mum deviation, differentiate Eq. (432) and place equal to zero,
and we have
dr ‘
1—(n+ 1) =0. (433)
From sint =p sin7, (434)
we have oF yy (435)
di wcosr’
which substituted in Eq. (433) gives
COs 7
1— (+1) = 93 (436)
whence
pe? cos? r = (n + 1)? cos? 4. (437)
2S ELEMENTS OF WAVE MOTION.
Eliminating cos? r by Eq. (434) we have
pe = 1 + (v2? + 2n) cos? 7; (438)
whence =
Bunt =.
cost = /é oh (489)
365. For water p is about 4, and hence p? — 1 is $3 1? + 2n Is
at least 3; therefore the value of 7 is always admissible, and what-
ever may be the number of internal reflections, there always exists
for a given color a single value for the angle of incidence for which
the deviation isa maximum or minimum. Rays which have this
angle of incidence or differ but little from it, are called when they
emerge efficucious rays. To find whether this value of 7 corre-
sponds to a maximum or minimum deviation, we have
Po er
qa = 2+ ge (440)
3 i PO ,
from which we see that the sign of qa 8 always contrary to
(er
that of dz
From the relation,
dr cos 7
di pcosr’ ee)
we have
fr — cov rsint +psinr cos i — (1—p*) sin? |
diz 13 cos? 7 ~ cost” te)
ar . 4 Po .
whence we see that ae always negative and hence a 8 always
positive. Therefore the deviation of the efficacious rays is always
a minimum, whatever be the number of reflections.
866. To determine the position of the point of emergence of
any ray SI with reference to the point of emergence of the normal
rav SA, we have (7 +1)7 for the value of the angle between
A and the point of emergence of SA after any number of reflec-
tions ; and fur the corresponding angle between the point of emer-
gence of ST and I the expression (m + 1) (7 — 27); representing
the required angle by 6, we have
RELATING TO SOUND AND LIGHT. 239
6=2(n+1)r—i4. (443)
Applying the test for maxima and minima we have
dé dr enn
Ga tal =0; (444)
whence by reduction
25 = a Wh
cos? t = Oe ip 1 (445)
The value of cos i for one reflection is Wz}; a very small
quantity which diminishes very rapidly as 2 increases ; besides as
we hare
CO Pr
‘ er . _ @O. :
and since ae 1s always negative, ae 1s negative and corresponds
to a maximum for the value of cos 7 above.
367. The angular distance 6 of the point of emergence of the
ray SI to the point of emergence of the normal ray SA is zero
when SI coincides with SA, then increases with an increase in 7,
and attains a certain maximum when the angle of incidence is near
90° (¢ = cos“! W745), that is when the incident ray is nearly tan-
gent to the spherical drop.
868. The primary bow corresponds to one and the secondary to
two internal reflections. Assuming from the table the values of p
1.331 and 1.344 for the extreme red and violet rays for water, we
have for the primary bow,
for red i = 59° 32’, ry = 40° 21’, 6 = 187° 40’.
Angular radius of red = 7— 0 = 42° 20';
for violet ¢ = 58° 44’, r = 89° 30, 6 = 189° 28’.
Angular radius of violet = 7 — 6 = 40° 32’;
and therefore the angular breadth of the bow for the rays proceed-
ing from one point of the solar disk,
42° 20’ — 40° 32’ = 1° 48".
240 ELEMENTS OF WAVE MOTION.
Considering the pencils proceeding from all points of the solar
disk, this breadth would be increased by the sun’s apparent diame-
ter, or about 32’, and therefore the whole breadth of the primary
bow would be about 2° 20’,
For the secondary bow we would have in the same way,
for ved i = 71° 54’, r = 45° 34, 5 = 280° 24’.
Angular radius of the red = 6 — 7 = 50° 24’;
for violet 7 = 71° 29', pr = 44° 52’, 0 = 233° 46’.
Angular radius of the violet = 6 — 7 = 53° 22’;
and for total breadth, 53° 22’ — 50° 24’ + 32’ = 3° 30’.
369, Now if the direction of any efficacious ray due to one in-
ternal reflection IR, Fig. 86, be produced backward, and be sup-
posed revolved about the axis of the bow, the arc marked out by
its extremity will be the are of the bow for that color. All rain-
drops which meet the conical surface described by the revolution of
this ray, contribute in passing, their few rays of that particular
color which reach the eye of the observer, supposed at the vertex of
the cone. And since the semi-angle of the cone is 42° 20’ for red,
and varies by continuity for the other colors of the spectrum to
40° 32’ for violet, therefore in the primary bow the red is without
and the violet within. When the effica-
cious ray I'’R, Fig. 87, due to two in-
ternal reflections is in like manner re-
volved, the semi-angle at the vertex for
violet is 53° 22’, and for red is 50° 24’,
and therefore the violet is without and
the red within in the secondary bow. Figure 87.
All rays either once or twice internally
reflected while the drops are passing within the bows, pass either
above or below the eye, and hence, this space sending fewer rays, to
the observer appears less bright than the rest of the sky.
370. This explanation of the rainbow, with the values of its
dimensions, is based on the supposition that the positions of the
emergent efficacious rays correspond to the only maximum. Let
us construct, as in Fig. 88, the paths of rays incident above the
RELATING TO SOUND AND LIGHT. 241
central ray, which undergo one internal reflection. Those that are
incident between A and I are first reflected between A’ and I’, and
emerge between A.and I”; those incident between I and the tan-
gent ray near T are reflected between I’ and a point near A’, and
emerge between B and I”. The consecutive intersections of the
emergent rays therefore form a caustic of two branches, one A’C’
normal to the spherical drop at A’, and the other BC tangent to the
drop at B, very nearI”,as shown in the figure. The efficacious ray
after emergence I’E is an asymptote to these caustics. The
emergent wave is the development of the caustic surface formed by
revolving the caustic curves about the ray SA, which is undeviated.
Figure 88.
A section of the emergent wave by the plane of the figure is 2.
curve having a point of inflection at m on the efficacious ray. The
intensity of the luminous effect in the neighborhood of a caustic
has been thoroughly discussed by Airy, and the numerical values of
his integrals, when applied to the theory of the rainbow, are in per-
fect accord with. the observed phenomena. It results that for posi-
tions near, but to the right of the efficacious ray, as shown in
the figure, there are series of maxima and minima luminous in-
tensity, of which the first maximum not exactly coincident with
IE corresponds to the primary bow; the other maxima, still far- _
16
42 ELEMENTS OF WAVE MOTION.
ther from IE, belong to the supernumerary bows. The deviation
of the primary bow is less than that determined by the Cartesian
theory above, and is found to decrease slightly with the decrease in
the diameter of the rain-drops. The same modifications oceur in
_the secondary, and in its supernumerary bows, located above it.
“371. Interference of Light. The principles enunciated
in Part I, Arts. 64 to 72, being applicable to all undulatory motion,
will therefore be assumed to govern in the succeeding discussion.
To ascertain the circumstances under which two luminous waves
will interfere, let 8, in
Figure 89, be a lumi-
nous source from which
homogeneous rays of
waye length 4 proceed ;
M and M’ two plane
nirrors which intercept
and deviate the rays,
and whose inclination
is indicated by 7, as in
the figure. Then, by
construction, s and s’ are the virtual foci of 8, and the path of the
rays after deviation by the mirrors will be the same as if, starting in
the same phase, they had proceeded from two separate luminous
origins s and s’.
Join A the centre of ss' with C by a right line, and extend it
indefinitely to 0; let N be any near point to O on LO, drawn at
right angles to CO, and let soc =a and CO= 0b. Since 7@ is the
inclination of the mirrors, the angle sCs’ is 27.
Figure 89
372. To find the resultant displacement of a molecule at N,
due to the two undulations proceeding from s and s’, we will assume
the general expression for the displacement, Eq. (69),
6 = asin | (vi Sat 4'|, (447)
in which ¢ is the displacement at any time ¢, « the maximum dis-
placement, 4 the wave length, V the velocity of propagation, z the
distance of the molecule from the sources, and A’ the arbitrary arc
RELATING TO SOUND AND LIGHT. 245
which determines the phase. If c is the value of the maximum dis-
placement at a unit’s distance from the origin, we have, by Art. 72,
c
emt
x
Substituting this value for «, aa 2 ~ A for A’, the displace-
ments of N due to the waves from s au s' will be, respectively,
é ., | aa x
\ (448)
j an '
c= oy sin 2 (v1 — aw 4) |;
and, since AO may replace sN or s’N in the coefficient without ap-
preciavle error, the resultant displacement, by Art. 65, at N will be
c . | 27 . | 27 F
io} sin| 27 (ve — an + 4) |+sin[ (reo + 4) | .
449
which may be written, Ge)
; _ | 27 sN + s'N
a cos % (en — 9) | sin | 2 (re = + 4) |
(450)
The maximum resultant displacement is therefore
c TIN — oN
xo °° E (sN —s x) | ‘ (451)
and the maximum intensity of the light is
4e2 gh ;
aor cos |= (sN —s ») | (452)
From the figure we have
=? a? 7
sN° = AO* + (sA + ON) (453)
= (acosi+ 6)? + (asin? + ON)
ase
244 ELEMENTS OF WAVE MOTION.
1 (@sini + ON)?
acost +b
1 (asin i— ON)?
acosi + b
* sN = acost+0 +5 » very nearly ;
as
s'N=acosi+d+ 5
» very nearly ;
2a sin t_ N
acosi+b
_ 2a sin 4
~ ato
sN—s'N =
(455)
ON, very nearly, since 7 is very small ;
and the intensity at N will be therefore
40 cos? (=. asin @
(a + bP a a+b
which varies for all points N with the distance of N from 0.
Making ON in succession equal to
on), (456)
at+ba a+b 2a a+b _ 8A
——. 8a ——: ete.
asina +’ asint 47 asint 4’
0,
We will have, for the corresponding intensities at the points N,
= as
(a+ 6)? (a + bY?’
and generally, for points for which
asint 4”
0, ete;
2
; o integer, the intensity will be aap and for points
or whic
a+) Qn+1)?
a rar 4
the intensity will be zero, or there will be darkness. Intermediate
values of ON will correspond to points at which all degrees of
brightness will exist, from darkness to the maximum illumination.
The same will be true for points in the plane OL, normal to OC,
and there will then be a series of lines of alternate darkness and
brightness, which are called interference fringes.
RELATING TO SOUND AND LIGHT. 245
373, If from s and s’, as centres, arcs of circles be described
whose consecutive radii differ by a we readily see, since the waves
start in the same phase, that at points where the difference of route
is any even multiple of 2 , the component displacements meet the
molecule in the same phase and add their effects, while at points
where this difference is an odd multiple of = the displacements
meet in opposite phases and cause mutual destruction. The points
of maximum brightness are indicated in the figure by the intersec-
tions of two full or two dotted curves, and the points of darkness
by the intersection of a full and a dotted line. The locus of each
of these series of points is an hyperbola, whose foci are s and »’,
since the difference of distance of any point of the locus from s and
A : ‘
s’ is equal to oi n being odd for the dark lines and even for the
bright lines.
374. If instead of homogeneous light, white light is employed,
the fringes corresponding to each constituent will be in sequence
nearer to OC, according as the values of A are smaller. Therefore
the violet fringe will be the nearest, the blue next, and so on in suc-
cession to the red. At O all will be united, and their combined
effect will be white. The position of the first fringe for any con-
stitnent, as violet, is determined by the value of
a+b Nn,
asini 2?’
ON =
and its intensity would be measured by
rae
(a + bP
At the same point the intensity for the blue would be
4c A
a epee td
(a az by? . BW?
and hence we see that the dark spaces would soon be encroached
upon by the various colors, due to their difference of wave length,
and therefore the number of distinct fringes in white light is
246 ELEMENTS OF WAVE MOTION.
limited. The wave length A is so small as to be incapable of direct
measurement, or even appreciation by the eye, but the value of the
expression
i} a+ b a
asini 2
can be made large by making 7 small, and since a, 6, and 7 can be
measured directly, the value of 4 can be determined for each con-
stituent of white light.
875. This crucial experiment is one of fundamental importance
in the undulatory theory of light. If light be not the result of un-
dulatory motion, no rational explanation can be given of the fact
that two lights added together do produce darkness, and that this
result is due to the concurrent effect of both is satisfactorily shown
by the fact that if one of the mirrors be covered or removed, the
fringes at once disappear, and uniform illumination results.
An apparent difficulty arises from the fact that this property of
interference does not occur when the two sources of light are inde-
pendent and distinct; but this is satisfactorily accounted for by
considering the impossibility of any two independent luminous
sources sending out waves with exactly the same phase for even the
smallest appreciable time. We may conceive this to happen for a
few or even several thousand vibrations, but the probability that
this should continue for millions of millions of vibrations, when the
variations that affect the luminous sources are entirely independent
of each other, is beyond conception. Now by many repeated ob-
servations, the mean wave length 4 is found to be a little greater
than 0.0000005 m., and since the velocity of light in air is about
300,000,000 m., it follows that the number of waves sent out from
a luminous source of homogeneous light having this periodicity is
about 600,000,000,000,000, in each second of time. Hence, in order
that the waves from these two sources should interfere and for a
second of time permit the fringe to be visible, this enormous num-
ber of waves must be sent out from each, without any variation as
to phase, no matter what may be the variation in the causes that
serve to keep the sources active during this period.
376. To explain this analytically, let 6 and 6’ represent the
molecular displacements due to two waves proceeding from differ-
ent sources, and a’ the distances of m from the sources, and ¢ and
RELATING TO SOUND AND LIGHT. 247
g’ the phases of the vibration at any time ¢ from the epoch ; then
we have for the displacements of the molecule due to each wave,
‘ t @
6 =e sin an(=—$ +o),
cé=a' in an (£ = ‘); °
=a8 ge tS
whence the resultant energy would be measured by
/
ce? +. @'® + 2ae' cos 27 a= +¢'— 0).
The value of this expression depends as:much upon ¢' — @ as
upon -—*, and therefore as the difference of phase ¢' — ¢ of
the concurrent waves will be affected by all the changes that are
constantly affecting their sources, we see the absolute necessity of
employing a single source in exhibiting the interference fringes.
Otherwise, the expression ¢’ — @ will have in general an indefinitely
great number of different values, and the expression into which it
enters will likewise have in an indefinitely short time correspond-
ing values, which will be limited on the one side by its maximum
(@ + «’)*, and by its minimum (« — @')* on the other. Therefore
the intensity of the light due to their concurrent effect, as estimated
either by photometric measurement or by the eye, will be repre-
sented in any appreciable time by the mean of these values, or
«+ a, that is, by the sum of the separate intensities. When,
however, the source is single and interference occurs, the intensity
is either the square of the sum of the two maxima (« + a’)’, or of
their difference (a — a')?, depending on the point considered, and
if as is usually the case, « = «’, these become respectively 4a? and
zero; that is, an intensity four times greater than that due to
either or none whatever.
377. Instead of the mirrors, a double prism or two prisms
united at their bases, having very small refracting angles, can be
used to exhibit the interference fringes. In this, as in the pre-
vious case, the source is usually a line of light formed at the focus
of a cylindrical lens, from which rays fall nearly normal to the
back of the prism, and after deviation proceed as if they had
245 ELEMENTS OF WAVE MOTION.
originated at the two virtual foci of the biprism. The distance of
the foci s and s' from A in this case is equal to that found for the
mirrors multiplied by (“—1), » being the index of refraction
of the prism. Hence, all other things being unchanged, the inten-
sity and distance apart of the fringes are represented by
Gai cost [27-4 9* (uy —1) ow], (457)
Gop ies 45
and ine ee respectively. (458)
The breadth of the fringes for different colors will therefore
depend on (% — 1) as well as on A, and since y varies with A, these
fringes for different colors will be more unequal in breadth than in
the case of the mirrors; consequently the colors by overlapping
will lose distinctness at a short distance from O. The few, however,
which enclose the bright white line at O are much more intense
and distinct than those of the mirrors, because of the greater
intensity of the light transmitted by the prism.
378. If in either of these cases a piece of thin glass or other
transparent solid or liquid medium be interposed, so as to make
the route of one of the interfering beanis optically longer than the
other, the fringes will be found displaced towards the correspond-
ing side. By this means the velocity of light in air has been shown
to be unquestionably greater than that in glass, and to be in the
proportion of 1.
379. Colors of Thin Plates. The colors displayed in
soap bubbles, thin films of coal tar on the surface of water, frac-
tures in glass or in crystals of various kinds, and in other like
phenomena, are due only to the principle of interference. If a
lens of small curvature be placed on a piece of plane glass, the thin
film of air of varying thickness between the two surfaces will cause
incident light to interfere and produce colored rings of varying
diameter. When homogeneous light is used the rings are alter-
nately dark and bright when seen by reflected light, and bright
and dark by transmitted light. With incident white light the
rings are variously colored and form a perfectly determinate and
easily reproduced serics, so that they can be used as a reliable
RELATING TO SOUND AND LIGHT. 249
standard for color. They are known as Newton’s rings, and the
succession of colors is called Newton’s scale.
380. It is easily seen that the thickness of the plate of air
increases from zero at the point of contact of the lens with the
plate, and is always the ver-sine of the arc; it therefore varies as
the square of the diameter of the ring. By accurate measurement
Newton found that the squares of the diameters of the rings form
an arithmetical progression, and hence the thicknesses of the thin
plates for rings of the same color are in the same ratio. To
explain in an elementary way their
formation, let SI, S'l’, ete., be inci- Rg s
dent rays of homogeneous light, and
suppose the two rays that move along
I'R’ to be in a condition to interfere;
part of S’'I’ being directly reflected at
I’, and part of SI] heing refracted NS Co CG
= reflected along II"TR’. To find 7
he difference of route so as to deter- R
mine the change of phase, draw the Figure 90,
normal I'K. It is evident that the
rays moving from I’ along I’R’ will be simultaneously in the same
phase as that moving along IR at K, since these points are on the
same plane normal to the reflected wave front; and since the
velocities in air and glass are as #:1, the distance KI reduced to
its equivalent route in air is #XI. Again, from mechanical princi-
ples and from the analogy of reflected waves of sound in air, Arts.
228 and 229, we see that a difference of phase of 180°, or its equiva-
lent - in route, must be added or subtracted from each reflection
2
of a wave incident on a surface separating a rarer from a denser
medium, while no difference is to be allowed, when reflection takes
place on a surface separating a denser from a rarer medium. We
therefore have for the difference of route for the two inter-
fering rays,
j a
I” +0" +3—pRI = ans, or = (Qn +1)5, (459)
for the maximum and minimum intensity respectively, in which »
represents any integer marking the number of the interference.
250 ELEMENTS OF WAVE MOTION.
Representing the thickness of the film of air by ¢, the angle of
incidence, AI'T by 7, and the angle of refraction $8’I'R’ by 7, and
since l'I’ = I'l, we have
Ql'l’ —pKI = @n—1) = or 2n ‘ (460)
But we have from the figure II" = se, and
cos 2
KI = ITcos KI’ = 2AI sin r 461
= 2]]" sin 7 sin ry = 2¢ tan7 sin im : (461)
Whence after reduction we have
A
t= i (Q2n—1) sect, or = 72m sec t. (462)
Whence we see that the thickness corresponding to the bright rings
are as the odd numbers 1, 3, 5, 7, etc., and to the dark rings are as
theeven numbers 0, 2, 4, 6, etc.; that the thickness for a ring of any
given order varies with 4, and hence varies with the color and with
the nature of the substance forming the film; and finally, that the
thickness varies with the incidence 4.
381. The transmitted rays R,, etc., are also capable of interfer-
ing, and a similar discussion would show that the interference
rings formed by refraction are complementary in color to the
reflected rings, a deduction which is confirmed by experiment.
382. Diffraction. In the cases of interference above re-
ferred to, since the routes of travel of the interfering rays must
differ by some odd multiple of . either reflection or refraction
must take place upon a portion at least of the rays which setting
out at the same phase proceed from a single origin. The principle
of Huyghens is applicable, however, in all cases belonging to diffrac-
tion, By this principle the disturbance at P’, Figure 10, page 50,
may be considered as the resultant of all the disturbances sent to it
from every point of the wave front in any one of its anterior
positions as DAC. Assume a series of distances beginning with
P’A and successively increasing with the constant value a with
these distances as elements conceive conical surfaces described
RELATING TO SOUND AND LIGHT. 251
having P’ for their common vertex and P’A for their common axis.
The zones which they intercept on the spherical wave front are
annular surfaces slightly increasing in area as we proceed outward
from A. From these zones of Huyghens, as they are called, second-
ary waves proceed which are all in the same phase at starting, but
which, owing to the difference of route, meet at P’ in various
phases. The secondary waves due to each zone vary by insensible
degrees from phases of complete accord to phases bordering on
complete discordance. The displacements of P’ due to waves from
any two contiguous zones are therefore of opposite signs, and were
the number of these waves exactly equal, their rays parallel, and
the displacements of the same amplitude, mutual destruction of
each other’s effects on P’ would ensue. But while this equivalency
is not exact, it is approximately so with respect to adjacent zones
at a very short distance from A, and the intensity of the light at P’
will be represented by the decreasing converging series,
Teddi ee, (463)
in which the terms depend on the aggregate displacements of the
molecule at P’ due to each zone. Hence the resultant intensity of
light at the point P’ is sensibly due to the concurrent effect of but
a few of the zones, and whatever be the number considered, this
intensity has always a value between I and I—I’. Therefore the
maximum effect is that due to the central zone, and since 4 is
exceedingly small, we may conclude that light is propagated sen-
sibly in right lines from its source.
383. The consequences of the above reasoning, which have been
repeatedly confirmed by experiment, are
1°. That if by any means the effect of the even numbered zones
2, 4, 6, 8, etc., be stopped out as by opaque screens, the intensity
of the light at P’ due to the zones 1, 3, 5, 7, etc., is very much
increased. By a carefully contrived experiment, the light of the
second zone has been obstructed and the intensity of the light
which remained was found to be nearly five times as great as
before.
2°. If a small circular opaque disk bé interposed in the cone of
rays proceeding from 0, the intensity of the light at the central
point of the geometric shadow is the same as when the disk is
removed.
252 ELEMENTS OF WAVE MOTION.
3°. A small circular aperture being interposed in the cone of
rays, and properly placed with respect to the screen, will exhibit a
perfectly dark spot at the projected centre of the aperture.
The analytical proof of these consequences, as well as of many
others, all of which are easily confirmed by experiment, is beyond
our assigned limits and is therefore omitted.
384, We have seen in Art. 251 that the extent of the luminous
source has a marked influence in softening the shadows cast by
opaque bodies. When the source is a mere point ora right line,
the contour of the geometric shadow is not the true boundary
between the light and darkness, but there are certain luminous
effects both within and without the geometric shadow, which are
perfectly explained by the principles upon which this subject of
diffraction is founded. Let the parallel beam of homogeneous
light S be brought to a focus at O by a lens of short focus, and
suppose an opaque screen PM be placed in the diverging cone with
its edge P in the axis OQ. The geometric shadow of this edge on
QQ is at Q. Below this line Q a faint illumination exists within
Figure 91
the shadow, which fades away gradually from Q. Above Q and
parallel to it a series of alternately bright and dark bands or fringes
are found whose distances apart vary with the position of the
screen and with A. Careful measurements show that their loci are
hyperbolas whose vertices are at P and centres midway between O
and P. If the incident light be white, the fringes are iris colored,
the violet of each fringe being nearest the geometric shadow and
red the farthest.
385, If the obstacle be very narrow, such as a fine wire or
horsehair, the whole shadow is occupied by a series of very fine
alternately bright and dark lines parallel to the wire; these lines
RELATING TO SOUND AND LIGHT. 253
becoming wider as the diameter of the wire is diminished. If a
rectangular aperture be formed whose width is small compared to
its length, the diffraction fringes will be found in the luminous
portion, which will become wider as the aperture becomes nar-
rower. A series of rectangular slits, separated by opaque intervals,
may be formed by ruling on glass a number of equidistant lines, by
means of a diamond point. Such an apparatus is called a diffraction
grating. When a beam of light falls upon it a direct image of the
luminous surface is formed by reflection or transmission, flanked
by a series of spectra each of which is separated from the preceding
by a space devoid of light.
386. To explain the formation of the diffraction spectra by
means of a grating, let the diaphragm MN be placed normal to
the parallel beam of homogeneous light SS’, and suppose that bac
be a rectangular opening whose width is very small compared to
its length. Considering alone the effects on a screen PP’ in the
plane MP’, we see that by the principle of Huyghens all the points
of bc may be considered as centres of secondary waves from which
rays proceed in all directions beyond MN. By the preceding prin-
,
Ss Ss
M db} a |c N
he
fare! a a bier
Figure 92
ciples the maximum disturbance will occur at a’, and provided the
difference of route bc’ — cc’ be less than 5 the space 0'c’ will have
less illumination, but with inappreciable difference of intensity on
either side of a’. Now consider the effects at positions on the right
of the axial line aa’, since whatever is proved true for these will be
equally true for corresponding points on the left, and we will find
a space dd’ such that points within it have no disturbance due to
vot ELEMENTS OF WAVE MOTION,
rays bd ...ed' whose angle of diffraction @ is such that dh is equal
toa. For, beginning with a ray from 0, we have a ray from a
; : A :
whose difference of route is exactly 5, and for each consecutive
point in da we have a corresponding point in ae, such that rays
from each of these pairs neutralize each other within the space dd’.
Still farther to the right a space %"c”’ will be found such that the
difference bh will be equal to 34, and if we consider the centres in
bac separated into three equal groups, the rays from the first group
will completely neutralize those of the second, and therefore permit
those of the third to produce concurrent effects in 0'c'’; hence
é"c" will be illuminated with an intensity much less, however, than
that of b’c’; and so on for other alternate dark and bright spaces.
387. If now a bright line of light be used from which parallel
ravs proceed and a grating be interposed in their paths, the open
spaces will permit the light to pass while the furrows will
intercept it.
Let AB be such a grating, ad,
cd, ete, the clear spaces, bc, de,
ete., the furrows, and from what
precedes it is evident that if ah
be equal to wa, that the rays at
ab’ will unite their effects and
produce a fringe of the 2 order
of wave lengtha. If ah, however,
differ by any small fraction of A,
1 ‘
us —, then the beam passing ‘ ‘ oor
wh Figure 93
mth
through the “a fpace from ab will be in discordance with that
passing through aé, that passing through the G + 1) space with
that through the 2d, and soon. Hence the angle ¢@ is determi-
nate for ve rays of a particular color. Therefore to determine A for
any fringe as the n”, representing the width of a space and a fur-
row (ab + be) by w, we have
dames (464)
Ww
RELATING TO SOUND AND LIGHT. 255
2
388, If white light be employed, since 4 varies for each con-
stituent, the violet fringes will be found nearest the central line
and the others arranged in the order of their wave lengths. The
finely constructed gratings of Nobert, Rutherfurd, and others, give
such pure spectra that the fixed lines of Frauenhofer are clearly
defined, and thus afford means of measurement for the wave
lengths corresponding to their refrangibility. Knowing the num-
ber of ruled lines to the inch, the value of » is known; the angle
@, is measured by the spectrometer and therefore 4 results from the
solution of the above equation. The advantage of this method of
determining 2 consists in the fact that the distances of the fixed
lines in the diffracted spectra are always proportional whatever be
the substance of the grating, while in the prismatic spectrum they
vary with the dispersive power of the medium of the prism. Thus
since sin @ is very small, it may be replaced by @ sin 1’, and if
¢:, $25 %3, represent the deviations corresponding to any three
points of the n” spectrum, and 2,, 4), 43, the corresponding wave
lengths, we have
A A» A3
Pe ts Siege = 2 Aovd = ea (465)
ee a Ay — AL,
B---.Ds aD — A3— Ay : eee)
whence
but @— 9, and ¢,—¢, are the distances of the lines A, and A,
from A, in the n® spectrum, and are by Eq. (466) directly propor-
tional to the increase in wave length of A, and A, over that of 4,.
Since the latter are constant for given lines of the spectrum, the
ratio of the increase of ¢,—¢, to ¢,—¢, must be constant. We
also see that the consecutive diffraction spectra increase in length
from the centre and will soon overlap, so that the red of the n™ will
fall on the violet of the (m + 1)”.
389, By the above method the values of the wave lengths have
ben determined with the greatest accuracy by many observers.
The agreements in measurement extend to the units of the mil-
lionths of the millimetre ; those adopted in the following table are
from the investigations of Van der Willigen in 1868, as given in
the Archives of the Musée Teyler.
256. ELEMENTS OF WAVE MOTION.
Wave LENGTHS OF THE FIXED LINE? IN THE NORMAL SPECTRUM.
plage | CE | evo ee [a tron, lene
“" |Dgcmmats oF Ma.|1 Mu. or LENGTH. pacasiresl oh Mn- He ;
Re suGias 0.00076339 1310 393. 1.60756
es 0.00071895 1391 417.3 1.60926
Bec 0.00068748 1455 436.5 1.61079
ase 0.00065655 1523 456.9 1.61252
D.......} 0.00058956 1696 508.8 1.61728
ae 0.00052704 1897 569.1 1.62353
ke tates 0.00051860 1928 578.4 1.62495
F.......| 0.00048840 2056 616.8 1.62917
Cee 0.00043113 2319 695.7 1.64006
Pe ocaivns 0.00039714 2518 155.4 1.64969
‘ 390. Polarized Light. While we have assumed in Part I
that the vibrations of the molecules of the medium which transmit
the undulations of light are transversal, nothing in the subsequent
discussion of light has so far indicated the importance of this
assumption. And were it not for the phenomena cf polarization,
we might with equal indifference have assumed these vibrations to
be in any direction with respect to the Juminous ray. Any oblique
vibration, however, can be resolved into its components, one longi-
tudinal and the other transversal, and the question of direction is
necessarily limited to a choice between them, provided the proper-
ties of polarized light demand a decision in this respect. By direct
observation we have established that the sound waves which reach
the ear are due to the longitudinal vibrations of the air, whether
the sounding body itself vibrates transversely or longitudinally.
But the waves of light are so minute, that we can never hope to
discern by direct means the infinitesimal motions of the molecular
ether. Nevertheless the marked differences in the effects produced
by the superposition of two beams of natural light and by two of
polarized light, leave no alternative but the acceptance of the
principle of transversal vibrations as a real cause of all optical
phenomena.
891. The theoretical and practical study of polarized light are
quite separate and distinct, and either may take precedence of the
RELATING TO SOUND AND LIGHT. 257
other. But as there are certain advantages gained by clearly appre-
ciating the practical differences that exist between natural and
polarized light, we will first briefly refer to some of the more
prominent characteristics of the latter.
Theoretically, light is said to be polarized when the molecular
vibrations of the ray are rectilinear and perpendicular to a plane
called the plane of polarization. The operation of polarization
consists in bringing all the molecular vibrations .n‘o this position,
or by stopping out those that do not have this direction. When
this condition is exactly fulfilled, the light is said to be wholly
polarized or plane polarized ; when otherwise it is said to be more
or less partially polarized, according to the degree of its modifica-
tion. Practically we define polarized light to be that which pos-
sesses certain peculiarities, hereafter to be described, by which it
may readily be distinguished from natural light by the application
of certain tests. To the unassisted eye there is no distinction
between natural and polarized light, and we now proceed to indi-
cate in what manner these peculiarities can be manifested.
892, Any apparatus used to change natural into polarized light:
is called a polarizer, and that which is used to ascertain whether:
light be polarized or not is called an analyzer. These are essen-.-
tially the same and are perfectly interchangeable, deriving their
names only from the particular use to which they are put.
398. The usual methods by which common or natural light is
polarized, are double refraction, reflection, and refraction.
Polarization by Double Refraction. 1°. Iceland
Spar. The wsual form of a natural crystal of Iceland spar is that
of an oblique rhombohedron, whose adjacent faces are unequal.
The oblique and acute angles of the faces are 101° 55’ and-78° 5”
respectively. The regular form of the crystal is obtained by cleay-.
age, and it is in this form when all the edges are of equal length.
The right line joining the two obtuse solid angles when the crystal
is in the regular form is called the optic aris of the crystal. The
plane normal to any face, containing the direction of the axis, is the
principal plane. If a crystal of Iceland spar be placed in the track
of a beam of solar light in any direction except that of the optic
axis, a separation of the beam into two components of equal inten-
sity will take place, and these two beams will emerge from the
17
258 ELEMENTS OF WAVE MOTION.
opposite face parallel to each other and to their original direction.
This property is called double refraction.
Let us suppose a crystal of the regular form to be so mounted
that its principal plane, at first vertical, may pass by all azimuths
around the direction of a ray. We will notice that a horizontal in-
-eident beam of solar light will always be separated into two beams,
one of which always follows the ordinary laws of refraction, Art.
79, and which for this reason is called the ordinary beam, while the
vther is not always so controlled, and is therefore called the extra-
ordinary beam ; these will be represented hereafter by the letters O
and £. On rotating the crystal we will find that in every position
of the principal plane these two beams will have equal intensities, ©
and both will always lie in the principal plane.
Now if a second crystal exactly like the first be similarly placed
near it, and the beam # be stopped out by a diaphragm after it
emerges from the first crystal, we will notice the following phenom-
ena: Calling « the azimuth angle between the principal planes of
the crystals, and supposing the second crystal only to rotate, when
« = 0 there will be but one beam passing the second crystal, whose
displacement is double that due to either crystal; as « increases
from 0 to 45°, two beams of unequal intensity will pass the second
erystal, which we will designate by O, and O,. O,, at first of
greater intensity, will gradually diminish, while O, will increase,
both being equal when « is 45°; as « increases from 45° to 90°, O,
will decrease to extinction, and QO, will increase until it becomes as
bright as O, was formerly ; as « increases from 90° to 180°, similar
changes will occur, so that O, will increase to a maximum, and 0,
decrease to extinction; as « increases from 180° to 270°, the
changes of the first quadrant will recur, and from 270° to 360°
those of the second quadrant will be reproduced. If now O be
stopped at emergence from the first crystal, and H be permitted to
pass, and we represent its components by Z, and #,, we will find,
on rotating the second crystal, that Z, will be zero at 0° and 180°,
and Z, a maximum; at 90° and 270°, &, will be a maximum and
E., zero 3 at intermediate azimuths, #, and #, will differ in inten-
sity, except at 45°, 135°, 225°, and 315°, where they will be equal.
394, In these results which are aratiged below, I represents
the intensity of the original beam, and « the angle included be-
tween the principal planes of the two crystals:
RELATING TO SOUND AND LIGHT. 259
i _ § 0, = $l cos a,
Sa {OE trem, |
LH, = 41 sin? «,
H, = 41 cos? a.
l= (467)
B=77s |
Neither O nor # is similar to natural light, for when either is
received on the second crystal, two emergent beams do not always
result; with natural light, two beams are always produced. Neither
are Oand £ similar to each other, for in certain positions of the
second crystal, O produces only an ordinary beam, and 2 only an
extraordinary beam ; in certain other positions, O produces only an
extraordinary beam, and # only an ordinary beam. They therefore
have peculiar properties depending on the position of the crystal.
But we see that O and /# have properties of the same kind with
yeference to two planes passing through their direction, the first
being the principal plane and the second at 90° from it. For these
reasons the beams O and £, and their components O,, O., #,, £.,
are said to be polarized, the O’s in the plane of principal section,
and the #’s in the plane at right angles to the principal section.
395, 2°. By a Plate of Towrmalin. All crystals which, like
Iceland spar, have but one axis of symmetry, act in the same man-
ner upon light. Tourmalin is such a crystal, and when cut into a
thin plate parallel to the axis, can be used to polarize light. We
should therefore have, 1°, an extraordinary beam £, equal to 4J,
but which by absorption is reduced to $/u‘, polarized perpendicu-
lar to the principal section; and 2°, an ordinary beam O, polarized
in the plane of principal section, and whose intensity should be
4Ju't; u' is however so small, that for very little thicknesses the
intensity of O that passes is practically negligible, and hence #
emerges sensibly free from O. If two of these plates be superposed,
E = flu* cos’ « ;
hence, when the axes are parallel,
#H= tlu*,
and when perpendicular, £ is zero.
396. 3°. By a Nicol Prism. If a crystal of Iceland spar,
having equal width and thickness, and a length three times its
260 ELEMENTS OF WAVE MOTION,
width, be cut along the plane containing the longer diagonal, and
the surfaces be polished and reunited with a thin layer of Canada
balsam, in their original position, the crystal is called a Micol prism.
The refractive index of the balsam, 1.549, lies between 1.654, that
of O, and 1.483, that of #. Hence, an incident ray of natural
light being separated into two rays O and # within the crystal, the
former will meet the surface of the balsam at an angle of incidence
such that it will be totally reflected to the side of the crystal, where
it is absorbed, while the # ray will pass through unchanged. A
layer of air serves the same purpose as the balsam, with the advan-
tage of requiring a shorter length of crystal for the prism. The
emergent ray /’ is polarized in the plane perpendicular to the prin-
cipal section of the prism.
397. By Reflection. If a small beam of natural light be
incident on a black glass mirror, the reflected light will be that due
to reflection from the first surface. If the angle of incidence be
about 54° 35‘, and the reflected beam be examined on ail sidcs by
another mirror at the same angle of incidence, we will observe that
when the planes of the two mirrors are parallel or perpendicular to
each other, the intensity of the second reflected beam is a maxi-
mum ; at positions 90° from these, the intensity is zero, and at in-
termediate positions the intensity is variable. The law which
governs the variation in intensity is given by the equation
I' = I cos’ a,
in which J’ is the intensity of the second reflected beam at any azi-
muth, Z the intensity of the same beam at the position correspond-
ing to @ = 0, and « is the angle included between the planes of
incidence of the two mirrors. The plane of first incidence is called
the plane of polarization of the beam under examination.
398. By Refraction. If the beam of natural light be inci-
dent on a bundle of thin glass plates at the same angle, 54° 35’,
and the transmitted beam be examined in the same way, it will be
found to possess similar properties, except that the equation for in-
tensity will be
f' = Tsin® «,
‘
RELATING TO SOUND AND LIGHT. 261
and the plane of polarization will be perpendicular to the plane of
first incidence.
By either of these tests, therefore, a beam of light may be ex-
amined, its properties in the above point of view ascertained, and
it may then be classified either as natural light, or polarized light,
either wholly or partially.
399. The differences between natural and wholly polarized light
are thus contrasted :
Naturau Lieut. PoLaRizED LIGHT.
Iceland spar.
Always two beams of equal
intensity transmitted in al/
azimuths of the principal
plane.
Always two beams of rarying inten-
sity transmitted, except in the
azimuths 0°, 90°, 180°, and 270°
of the principal plane, at which
there will be but one beam. In
0° and 180°, if beam is 0; andin
90° and 270°, if beam is Z.
Tourmalin and
Nicol Prism.
Always one beam of uniform
intensity transmitted at all
azimuths of the axis of the
crystal.
Always one beam of varying inten-
sity transmitted, except at azi-
muths 0° and 180°, if beam is O ;
and 90° and 270°, if beam is #.
Reflection by black
glass at an inci-
dence of 54° 35’.
Always a beam of uniform
intensity reflected at all
azimuths.
Always a beam of varying intensity
reflected at all azimuths, except
at 0° and 180°, if beam is #; and
of 90° and 270°, if beam is 0.
Refraction through
a bundle of glass
dence of 54° 35’.
plates at an inci- | -
Always a beam of uniform
intensity refracted through
the bundle at all azimuths.
Always a beam of varying intensity
refracted at all azimuths, except
at 0° and 180° for O, and 90° and
270° for Z.
400. Experiment shows that, when by any of these or other
methods light polarized in one plane is produced from natural light,
there is always produced at the same time light polarized in a per.
pendicular plane; and that light polarized in one plane cannot be
made to give light polarized in a perpendicular plane. Experiment
further shows that whenever two plane polarized beams are pro-
duced from the same source, and we derive from them by any
means other plane polarized beams whose planes are coincident,
interference takes place exactly as in the case of natural light; but
262 ELEMENTS OF WAVE MOTION.
light polarized in one plane cannot be destroyed by light polarized
in the perpendicular plane.
If light be due to longitudinal vibrations, no reason can be given
why interference should not take place independently of the posi-
tion of the planes of polarization of a ray; for the velocity of the
molecule in the direction of the ray cannot be influenced in any
way by any azimuthal change of the plane of polarization around
the ray. If, however, we suppose the vibrations transversal, it is
evident that any azimuthal change of one of the planes of polariza-
tion will give a component displacement in the direction of that due
to the other, so that the former may be combined with the latter,
and thus determine a resultant displacement of greater or less mag-
nitude, depending on the difference of phase of the components.
401. Analytical Proof of the Transversality of
the Molecular Vibrations. Let us consider a molecule of
ether on the axis of z, along which a plane polarized ray is moving
whose plane of polarization coincides with zy. Whatever be the
rectilinear vibration of the molecule, the molecular displacements
in the direction of the co-ordinate axes a, y, z, may be represented by
as t i
u = asin 2n (= — 4),
v = bsin Qn (é—¥), (468)
‘ tx
= 97 (2%
w= c sin 27 (! x),
Suppose another plane polarized ray, having a different intensity.
and difference of phase 6 at the time f, but the same plane of polar-
ization also moving along the axis of x; the component displace-
ments of the molecule will be
. t +6
u, = ma sin 27 (- — or),
: t +4
v, = mb sin 27 CG aut"), (469)
), L (470)
w = mb sin 27 (¢ _ et"), |
Combining the displacements in the same directions by Eq. (61),
then the sum of the resulting squares will be the measure of the
resultant intensity due to the concurrent effect of the two plane
polarized rays. By the application of the method of Art. 67, this
resultant will be
6
T= CLP + mee + m2? + mic? + 2a’? cos 27 i
— —v+d
— 2mbe cos grt ete -- 2mbe cos 27 ir
5 (471)
= (a + B+ &) (1 + m’) + 2me? cos ar >
gO ee.
+ 4mbe sin 27 i sin 27 ye J
402. Since experiment shows that two perpendicular plane
polarized waves can never interfere, this intensity is constant, what-
ever be the value of the difference of phase 0, which therefore exacts
as simultaneous conditions that
a=0 and ldesin2n i = 0, (472)
264 ELEMENTS OF WAVE MOTION.
The first condition shows that the displacement in the direction
of propagation is zero, and consequently that the vibrations due to
light are transversal. The second condition can be satisfied in one
of three ways, viz., either
eS aa
6=0, or e=0, or sindn =
If we suppose either 0 or ¢ is zero, the vibrations are rectilinear, and
either parallel or perpendicular to the plane of polarization; both
% and c cannot be supposed zero at the same time, without suppos-
ing both rays to be of zero intensity. The third supposition gives
as a consequence, :
a
x-y= M5»
n being any whole number, and therefore for the displacements in
the direction of y and z for the first ray,
v= O6sin an (£—¥),
(473)
paca 4, |
a
We have as a consequence of this supposition that the vibrations
are rectilinear, and parallel to a right line in the plane yz, which is
inclined to the axis of y by an angle whose tangent is -+ i This
conclusion is not admissible, because observation proves that there
is a complete symmetry of all the phenomena of polarization with
respect to the plane of polarization zy. This supposition must then
be discarded, and either one of the other two adopted ; whence we
conclude that the molecular vibrations are always transversal, and
in a plane polarized ray are either parallel or perpendicular to the
plane of polarization. The best authority as well as the strongest
experimental evidence are in favor of the latter conclusion, and
therefore we will assume the direction of the molecular vibrations
to be perpendiculay to the plane of polarization. Ag the latter
plane is readily determined, the particular direction of the molecu-
lar vibration follows as a consequence.
RELATING TO SOUND AND LIGHT. 265
408. Distinction between Natural and Polarized
Light. Let the direction of the plane polarized ray be along the
axis of 2, and the molecular vibration in the plane yz Whatever
be the direction and amount of the displacement, it may be resolved
into its components along y and z, which will always bear a con-
stant ratio to each other. Thus, if the displacement be
asin 2n4, (474)
its components along y and 2 will be
‘ t
a cos & sin ane,
: (475)
a sin @ sin Q7-, J
« being the angle which the displacement makes with y. The in-
tensity of the ray will be. a, and that of its components,
e@cota and asin? «a
Conversely, if we have two component displacements, having the
same phase at the time ¢, along y and z,
bain an, |
(476)
s t
¢ sin 27-,
=
their resultant d sin ant will be definite, and its direction makes
an angle whose tangent is ; with the axis of y.
404, Now since natural light is always decomposable, as shown
by experiment, into two plane polarized beams of equal intensity,
whose vibrations are in right lines perpendicular to each other, it
would seem from the above that their recomposition should give a
plane polarized beam whose molecular vibrations should be along a
right line in the azimuth of 45°. Experiment shows that such is
not the case, but that their recomposition gives natural light again.
In a ray of natural light, the amplitudes and phases must of neces-
266 ELEMENTS OF WAVE MOTION.
sity undergo abrupt changes after certain periods of regularity,
which are insignificant when measured by sensation, but are great
when compared with 7, and the vibrations may be sometimes along
y, sometimes along z, and sometimes in intermediate directions,
When along any right line, we may suppose the displacement re-
placed by its components, @ cos « and @ sin «, which by recompo-
sition would give the supposed displacement. But as the line along
which this is made varies each instant, each pair of components
would reproduce a displacement continually varying in azimuth ;
such are supposed to be the characteristics of natural light. But
when we consider originally the vibration of the molecule in a
polarized ray, along any right line, while its amplitude and phase
may be variable, its direction of vibration must be constant, and
any recomposition of its components will always give a constant
azimuth angle, since the ratio of the components at any instant is
the tangent of this angle. Therefore when we wish to exhibit the
effects of interference between two polarized beams, they must be
obtained from the decomposition of a single plane polarized beam,
since all the perturbations that arise in these components coming
from the common resultant, must themselves be common and con-
current in the components. The intensities and phases may be as
variable as possible, but the ratios of these intensities and the dif-
ference of phase will be constant. The analogy in this respect is
as complete as in the case of the interference of natural light, where
it was shown that these phenomena are only exhibited by two
beams which arise from a single origin, and are made to travel over
A
routes which differ by an odd number of times 3
meeting.
to the place of
405. Construction of the Refracted Ray. Resuming
‘the discussion of plane waves of homogeneous light in different
media, since the velocity of propagation varies directly as the square
root of the elastic force developed by the displacement, Eq. (96),
when the elastic forces are known we can readily determine the
velocities in the three cases to be considered.
1°. Isotropic Media. In such media, a = 6 =, and the
wave surface becomes a sphere whose equation is
C+P+t P= a (47%)
RELATING TO SOUND AND LIGHT. 2607
Let us suppose the incident medium in all cases to be air, and
the velocity of light in it to be unity. To construct the refracted
ray, let MM’ (Figure 94) be a
plane deviating surface sepa- s t
rating air from the isotropic od
medium considered ; let SI be 1
the incident ray, and with
radii of a@ and wnity describe
civcles IR and IA about I as a
centre. Prolong SI to A, and
from B, where the tangent at ee 94.
A intersects MM’, draw BR;
IR will be the refracted ray and BR the refracted plane wave. For
from the figure we have
: 1 ‘ a iis
sin IBA = IB and sin IBR = IB: (478)
sin IBA =“ sin IBR, or sing =psing’, (479)
Hence, in all isotropic media it is readily seen that the refracted
ray obeys both of the laws of Descartes.
406. 2°. Uniaval Crystals. Let us suppose 6 = c; then,
Art. 110, the wave surface becomes a surface of two nappes, one
being a sphere with radius 6, and the other an ellipsoid of revolu-
tion about the axis x, given by the equations,
+P P= BB ]
ae + BP (y? + 2) = ee. \ (H0)
In every direction, except along the axis of 2, there will be a
separation of a single incident ray into two refracted rays. each
moving with different velocities. The axis of 2 is called the optic
axis of the crystal, and because of this one direction of single ray
velocity, these crystals are called wniaral. Whenever the plane of
incidence is parallel or perpendicular to the principal section, the
two refracted rays will be found in the plane of incidence, and the
first law of Descartes will be obeyed by both rays; in every other
plane of incidence, one of the refracted rays will lie outside of the
26S ELEMENTS OF WAVE MOTION.
plane of incidence and obeys neither of these laws. In the first
case, the construction of the refracted rays is a problem of Plane
(zeometry, and in the second of Solid Geometry. Both refracted
rays are plane polarized, in planes at right angles to cach other;
that one which is polarized in the plane of principal section is the
ordinary, and the other is the extraordinary ray.
407. To illustrate the construction in the simpler cases, let the
plane of incidence contain the
optic avis PP’, Fig. 95, and let
SI be the incident ray. From I
as a centre, with radius unity,
construct as before the circle
TA. and with radius 6 the ordi-
nary wave 10. About PP’ as
an axis, construct the section of
the extraordinary wave IE; it
will be an ellipse, whose semi- Figure 98,
axes ure Banda. Drawing tan-
gents from B to O and E, we have IO and IE, the refracted ordi-
nary and extraordinary rays respectively, and for the plane waves
BO and BE respectively.
408. Let the plane of incidence be perpendicular to the optic
axis. Then, in Fig. 96, the sections of the wave surface are circles
with radii 6 and @ respectively,
and both O and E obey both :
laws of Descartes; but the index Se
i
of refraction of E is here tom i! —B vt
7? ” Re
which is less than : or that of O.
The index of refraction of E
!
!
|
'
!
1
|
'
!
varies between its maximum d
b’ Figure 96.
‘ = 1 :
and its minimum a depending upon the position of the incident
ray: but since when @ is known its value for any particular inci-
dence can he readily found, the extraordinary index is generally
designated by *.
RELATING TO SOUND AND LIGHT. 269
409. Let the optic azis lie in the plane of incidence and in the
deviating surface ; then the construction is shown in Fig. 97. In
this, as well as in the preceding
cases, the ordinary ray is nearer s |
the optic axis than the extra-
ordinary ray. In all uniaxal «| !
erystals in which the spheroid —
is oblate, the ordinary ray will
lie nearer the optic axis than
the extraordinary ray, and they
are called negative crystals; in
all other uniaxal crystals, in Figure 97,
which 8 > a, and hence the
spheroid becomes prolate, the contrary will be the case, and such
crystals are called positive crystals.
410. 38°. Biaxal Crystals. These comprise all crystals in
which the elasticities differ in three rectangular directions. The
wave surface is that given by Eq. 146, whose more important prop-
erties have been deduced in Part I. When the plane of incidence
is normal to one of the three axes of elasticity, one of the sections
of the surface by this plane is a circle, and hence one of the rays
follows the law of the sines; since three such planes may be passed,
the three indices of refraction, , so obtained are called the
il
a’ b?
principal indices of the crystal. They can be experimentally de-
termined by the method of minimum deviation, by means of three
prisms cut parallel to the three axes of the crystal, and when once
obtained suffice to define the optical qualities of the crystal.
411. Interior Conical Refraction. Possibly the most
remarkable example of the complete accord of the predictions of
theory with the results of experiment is the deduction of Sir Wil-
liam Hamilton, from a mathematical study of the properties of
Fresnel’s wave surface, It is seen (Fig. 18) that the tangent planes
MN, etc., are each tangent to the wave surface along the circum-
ference of a small circle, parallel to the circular sections of the
ellipsoid, Eq. (180), and hence it follows that a single ray of natural
light, incident externally in a direction such that after refraction it
coincides with the optic axis OM, it should be separated into an
270 ELEMENTS OF WAVE MOTION.
infinite number of rays within the crystal, taking the form of a
hollow cone. Therefore, should the emergent face of the crystal be
parallel to the incident face, this cone of rays should emerge as a
hollow cylinder of rays, parallel to the original incident ray.
412. Exterior Conical Refraction. Again, when any
two rays pass into a biaxal crystal in a single direction, their veloci-
tics are in general different, and are represented by the radii-vectores
of the wave surface, while their directions on emergence are deter-
mined by the positions of the tangent planes to the two nappes of
the wave surface at their points of intersection. But in the case of
the ray OI, the two radii-vectores unite, and the rays have the same
velocity. There are, however, an infinite number of tangent planes
to the wave surface at the point I, and therefore a single ray of
natural light, taking the direction IO within the crystal, should
emerge into the air as a cone of rays, whose directions and planes of
polarization depend on the position of the tangent planes.
Dr. Humphrey Lloyd completely established the truth of these
predictions, by experiment upon the biaxal crystal arragonite, and
thus furnished one of the most unexpected proofs of the general
correctness of the basis upon which Fresnel had established the
laws of double refraction. Fig. 98 illustrates the first case, or that
of interior conical refraction, and Fig. 99 that of exterior conical
refraction.
s Figure 98,
413, Mechanical Theory of Reflection and Re-
fraction. Fresnel, to whom we owe the solution of the problem
of determining the intensity and position of the plane of polariza-
tion of reflected and refracted light, has based his analysis upon
the following suppositions, viz. :
RELATING TO SOUND AND LIGHT. 271
1°. That the motions of the molecular ether in and near the
surface of separation of two media are the same in each, and there-
fore that the velocities parallel to the separating surface are equal.
2°. That the kinetic energy in the incident wave is equal to the
sum of the kinetic eiiergies of the reflected and refracted waves.
3°. That the elisticity of the ether in and near the surface of
separation is the same for both media, and therefore that the densi-
ties of the ether molecules in the two media are directly as the
squares of the refractive indices.
Although there is no sufficient reason a priori why these suppo-
sitions should be accepted unquestioned, the experimental verifica-
tions of the deductions which flow from them undoubtedly furnish
ample grounds @ posteriori of the correctness of the assumptions.
Several of the most important laws, which had before been reached
by experiment, are shown to be the direct consequences of the
theory.
414, 1°. Reflection of Light polarized in the Plane of
Incidence. The motion of the molecules are in this case normal
to the plane of incidence. Let w and v represent the velocities of
the molecules in the reflected and refracted rays respectively, and
let the velocity of the molecules in the incident wave be unity. By
the first principle we have
lu =v; (481)
and: by the second, m (1)? = mu? + m'v*; (482)
m—im' 2m
whence, =e a Dy ac ae? (483)
which are identical with the expressions for the
velocities of two perfectly elastic balls of mass m
and m’ after impact, unity being the velocity of
m before impact. Let AM and MC (Fig. 100)
represent the velocities and directions of the inci-
dent and refracted rays in the two media. The
corresponding volumes of the ether put in motion Figure 100.
by them in the two media are directly as the areas
MB and MD; MB being that for the incident and also for the re-
flected wave, and MD that for the refracted wave.. Or, if ¢ and 9’
der. i? '
252 ELEMENTS OF WAVE MOTION.
represent the angles of incidence and refraction, these volumes are
directly as .
sing@cos@ and _ sin ¢’ cos @’.
By the third supposition, the densities of the ether of the two
media are as ; ;
1:2, oras_ sin? ¢’: sin? ¢.
Whence, for the ratio of the masses, we have
sin cos ¢ sin? ¢’ : sin @' cos @' sin? @ :: mim’;
m' _ tang o.
m — tan ¢" (484)
or
Substituting this value in Eq. (482), and eliminating v by Eq.
(481), we have
l1—wu _ tan@
1+u~ tan ° a
_ _ sin (@—@')
whence, Wo gin (ee)? (486)
2 2 cos @ sin o
See ae eae ‘sin (@+ 4’)
415, 2°. Reflection of Licht polarized Perpendicularly
to the Plane of Incidence. In this case, since the molecular
vibrations are all in the plane of incidence, and are besides normal
to the directions of the rays themselves, we have, representing by
‘and v’ the molecular velocities in the reflected and refracted
waves, :
1 =
(1 + uw’) cos ee. (481)
m (1)? = mu? + m'v?;
1—w! uw’ sin @ Cos d
1+’ ~ sin o' “COs d
whence, ne (488)
and therefore, w’ = tan (® — ¢')
~~ tan (9 + ¢')’
"i 2 sin ¢’cos
sin (@ + $') cos (6— @’)
RELATING TO SOUND AND LIGHT. 273
416. Let the intensity of the incident light in the two cases be
represented by wnity ; then the intensities of the reflected and re-
tracted components will be w? and 1— wv, and w? and 1— uw’,
respectively. Developing the values of « and wu’, and replacing
sin ¢’ and cos ¢' by their values in terms of @ and p, we readily
obtain
ue
= sin? (¢—9') __ (4 — sin? @ — cos S), (490)
sin? (@ + ¢’) V2 — sin? @ + cos o/”
2(d@ — a 2 eine & a 2
fe ee). (ata fae ). (491)
tan? (¢ + 9’) V/ 2 — sin? @ + w2 cos o
from which we can obtain the values of 2 and 2’ for all values of ¢.
For ¢ = 0, we have
=CEi) at = Sy)
w= (SO and ow? = ei
therefore, for normal incidence, the quantity of reflected light is
the same in the two cases. As @ increases, uv? will continually in-
crease, until when ¢ = 90°,
wool;
whence the reflected light is equal to the incident, when the inci-
dent beam is polarized in the plane of incidence. But 2’? does not
increase continuously for an increase of @; it decreases from
es 2
(; ¢ ‘) for @¢ = 0 to zero for ¢ = tan! «, and then increases to
unity as @ increases to 90° from this angle. ‘There will then be no
reflected light when the incident light is polarized perpendicularly
to the plane of incidence and the angle of incidence is tan yp. We
also see, since for @ = 0,
2— 42 —.
2 =u? = (: 7?
that the intensity of the reflected light is at this incidence inde-
pendent of the plane of polarization of the incident light.
417. 3°. Reflection of Natural Licht. Natural light is pro-
duced by the vibratory motions taking place successively in every
direction on the surface of the wave. Each of these motions may
18
274 ELEMENTS OF WAVE MOTION.
be resolved into two others polarized in planes at right angles to
each other. However these components may vary at every instant,
the rapidity of these variations is such that in any appreciable time
the sums of the component intensities are exactly equal in the two
planes, and therefore we may regard a ray of natural light as being
equivalent to two rays of equal intensity polarized in planes at right
angles to each other. If we consider the component rays to be
polarized, one in the plane of incidence and the other in a perpen-
dicular plane, we may apply the conclusions deduced above to these
separately, combine the results, and then find the intensities of the
reflected and refracted components of the natural ray. We will
therefore have, for the intensity of the reflected components,
1, 1,si (O—9)
2° 2 sin? (¢ + ¢’)’ (492)
1 1 tan®(6—¢’),
and a= 2 tant(@ +o)?
and for the intensity of the reflected component of the natural ray,
; (wv + w) ;
and similarly, for the intensity of the refracted component,
5 (08 +o?) = ; (1 — w*) +- 5 — wu’).
Since u > uw’, we see that the reflected ray is not of the same
nature as natural light, and for the same reason neither is the re-
fracted ray. In the former there will be an excess of light polarized
in the plane of incidence, which is measured by the difference of
the component intensities, $(w? — w?), and in the latter there will
be an excess of polarized light in the plane perpendicular to the
plane of incidence, measured by the same difference. We therefore
see that the reflected and refracted rays of an incident natural ray
contain equal quantities of polarized light, but polarized in planes
at right angles to each other. This conclusion, which has been ex-
perimentally demonstrated by Arago, is therefore a consequence of
Fresnel’s theory. The operation of reflection and refraction then
results in partial polarization of natural light.
RELATING TO SOUND AND LIGHT. 275
The reflected light will be completely polarized when one of the
portions of which it consists disappears. But the first portion
which is polarized in the plane of incidence cannot disappear, for
we have shown that the value of its intensity constantly increases
—12
as ¢ increases, from (—) to unity when ¢ = 90°. But the
second portion vanishes when (¢ + ') = 90°; this relation gives
cos @ = sin ¢’ = = or tan@g=up
This law, that the tangent of the angle of maximum polar-
ization is equal to the index of refraction of the mediwm,
was experimentally discovered by Brewster, and is another conse-
quence of Fresnel’s theory. For glass this angle is about 54° 35’,
and is the same that was referred to in Art. 397, in polarization by
refléction.
When ¢ +9’ > 90°, the expression for w’ changes its sign,
which is equivalent to a change of phase of 180° as the angle of in-
cidence passes the polarizing angle.
418. Change of Plane of Polarization. If « repre-
sent the angle made by the plane of polarization of a plane polarized
ray with the plane of incidence, we may suppose the vibration
resolved into two components, one in the plane of incidence © cos a,
and the other, w sin «, perpendicular to it. After reflection, these
become
sin (p — ¢’)
— @ COS & sin (@ + @) ’ ,
tan (9 — 4 me
and —wo sine —-—
fan (6 + 9°)’ )
respectively. Calling «' the angle of the resultant vibration, we
have (
cos ($ + #) 9
cee ey 7H)
from which the value of «' can be determined. If « = 0, then
«' = 0; andif « = 90°, «' = 90°; therefore, when the plane of
polarization of the incident ray either coincides with or is perpen-
dicular to the plane of incidence, the plane of polarization of the
tan «e' = + tan e
276 ELEMENTS OF WAVE MOTION.
reflected ray is unchanged. When $+ ¢’ = 90°, then « = 0,
and the plane of polarization of the reflected ray coincides with the
plane of incidence, whatever be the azimuth of the incident ray.
In all other cases, the plane of polarization of the reflected ray ap-
proaches the plane of incidence, since in these cases e' << «@ Ina
similar manner, it may be shown that the plane of polarization of
the refracted ray recedes from the plane of incidence, the law of
this recession being expressed by the formula
cot «' = cot « cos (¢ — ¢’). (495)
When the refracted ray meets a second surface parallel to the for-
mer and emerges into the air, the azimuth of the plane of polariza-
tion of the emergent ray is given by
cot «'’ = cot «' cos (¢' — ¢) = cot « cos? (o —'). (496).
The methods of complete polarization by successive reflection and
refraction are therefore simple consequences of Fresnel’s theory.
419. Elliptic Polarization.— Let us suppose the direction
of two plane polarized rays to be along the axis of # and their
planes of polarization to be at right angles to each other, and their
phases when they reach any molecule to differ. Then
y = Osin (an! —8), z= e¢sin (an=—y), (497)
will represent their amplitudes at any time ¢. From these we have
sin = ane —fB, and sin = ant —y3 (498)
ee
whence (8 —y) = sin: — sin
c
Taking the cosines of both members and reducing we have
e
2
pt a— 2% cos (8 — y) = sin? (6 — y); (499)
be
whence we sce that the molecular orbit is an ellipse referred to its
centre and a pair of rectangular co-ordinate axes. If the compo-
RELATING TO SOUND AND LIGHT. 277
nent rays are in the same phase or in phases differing by 180°,
we have
7
Seo i+ af = 0; (500)
y_ 2
whence = + o (501)
or the orbit is a right line passing through the place of rest of the
molecule, aud the resultant ray is plane polarized.
When the component rays differ in phase by 90° and the vibra-
tions are of equal amplitude, the orbit becomes a circle given by
Y+te=eR (502)
The polarization in these cases is said to be elliptical, plane,
and cireular respectively. In elliptical polarization the direction
of the axes of the ellipse and the ratio of their lengths can be
determined by observation. Thus, when such a beam is analyzed
by a double refracting crystal whose principal section coincides
with one of the axes of the ellipse, the component plane polarized
beams have their maximum and minimum intensity; and when
the plane of principal section is inclined to either axis by an angle
of 45°, the two components are of equal intensity.
When a plane polarized ray undergoes reflection, the reflected
ray is generally elliptically polarized ; for the incident plane polar-
ized ray may be supposed resolved into two rays, polarized respec-
tively in the plane of incidence and in the perpendicular plane ;
the effect of reflection changing in general the phase of the vibra-
tion of the components unequally, when these are united the
resultant ray is in the condition of that discussed above ; that is,
their vibrations are rectangular and differ in phase on the molecule
at the place of meeting.
420. Reflection of a Polarized Ray at a Surface
separating a Denser from a Rarer Medium. Resum-
ing the expressions for the reflected molecular velocities for rays
polarized in and perpendicular to the plane of incidence,
sin (o— @ ?') st tan (¢ — o —¢), (503)
sand w= — ——-
~ sin (p + 4)’ tan (@ + 9')’
27 ELEMENTS OF WAVE MOTION.
wD
and considering the incident ray in the denser medium, we see
that when ¢' = 90° both w and w' become equal to unity. In this
case, since there is no refracted ray, the intensity of the reflected
ray should be equal to that of the incident. But when ¢@ has
greater values than that corresponding to @' = 90°, or that of total
reflection, these expressions become, by substituting « sin @ for
sin @ and /—1-/(p? sin? @ —1) for cos ¢’.
__ wsin ¢ cos ¢ — sind V—1- Vv? sin? ¢ — 1)
~ psin d cos ¢ + sino V—1- V(u? sin? ¢ — 1)
— (504)
sin ¢ cosp+psin ¢ /— 1+ /(p? sin? ¢ — 1)
sing cos@+usingd /—1- V (2 sin? ¢ —1)
2 gin? @ —
Placing tan 0 = vas ue
ft COS s
Be , (505)
and in = EO wit ang 1)
cos =
the above expressions may be written
wu = cos 26 — /—1-sin 20 )
(506)
== a
au’ = cos 26’— /—1- sin 26'J
421, These imaginary expressions suggest of themselves no
reasonable interpretation. Fresnel, however, from the analogy of
certain geometric cases in which the multiplication of the length
of a line by the imaginary expression ~/— 1 indicates a change of
direction of the line equal to 90°, assumes in these cases that the
phase of the vibration into which »/—1 enters as a factor has
undergone an increase of 90°, and therefore the expression
ean 9-
(cos 26 — »/—1 sin 26) sin = (vt — x) (507)
is to be interpreted as
lacs tei ; « [ar é
cos 26 sin a (vt — z) — sin 20 sin [= (vt — x) + 90 ]608)
RELATING TO SOUND AND LIGHT. 279
which may be written
. [27
sin [= (vt — 2) — 26|. (509)
Similarly the other expression may be written
. far ;
sin |= (vt — 2) — 20 | (510)
Representing the difference of phase of the component rays by 6,
we have
es _ cos @ V/(u? sin? 6 — 1)
tan 40 = tan (6 _ 6) = ao ; (511)
L—tan? 30 sint @ — (1 + #’) sing +1
1+ tan? 46 (1 + pw) sin? od — 1
cos 6 = 3 (512)
whence 6=0 when sing = : or =1.
If we assume 6 == 45°, we have
Que = ao
(1 + #) cosec? @ — cosec! @ 1+ V43 (oe)
and taking » = 1.51, the refractive index for crown glass,
@ = 54° 87' 20", or 48° 37’ 30".
422. Hence, if light be incident internally on
the surface of crown glass at either of these
angles, the phase of the vibrations in the plane
of incidence is accelerated more than that of the
vibrations perpendicular to this plane by 45°.
And if light be twice reflected in the same plane
of incidence, the difference of phase will be 90°.
Fresnel’s rhomb, Fig. 101, is constructed of crown Figure 10t.
glass, having the angles A and C each 54° 37’.
Then light incident perpendicular to the face AB will be totally
reflected at E and F and emerge normal to CD, or parallel to its
original direction. The two reflections at E and F will accelerate
the phases of the vibration parallel to the plane of incidence 90°
more than those perpendicular to the same plane.
280 LEMENTS OF WAVE MOTION.
423. Let.« be the angle made by the plane of polarization of
any incident plane polarized ray, with the plane of incidence in
Fresnel’s rhomb. Then the vibrations perpendicular to the plane
of incidence are represented by
«
a cos « sin = (514)
and those parallel to the plane of incidence, since they are acceler-
ated by the angle 9, are
asin « sin (= + 6). (515)
Let y be the ordinate of the disturbed molecule in the plane of
incidence, and z the ordinate in the perpendicular direction, both
measured from the place of rest of the molecule. Then,
1°. If « = 45° and 0 = 90°, we have
P= ero, z= avy sin =, (516)
2
and Pres 5 ; (517)
hence the resulting ray is circularly polarized.
2°, If « have any value and d = 90°, we have
p art
y = asin « cos a “= acos @ sin ae (518)
2 2
and mee See ay
o sive @cot a yy en)
hence the resulting ray is elliptically polarized, and the axes of the
ellipse are w sin & in the plane of incidence and a cos « perpen-
dicular to that plane.
3°, If « and 6 have any values in general, we have
3 =
y=asin« (sin = cos 6 + cos - sin 5) ; (520)
Z=ac0Sesin 2 ; (521)
7
RELATING TO SOUND AND LIGHT. 281
whence we have by combining and reducing,
(y — z tan « cos 0)? = a@* sin? « sin? d — 2 tan® @ sin? d; (522)
which is the equation of an ellipse whose axes are inclined to the
plane of reflection depending on « and 0.
4°. If « = 0 or « = 90°, the reflected ray retains its plane of
polarization unchanged.
Therefore we see that by means of a Fresnel’s rhomb, whenever
the plane of polarization of the incident plane polarized ray makes
angles with the plane of reflection of 0°, 90°, 180°, or 270°, the
plane polarized light is not altered ; when this angle is 45°, 135°
225°, or 315°, the emergent light is circularly polarized, and for alli
other values of a the emergent light is elliptically polarized.
424. Circularly Polarized Light. Tf circularly polar-
ized light be examined by an analyzer, it is evident that since it
can be resolved into two equal vibrations parallel and perpendicular
to any plane, it will present from this method of examination none
of the peculiarities of ordinary polarization ; indeed it can not in
this way be distinguished from natural light. Llliptically polar-
ized light would, however, present different intensities on different
sides, although in no position would the reflected ray reduce to
zero intensity. If circularly polarized light be made to undergo
two more total reflections in the same plane and at the same angle
by transmitting it through a second rhomb placed parallel to the
first, it will emerge plane polarized and its plane of polarization
will be inclined 45° on the other side of the plane of incidence.
Since the two additional reflections increase the difference of phase
of the two portions from 90° to 180°, and the resultant of two
equal vibrations at right angles to each other, whose phases differ
by 180°, is a rectilinear vibration whose direction bisects the sup-
plement of the angle formed by their directions.
If, in Fresnel’s rhomb, «@ = — 45°, the transmitted light will
also be circularly polarized ; but in this latter case the motion of
the molecule will be left-handed if that due to « = 45° was
right-handed, and the reverse. Two such beams are said to be
oppositely polarized, and the same distinction exists with respect
to elliptically polarized light.
282.~ ELEMENTS OF WAVE MOTION.
Ay 425. Interference of Polarized Light. If a plane
polarized beam traverses a uniaxal crystal cut parallel to the optic
axis of the crystal, it will in general be separated into two beams of
unequal intensity. These components will have different velocities,
and hence on emergence from the crystal will be in different phases.
If the plate be thin and its incident and emergent faces parallel,
the separation of the beams will be insensible. The two emergent
beams being polarized in perpendicular planes can not interfere,
but they may each be subdivided into two other plane polarized
beams by means of an analyzer, having their planes of polarization
coinciding with the principal section of the analyzer and at right
angles to the principal section. One pair of these components
that have the same plane of polarization will then conspire and the
other pair will be in opposition ; the particular color for which
this interference takes place will vary according to the difference
cf route travelled in passing the crystalline plate, and therefore
depends on the relation of its thickness to . for that color.
426. In Fig. 102, let PP be the trace of the plane of polariza-
tion of the incident beam on the face of the crystal, 8S the trace of
the principal section of the crystal, and
AA the trace of the principal section of
the analyzer on the same face ; let @ be the
angle PCS, and 8 the angle POA. Repre-
sent the amplitude of the vibration of the
incident beam by unity; then the ampli-
tudes of O and E will be represented on
emergence from the crystal by cos @ and
sin « respectively. The components of O
by the analyzer will be
Figure 102
O, = cos « cos (« — 8),
and O, = cos « sin (¢ — 8),
and of E will be E, = sin « sin (« — 8),
and E, = — sin « cos (« — B).
The diagram of the decomposition of the vibration in the figure
will illustrate these values, if it be supposed turned in azimuth 90°
RELATING TO SOUND AND LIGHT. 288
around C, since the vibrations are normal to the planes of polariza-
tion of the beams. The components O, and H, are polarized in the
plane AA and form the ordinary emergent beam, while O, and E, are
polarized in the plane aa, forming the emergent extraordinary
beam. The beams O, and E, are not in the same phase, since the
former has undergone ordinary refraction in the crystal and ordi-
nary refraction in the analyzer, while the latter has undergone
extraordinary refraction in the crystal and ordinary refraction in
the analyzer; therefore, since the velocities of the ordinary and
extraordinary ray are unequal, the two beams on leaving the
analyzer will not in general be in accord. One will be retarded
with respect to the other, and depending on the amount of this
retardation, they may add or partially destroy each other’s effects,
or completely interfere. The whole of this difference is produced
while the beams O and E are passing through the crystal, for in the
analyzer both components considered, experience only ordinary
refraction. Let d be the equivalent difference of route in air, due
to the different velocities of O and E in passing through the crystal ;
then, from Eq. (76), we have for the intensity of the resultant of
the pairs O, and E,, and O, and E,, the general expression
« _— a? + a" 4+ Qee' ee" cos and
(523)
= (@ — a? + 2e/a" (1 + cos 27 “. }
Replacing @, a’, a, by I,, O,, E,, and by L, O,, E., we
have, after reduction .
I, = cos? 8 — sin 2e@ sin 2 (@ — 8) sin? + a. (524)
7 I, = sin? @ + sin 2@ sin 2 (@ — GB) sin’ a . (525)
427, These equations émbody all the principles of interference
of polarized light, and we readily deduce as consequences from them,
1°, That the intensities of the two beams are complementary,
since I, + I, is unity; and when the incident light is white, the
colors of the two beams are complementary, since their superposi-
tion gives white light.
2st ELEMENTS OF WAVE MOTION.
2°, That the two pencils are white, whatever be the retardation
within the crystal, when
sin 2e sin 2 («@ —B) = 0;
therefore, when « is 0°, 90°, 180°, or 270°, or the principal section
of the crystal is either parallel or perpendicular to the plane of
polarization of the incident beam, and when « — P is 0°, 90°, 180°,
or 270°, or the principal section of the analyzer is parallel or per-
pendicular to the principal section of the crystal.
3°. That the color is most intense when
sin 2e sin 2 (« —B) = 1,
or when « = 45° and B = 0° or 90°; that is, when the principal
section of the crystal is turned in azimuth 45° from the plane of
primitive polarization, and the principal section of the analyzer is
either parallel or perpendicular to the same plane.
4°. When @ and 6 are unchanged, I, and I, vary only with
being, for a given value of A,
but d is proportional to the thickness of the crystal, its value
(u, a 7) t,
in which p, and p, are the principal indices of the crystal, and ¢ the
thickness. When ¢ is increased so that d is increased by a whole
number of times A, the intensity resumes its original value. When
d contains 4 many times, or the crystal is thick, the colors become
superposed on each other, and the resulting light is white, just as
in the case of Newton’s rings. The thicknesses between which
these effects are displayed are in selenite, from gjgq to zy of an
inch, and similarly for other crystals.
428. Colored Rings in Uniaxal Crystals. Let
ABDC, Fig. 103, be a uniaxal crystal cut perpendicular to the axis,
and suppose a converging cone of polarized light be transmitted
through it. We will also suppose that after it emerges from AB it
is received by an analyzer, as in the previous case. Let PO be the
direction of the axis. The ray PO will not be separated within the
crystal, and emerges unchanged in polarization. If its plane of
polarization is parallel to the principal section of the analyzer, it
RELATING TO SOUND AND LIGHT. 285
will be transmitted, and if perpendicular it will not. All other rays
will be modified, depending on the angle they make with the optic
axis and the position of the
principal section of the
analyzer. Let the circle
MN’M' be the section of the
emergent cone of rays, and
MM’ and NN’ the traces of
the plane of primitive po-
larization and the perpen- Figure 103.
dicular plane respectively.
All the rays which emerge on these lines will not be separated into
two within the crystal, nor will their planes of polarization be
altered. If the principal section of the analyzer coincides with
either of these planes, the corresponding rays will be transmitted,
and if perpendicular will not be transmitted. In the former case
there will be a white cross, and in the latter a black cross. Rays
which emerge at other points, such as L, will be separated into two
within the crystal, whose planes of polarization will be parallel and.
perpendicular to OL, since such principal sections OL do not coin-
cide with the primitive plane of polarization. The analyzer reduces
each of these vibrations to the same plane, and interference will
take place. The character of the interference will depend on the
difference of phase, and hence on the interval of retardation.
429, The equation which expresses the law of the retardation
can be obtained from Eq. (194), by making it applicable to uniaxal
crystals. Thus, supposing a > 6 and =e,
wi =u’ =U;
and representing the velocity of the extraordinary ray by r, that of
the ordinary being 8, we have
2__ Be
5 2 oi gin? w. (526)
In the case considered, the angle LEO is very nearly equal to w,
and therefore all rays that meet the emergent face of the crystal,
making sin « constant, will experience equal retardations. Taking
all possible values of sin u as radii, we will have concentric rings
of color, separated by dark rings surrounding O, and intercepted by
286 ELEMENTS OF WAVE MOTION.
the bright or dark crosses referred to above, depending on the posi-
tion of principal plane of the analyzer.
430. Biaxal Crystals. Let us suppose such a crystal cut
perpendicularly to the mean line, that is, the lme bisecting the
angle made by the directions of equal wave velocity. The values of
the retardations of ray velocities is given for these crystals by Eq.
(194). or
pe
1 =e 1) ee
a @) Sin w sin 0". (527)
The curve corresponding to equal retardations will no longer
be a circle, but the lemniscate, whose property is that the product
of the radii-vectores drawn from two fixed poles to any point of the
curve is a constant.
Figure 104
Figure 104 represents the system of curves of this character sur-
rounding the poles I and I’. The directions of the vibrations of
the two rays at any point M, M, etc., of either of these curves is
readily seen, from Art. 130, to be in the bisectors of the angles
formed by joining I and I’ with the point M, and extending the
lines beyond M. The form of the dark brushes which cross the
entire system of rings is determined by the law which governs the
planes of polarization of the emergent rays. In general, two such
dark curves pass through each pole, which theory and experiment
show to be hyperbolas having a common centre on the mean line.
RELATING TO SOUND AND LIGHT. 287
431. Chromatic polarization furnishes valuable assistance to
crystallography. It enables the observer to determine whether the
crystal is uniaxal or biaxal, to find the positions and angles of in-
clination of its axes, and indeed almost to view the interior struc-
ture of the crystal itself. The properties of color by double refrac-
tion belong also to non-crystallized media when they have been
subjected to changes by which their elasticities have been altered in
certain directions. These changes may be brought about by me-
chanical pressure, or by sudden heating and cooling, and are exhib-
ited in unannealed glass, or glass subjected to pressure or heat. The
curves of color displayed when such substances are placed between
the polarizer and analyzer are similar to those of crystals, and re-
veal the localities of equal wave retardation. Such phenomena
belong to what is called aecidental double refraction.
432. Rotatory Polarization. When a homogeneous ray
of polarized light traverses a crystal of quartz in the direction of the
axis, its plane of polarization is found to be turned about the ray
from right to left,*or the reverse, depending on the particular speci-
men employed. This phenomenon is called rotatory polarizution,
and the crystal is said to be right or left handed, according as the
rotation is in the direction of or contrary to the motion of the hands
of a watch as we look toward the crystal. It is due to the inter-
ference of the two opposite circularly polarized rays into which the
plane polarized ray has been separated in its passage through the
crystal. It is easily seen from what precedes that a plane polarized
ray is equivalent to two circularly polarized rays of half the inten-
sity, in which the vibrations are in opposite directions. Each is
transmitted with a different velocity, and on emergence will unite
into a plane polarized ray with its plane turned through an angle, due
to the difference of phase of the components arising from the retarda-
tion which one of the components has experienced over the other.
It is found that the rotation of the plane of polarization is different
‘or different colors. Thus, fora plate of quartz 1mm. in thickness,
the angle of rotation of the extreme red was found to be 17§°, and
with an increasing value up to the violet, whose angle was 44°.
433. When white light is used, each simple ray on emergence
will have turned through an angle «, which is variable for each
color; and if B be the angle which the principal section of the
288 ELEMENTS OF WAVE MOTION.
analyzer makes with the plane of primitive polarization, the inten-
sities of the O and E components of each simple ray will be propor-
tional to cos? (@ —) and sin? (« — 8), respectively. The tint
formed by the mixture of these colors will be determined by these
intensities, and hence, if a double refracting analyzer be used, the
O and E components will be differently colored, but comple-
mentary. With a Nicol as an analyzer, among the tints which
succeed each other as it 1s turned, there is one called the ¢int of
passage or transition tint ; it is a pale violet blue, and can always
be easily recognized, for upon turning the Nicol the slightest de-
gree, a clear blue is displayed in one direction, while in the contrary
direction a vivid red is shown. By means of this tint the deviation
of the plane of polarization can be found with the greatest precision.
434, While it seems natural to admit that the cause of rotatory
polarization in quartz and other crystals in which this property
exists is due to the regular arrangement of their molecules, this
supposition fails to account for its existence in certain vapors and
liquids, and therefore it is supposed to be due in the latter to the
individual action exerted by the molecules themselves.
435, The application of rotatory molecular polarization has an
important field in determining the state in which the molecules of
an active substance is found, when dissolved in an inactive liquid;
a state in which chemical analysis is powerless to discern. One of
the most important practically, because of its commercial value, is
its application to saccharimetry. The rotatory power of saccharine
solutions furnishes a rapid and sure method of detecting the pro-
portion of cane sugar which they contain, even when this sugar is
mixed with other kinds whose rotatory power is different.
436. Our limits prevent a further allusion to these interesting
subjects. Those which are of necessity omitted will be referred to
in the accompanying lectures as far as time will permit. It is
hoped that the slight insight here given will at least excite interest,
and encourage subsequent study in those who may hereafter have
the proper opportunities offered them. We have seen how varied
and intricate are the phenomena, whose explanation flows from the
simple fundamental principles of Mechanics, and can recognize the
essential importance of gaining a more intimate knowledge of the
mathematical processes by which these principles are able to ex-
pound the wonderful phenomena of natural science.
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