Chapter XIII: Appendix: ⋄

NOTE A (see p. 21).

The distinction between the individual wave-velocity and a wave-group velocity, to which, as stated in the text, attention was first called by Sir G. G. Stokes in an Examination question set at Cambridge in 1876, is closely connected with the phenomena of _beats_ in music.

If two infinitely long sets of deep-sea waves, having slightly different wave-lengths, and therefore slightly different velocities, are superimposed, we obtain a resultant wave-train which exhibits a variation in wave-amplitude along its course periodically. If we were to look along the train, we should see the wave-amplitude at intervals waxing to a maximum and then waning again to nothing. These points of maximum amplitude regularly arranged in space constitute, as it were, waves on waves. They are spaced at equal distances, and separated by intervals of more or less waveless or smooth water. These maximum points move forward with a uniform velocity, which we may call the _velocity of the wave-train_, and the distance between maximum and maximum surface-disturbances may be called the _wave-train length_.

Let _v_ and _v′_ be the velocities, and _n_ and _n′_ the frequencies, of the two constituent wave-motions. Let λ and λ′ be the corresponding wave-lengths. Let V be the wave-train velocity, N the wave-train frequency, and L the wave-train length. Then N is the number of times per second which a place of maximum wave-amplitude passes a given fixed point.

Then we have the following obvious relations:—

_v_ = _n_λ, _v′_ = _n′_λ′, N = _n_ - _n′_ = _v_/λ - _v′_/λ′

Also a little consideration will show that—

L/λ′ = λ/(λ - λ′)

since λ is nearly equal, by assumption, to λ′. Hence we have—

1/L = 1/λ - 1/λ′; and also V = NL

Accordingly—

V = N/(1/L) = (_v_/λ - _v′_/λ′)/(1/λ - 1/λ′)

Let us write 2π/_k_ instead of λ, and 2π/_k′_ instead of λ′; then we have—

V = (_vk_ - _v′k′_)/(_k_ - _k′_) (i.)

And since _k_ and _k′_, _v_ and _v′_ are nearly equal, we may write the above expression as a differential coefficient; thus—

V = _d_(_vk_)/_d_(_k_) (ii.)

Suppose, then, that, as in the case of deep-sea waves, the wave-velocity varies as the square root of the wave-length. Then if C is a constant, which in the case of gravitation waves is equal to _g_/2π, where _g_ is the acceleration due to gravity, we have—

_v_^2 = Cλ, or _v_^2 = (_g_/2π) × λ

But λ = 2π/_k, hence—

_vk_ = 2πC/_v_

Hence if we differentiate with respect to _v_, we have—

_d_(_vk_)/_dv_ = -2πC/_v_^2

Again, _k_ = 2π/λ = 2πC/_v_^2; therefore—

_d_(_k_)/_dv_ = -2(2πC/_v_^3)

Hence, dividing the expression for _d_(_vk_)/_dv_ by that
for _d_(_k_)/_dv_, we have—

V = _d_(_vk_)/_d_(_k_) = _v_/2

In other words, the wave-train velocity is equal to half the wave-velocity. This is the case with deep-sea waves. Suppose, however, that, as in the case of air waves, the wave-velocity is independent of the wave-length. Then if two trains of waves of slightly different wave-length are superposed, we have _k_ and _k′_ different in value but nearly equal, and _v_ and _v′_ equal. Hence the equation (i.) takes the form—

V = _v_

In other words, the _beats_ travel forward with the same speed as the constituent waves. And in this case there is no difference between the velocity of the wave-train and the velocity of the individual wave. The above proof may be generalized as follows:—

Let the wave-velocity vary as the _n_th root of the wave-length, or let _v_^_n_ = Cλ; and let λ = 2π/_k_ as before.

Then—

_v_^_n_ = 2πC/_k_, and _vk_ = 2πC/_v_^{_n_-1} = 2πC_v_^{-(_n_-1)}

also _k_ = 2π/λ = 2πC/_v_^_n_ = 2πC_v_^{-_n_}

Hence _d_(_vk_)/_d_(_k_) = (_n_-1_v_^{-(_n_-1)-1})/(_nv_^{-_n_-1})
= ((_n_-1)/_n_)_v_

or V = ((_n_-1)/_n_)_v_

That is, the wave-train velocity is equal to (_n_-1)/_n_ times the wave-velocity.

In the case of sea waves _n_ = 2, and in the case of air waves _n_ = infinity.

If _n_ were 3, then V = (2/3)_v_, or the group-velocity would be two-thirds the wave-velocity.

NOTE B (see p. 273).

Every electric circuit comprising a coil of wire and a condenser has a definite time-period in which an electric charge given to it will oscillate if a state of electric strain in it is suddenly released. Thus the Leyden jar L and associated coil P shown in Fig. 82, p. 271, constitutes an electric circuit, having a certain _capacity_ measured in units, called a microfarad, and a certain _inductance_, or electric inertia measured in centimetres. The capacity of the circuit is the quality of it in virtue of which an electric strain or displacement can be made by an electromotive force acting on it. The inductance is the inertia quality of the circuit, in virtue of which an electric current created in it tends to persist. In the case of mechanical oscillations such as those made by vibrating a pendulum, the time of one complete oscillation, T, is connected with the _moment of inertia_, I, and the mechanical force brought into play by a small displacement as follows: Suppose we give the pendulum a small angular displacement, denoted by θ. Then this displacement brings into existence a restoring force or torque which brings the pendulum back, when released, to its original position of rest. In the case of a simple pendulum consisting of a small ball attached to a string, the restoring torque created by displacing the pendulum through a small angle, θ, is equal to the product _mgl_θ, where _m_ is the mass of the bob, _g_ is the acceleration of gravity, and _l_ is the length of the string. The ratio of displacement (θ) to the restoring torque _mgl_θ is 1/_mgl_. This may be called the displacement per unit torque, and may otherwise be called the _pliability_ of the system, and denoted generally by P. Let I denote the moment of inertia. This quantity, in the case of a simple pendulum, is the product of the mass of the bob and the square of the length of the string, or I = _ml_^2.

In the case of a body of any shape which can vibrate round any centre or axis, the moment of inertia round this axis of rotation is the sum of the products of each element of its mass and the square of their respective distances from this axis. The periodic time T of any small vibration of this body is then obtained by the following rule:—

T = 2π√((moment of inertia round the axis of rotation)
× (displacement per unit of torque, or pliability))

or T = 2π√(IP).

In the case of an electric circuit the inductance corresponds to the moment of inertia of a body in mechanical vibration; and the capacity to its pliability as above defined. Hence the time of vibration, or the electrical time-period of an electric circuit, is given by the equation—

T = 2π√(LC)

where L is the inductance, and C is the capacity.

It can be shown easily that the frequency _n_, or number of electrical vibrations per second, is given by the rule—

_n_ = 5000000/(√((capacity in microfarads)
× (inductance in centimetres)))

For instance, if we discharge a Leyden jar having a capacity of ¹⁄₃₀₀ of a microfarad through a stout piece of copper wire about 4 feet in length and one-sixth of an inch in diameter, having an inductance of about 1200 centimetres, the electrical oscillations ensuing would be at the rate of 2¹⁄₂ millions per second.

Any two electrical circuits which have the same time-period are said to be “in tune” with each other, and the process of adjusting the inductance and capacity of the circuits to bring about this result is called electrical tuning. In the case of a vertical aerial wire as used in wireless telegraphy, in which the oscillations are created by the inductive action of an oscillation-transformer as shown in Fig. 82, page 271, the capacity of the Leyden jar in the condenser circuit must be adjusted so that the time-period of the nearly closed or primary oscillation P agrees with that of the open or secondary circuit S. When this is the case, the electrical oscillations set up in the closed circuit have a far greater effect in producing others in the open circuit than if the two circuits were not in tune. The length of the wave given off from the open circuit is approximately equal to four times the length of the aerial wire, including the length of the coil forming the secondary circuit of the oscillation-transformer in series with it.

FOOTNOTES

[1] The wave-velocity in the case of waves on deep water varies as √(_g_λ/2π), where λ is the wave-length. The rule in the text is deduced from this formula.

[2] If V is the velocity of the wave in feet per minute, and V′ is the velocity in miles per hour, then (V′ × 5280)/60 = V. But V′ = √(2¹⁄₄λ), and V = _n_λ, where λ is the wave-length in feet and _n_ the frequency per minute; from which we have V′ = 198/_n_, or the rule given in the text.

[3] The amplitude of disturbance of a particle of water at a depth equal to one wave-length is equal to 1/ϵ^{2π} of its amplitude at the surface. (See Lamb’s “Hydrodynamics,” p. 189.)

[4] This can easily be shown to an audience by projecting the apparatus on a screen by the aid of an optical lantern.

[5] See “The Splash of a Drop,” by Professor A. M. Worthington, F.R.S., Romance of Science Series, published by the Society for Promoting Christian Knowledge.

[6] See Osborne Reynolds, _Nature_, vol. 16, 1877, p. 343, a paper read before the British Association at Plymouth; see also Appendix, Note A.

[7] A very interesting article on “Kumatology, or the Science of Waves,” appeared in a number of _Pearson’s Magazine_ for July, 1901. In this article, by Mr. Marcus Tindal, many interesting facts about, and pictures of, sea waves are given.

[8] Lord Kelvin (see lecture on “Ship Waves,” Popular Lectures, vol. iii. p. 468) says the wave-length must be at least fifty times the depth of the canal.

[9] See article “Tides,” by G. H. Darwin, “Encyclopædia Britannica,” 9th edit., vol. 23, p. 353.

[10] The progress of the Severn “bore” has been photographed and reproduced by a kinematograph by Dr. Vaughan Cornish. For a series of papers bearing on this sort of wave, by Lord Kelvin, see the _Philosophical Magazine_ for 1886 and 1887.

[11] See Lord Kelvin, “Hydrokinetic Solutions and Observations,” _Philosophical Magazine_, November, 1871.

[12] “On the Photography of Ripples,” by J. H. Vincent, _Philosophical Magazine_, vol. 43, 1897, p. 411, and also vol. 48, 1899. These photographs of ripples have been reproduced as lantern slides by Messrs. Newton and Co., of Fleet Street, London.

[13] Some smokers can blow these smoke rings from their mouth, and they may sometimes be seen when a gun is fired with black old-fashioned gunpowder, or from engine-funnels.

[14] For details and illustrations of these researches, the reader is referred to papers by Professor H. S. Hele-Shaw, entitled, “Investigation of the Nature of Surface-resistance of Water, and of Stream-line Motion under Experimental Conditions,” _Proceedings of the Institution of Naval Architects_, July, 1897, and March, 1898. A convenient apparatus for exhibiting these experiments in lectures has been designed by Professor Hele-Shaw, and is manufactured by the Imperial Engineering Company, Pembroke Place, Liverpool.

[15] The French word _échelon_ means a step-ladder-like arrangement; but it is usually applied to an arrangement of rows of objects when each row extends a little beyond its neighbour. Soldiers are said to march in echelon when the ranks of men are so ordered.

[16] See Lord Kelvin on “Ship Waves,” Popular Lectures, vol. iii. p. 482.

[17] More accurately, as the 1·83 power of the speed.

[18] This figure is taken by permission from an article by Mr. R. W. Dana, which appeared in _Nature_ for June 5, 1902, the diagram being borrowed from a paper by Naval Const. D. W. Taylor, U.S., read before the (U.S.) Society of Naval Architects and Marine Engineers (1900).

[19] “Practical Applications of Model Experiments to Merchant Ship Design,” by Mr. Archibald Denny, Engineering Conference, Institution of Civil Engineers, May 25, 1897.

[20] Reproduced here by the kind permission of the editor of _Harmsworth’s Magazine_.

[21] See Lord Kelvin’s Popular Lectures, vol. iii., “Navigation,” Lecture on “Ship Waves.”

[22] See Professor W. F. Barrett, _Nature_, 1877, vol. 16, p. 12.

[23] This follows from the ordinary formula for the focal length _f_ of a biconvex lens, each surface having a radius of curvature equal to _r_. For then it can be shown that

_f_ = (r_/2) · (1/(μ - 1))

where μ is the index of refracture of the lens material. As shown later on, the acoustic index of refraction of carbonic acid, when that of air is taken as unity, is 1·273. Hence, μ - 1 = 0·273, and 1/(μ - 1) = 3²⁄₃. Hence, _f_ = 2_r_(¹¹⁄₁₂), or _f_ is slightly less than twice the radius of curvature of the spherical segment forming the sound-lens.

[24] We can, in fact, discover the ratio of the velocities from the amount of bending the ray experiences and the angle BAC of the prism, called its refracting angle. It can be shown that if we denote this refracting angle by the letter A, and the deflection or total bending of the ray by the letter D, then the ratio of the velocity of the wave in air to its velocity in carbonic acid gas (called the _acoustic refractive index_), being denoted by the Greek letter μ; we have—

μ = sin((A + D)/2)/sin(A/2)

[25] On the occasion when this lecture was given at the Royal Institution, a large phonograph, kindly lent by the Edison-Bell Phonograph Company, Ltd., of Charing Cross Road, London, was employed to reproduce a short address on Natural History to the young people present which had been spoken to the instrument ten days previously by Lord Avebury, at the request of the author. The address was heard perfectly by the five or six hundred persons comprising the audience.

[26] In the case of the paraffin prism the refracting angle (_i_) was 60°, and the deviation of the ray (_d_) was 50°. Hence, by the known optical formula for the index of refraction (_r_), we have—

_r_ = sin((_i_ + _d_)/2)/sin(_i_/2) = sin(55°)/sin(30°) = 1·64

For the ice prism the refracting angle was 50°, and the deviation 50°; accordingly for ice we have—

_r_ = sin((50 + 50)/2)/sin(50/2) = sin(50°)/sin(25°) = 1·88

See “Cantor Lectures,” Society of Arts, December 17, 1900. J. A Fleming on “Electric Oscillations and Electric Waves.”

[27] See Appendix, Note B.

INDEX.

A

Actinic rays, 254
Æther, the, 191
——, properties of, 192
—— wave radiation, range of, 262
—— waves, various kinds of, 234
Air, movement of, in a concert-room, 183
—— necessary for production of sound, 103
—— particles, mode of motion of, in case of sound wave, 112
—— waves, 103
—— ——, interference of, 139
—— ——, length of, 114
—— ——, nature of, 114
—— ——, speed of, 115
Alphabet used in telegraphy, 274
America Cup race, pictures of yachts entered for the, 94, 95
_America_ yacht, the, 93
Amplitude, 8
Anti-node, 159
Apparatus for detecting electric waves, 237
—— for exhibiting motion of air in case of sound wave, 109
—— for investigating the laws of falling bodies, 35
Atlantic waves, height of, 8
—— ——, length of, 9
Atomic theory, 226

B

Beam of sound focused by collodion lens, 131
Beats, Helmholtz’s theory of, 165
——, musical, 163
Billows, 1
Bore, 38
Branly, Professor, electric wave detector, 211
Breaking wave, 29

C

Canal-boat waves, 100
Canal wave, 32
—— —— velocity, 34
Capillary ripples, 44
Chromatic scale, 162
Clef, musical, 160
Closed organ-pipe, 171
Cloud waves, 30
Coherer, Lodge, 212
——, Marconi, 276
Column of air set in vibration by tuning-fork, 158
Conclusion, 285
Concords and discords in music, nature of difference between, 162
——, musical, 162
Conditions necessary for production of true wave in a medium, 15
Conductor, electric, 197
Conservation of energy, law of, 23
Conservation of matter, law of, 22
Convergence of an electric ray by a paraffin lens, 243
Cornish, Dr. Vaughan, 31
Corresponding speeds, Froude’s law of, 81
Cup Races, 93
Current, electric, 193

D

Dark heat, 250
Deep-sea waves, rule for determining speed of, 11
—— ——, velocity of, 11
Definition of a ripple, 42
—— of wave-frequency, 8
—— of wave-length, 7
Depth of water, effect of, on speed of canal wave, 34
Difference between electric conductors and non-conductors, 198
—— between velocity of a wave and of a wave-train, 20
Discords in music, 162
Dispersion of æther waves, 256
Distances at which sound can be heard, 120

E

Ear, power of, to analyze sound, 182, 183
Echelon waves made by a duck, 74
Eddies in liquids, 60
Eddy, 60
—— motion, 60
—— resistance, 68
Edison phonograph, 142
Elasticity of the air, 111
Electric circuit, open, 217
—— conductor and non-conductor, 197
—— corpuscles, 227
—— current, 193
—— ——, alternating, 194
—— ——, continuous, 194
—— —— energy, 215
—— ——, nature of, 193
—— displacement, 221
—— energy, mechanical analogue of, 216
—— force, 228
Electric index of refraction, 245
—— inductance, 214
—— inertia, 214
—— oscillations, 185
—— ——, apparatus for producing, 203
—— —— in open circuit, 217, 218
—— —— produced by discharge of Leyden jar, 200
—— radiation, 238
—— —— and light, identity in nature of, 251
—— radiation-detector (Fleming), 224
—— —— (Miller), 235
—— radiation, velocity of, 249
—— ray, reflection of, 241
—— ——, refraction of, 243
—— strain, 197
—— transparency and opacity, 239, 240
—— wave and air wave compared, 223
—— —— detector, 210
—— ——, nature of, 230
—— ——, production of, 209
—— waves, 185
Electrical or Hertz rays, 254
Electrodeless discharge, 205
—— ——, apparatus for producing, 206
Electro-magnet, 195
Electro-magnetic medium, 220
—— theory of light, 262
Electromotive force, 196
Electronic theory of electricity, 229
Electrons, 227
Energy, 21
——, kinetic, 25
—— of electro-static strain, 215
—— of motion, 25
—— of moving water, 32
——, potential, 25
——, two forms of electric, 215
Ether, the, 191
Experimental tank, uses of, in ship-design, 86
—— tanks, 85, 86
Experiments illustrating surface tension, 40, 41
Explosion of guns heard at great distances, 120

F

Falling bodies, laws of, 35
Fish, motion of a, 67
Flame, sensitive, 127
Flow of liquid in non-uniform tubes, 66
—— —— in tubes, 65
—— —— in uniform tubes, 65
Fog-signals, 123
——, influence of wind upon distance at which they are heard, 124
——, power absorbed in making, 125
Free period of vibration, influence upon force required to move a
body, 151, 152
Froude, Mr. William, 81
Froude’s experimental tank, 81
—— experiments at Torquay, 81

G

Gamut, 160
—— of æther waves, 260
Gravitation wave, 40
Ground swell, 31

H

Harmonic curve, 108
—— motion, 107
Harmonics, 156
Hele-Shaw, Professor, 62
—— ——, discovery of means of producing irrotational motion in
liquids, 63
—— ——, investigations on liquid motion, 63
Helmholtz’s investigation into nature of musical tones, 167
—— theory of discords and concords, 165
Hertz oscillation, 207
Hertz’s researches. Experiments with electric waves, apparatus for,
234
Hughes, Professor, investigations on electric waves, made by, 210
Human ear, the, 181
Hydraulic gradient, 65

I

Illustration of difference between wave-velocity and wave-group
velocity, 28
Inaudible sound, 140
Index of refraction, 54
Inductance, 214
Induction coil for wireless telegraphy, 267
Inefficiency of present methods of manufacturing light, 265
Inertia of the air, 111
Interference of air waves, 139
—— of electric rays, 248
—— of ripples and waves, 48
Irrotational motion, 59

K

Kelvin, Lord, investigations on ship waves, 77
Kinetic energy, 25
Krakatoa, eruption of, 116
——, sound produced by the eruption of, 116

L

Laplace, calculation of, concerning sound-velocity, 69
Law connecting velocity and pressure in liquid motion, 67
Length of wave, definition of the term, 7
Light, velocity of, 187, 189
Liquid flow in constricted tube, 66
Lodge, Sir Oliver, coherer invented by, 211
Long wave, 7
Longitudinal waves, 7
Luminous efficiency, 264
—— rays, 254

M

Magnetic force, 228
Major third in music, 161
Marconi coherer, 276
—— aerial wire, 266
——, experiments with wireless telegraphy across the Atlantic by, 280
—— system of wireless telegraphy, 276, 279
Matter, 22
Maxwell’s electro-magnetic theory of light, 262
Mechanical explanations of electrical phenomena, 222
Methods of manufacturing light, 265
Minor third in music, 161
Model illustrating longitudinal wave, 114
—— —— nature of an air wave, 113
Models, illustrating wave-motion, 5, 6
Morse alphabet, 274
Motion, harmonic, 107
——, irrotational, 59
—— of water in sea waves, 3
——, periodic, 107
——, rotational, 59
——, vortex, 59
Movement of the air in the case of a sound wave, 112
Music, theory of, 159
Musical beats, 163
—— scale, notes of the, 160
—— tones and noises contrasted, 110
—— ——, sharp and flat, 161

N

Natural period of vibration of a body, 148, 150
Node, 159
Non-conductor, electric, 197
Non-resistance to body moving through perfect fluid, 72

O
Octave, 160
Open electric circuit, 217
—— organ-pipe, 171
Optical proof that a sounding body is in vibration, 105, 106
Organ-pipes, construction of, 169, 171
——, distribution of air-pressure in, 172
——, overtones of, 173
——, relation between length of pipe and length of wave, 173, 174
Oscillations, electric, 185
—— of a stretched string, 154
Oscillator, Hertz, 207
Oscillatory electric sparks, photographs of, 202
Overtones, 156

P

Pendulum, isochronism of the, 149
Perfect fluid, 59
Periodic motion, 107
—— time, 9
Phonograph, action of the, 142
Photographic study of the production of waves, 16
Photographs of ripples on a mercury surface, 51
Plane wave, 55
Potential energy, 25
Power required to propel ships, 90, 91
Prism for refracting a beam of sound, 136
Production of a sound wave, 111

Q

Quality of sound, 115

R

Radiation, electric, 238
——, nature of, 263
—— of energy from the sun, 284
Rayleigh, Lord, an acoustic experiment with an open pipe by, 174
Receiver for wireless telegraphy (Marconi), 275
Reflection of a beam of sound, 132, 133
—— of an electric ray, 241
—— of a wave, 55
—— of ripples, 46, 47
Refraction, explanation of, 53, 54
—— of a beam of sound, 133, 134
—— of an electric ray, 243
—— —— by an ice prism, 242
—— of ripples, 52
Refractive index, 54
Relation of wave-velocity and wave-length, 9
Relay, telegraphic, 236
Resistance curves of ships, 92
—— to a body moving through a fluid, causes of the, 68
Resonance, 148
——, an experiment on, 158
Ripple and wave, distinction between, 40
—— mark, 30
——, reflection of a, 45, 46
——, scientific definition of a, 42
Ripples, 1
——, apparatus for producing, 43
——, interference of, 48
——, intersecting, 49
—— in the air, 103
—— on a lake, photographs of, 19
——, photography of, 51
——, by J. H. Vincent, 54
—— produced by stone thrown into water, 19
——, velocity of, 41, 42, 43
Rotational motion, 59
Russell, Scott, Mr., 81
—— experiments of, on canal-boat waves, 101

S

Scale, musical, notes of the, 160
—— of equal temperament, 162
Sea waves, 2
—— ——, motion of, 3
—— ——, relation of velocity and length in case of, 10
Semitone, 161
Sensitive flame, influence of sound upon a, 129
Severn bore, 38
_Shamrock II._, trials of, 97
Ship bow wave, mode of production, 75, 76
—— design, 87, 88
—— ——, the problem of, 73
—— models, the testing of, 85, 86
—— resistance, Froude’s law of, 82, 83
—— waves, 57
—— ——, complete system of, 78
—— ——, various systems of, 73
Short wave, 7
Sine curve, mode of drawing a, 108
Singing flame, 175
Siren, 123
Skin friction, 59, 68
—— —— for various classes of ships, 91
Soap film thrown into vibration by air waves, 145
—— solution for making bubbles, 144
Solitary wave, 26
—— —— and wave-train, difference between, 26
Sound, and music, 147
——, causes in variation in quality of, 142
—— due to air waves, 103, 104
—— lens, method of making a, 130
—— prism, 136
——, quality of, 115
—— signals, 123
——, velocity of, in various cases, 126
Sounding body is in vibration, 104, 105
Speed of a falling body, 35
—— of sound, 115
Stationary waves, 155, 156
Stone falling into water, photographs of a, 17
Stream-lines, 64
—— —— round an ovoid, 71
Stroh violin, 179
Structure of the human ear, 181
Surface tension of liquid, 40

T

Temperature, effect of, on sound-velocity, 118
Tesla coil, 205
Tidal wave, 38
—— ——, speed of, 39
Tides, 39
Time of vibration of a stretched string, 154
Tone, 161
Torpedo, motion of, in water, 67
Transference of wave-motion, 27
Transverse ship wave, 79
—— wave, 7
True wave, conditions for producing a, 15
—— ——, definition of a, 12
Tubes of flow, 64
—— —— in a liquid, 71

U

Utilization of the æther, 284

V

Various kinds of resistance to a body moving through a fluid, 68
Velocity of electric radiation, 249
—— of light, 189
—— of sea waves, rule for calculating the, 10
—— of sound, how affected by temperature, 117
—— —— in different gases, 126
—— ——, influence of specific heats upon the, 119
—— ——, measurements of the, 116
—— ——, theoretical determination of the, 118
—— of sound wave, 115
—— of waves in water, air, and æther compared, 283
Vernon Boys, Professor, instructions by, for making soap solution, 144
Vibration rates of musical tones, 160
—— —— of various æther waves, 252
Vibrations, forced, 149
——, free, 149
—— giving rise to musical tones, 160
Vincent, Mr. J. H., 51
Violin, improved by Mr. Augustus Stroh, 179
Violin, structure of, 177, 178
Viscosity of liquids, 58
Vortex motion, 59
—— ring in air, 61
—— ——, production of, in air, 61

W

Wave amplitude, definition of, 8
——, causes of breaking, 29
——, electric, nature of an, 185
—— frequency, 8
——, gravitation, 40
—— group, velocity of a, 20
—— length, 7
—— lengths of various kinds of æther waves, 257
——, longitudinal, 7
—— motion, 2
—— ——, definition of, 4
—— —— model, 5
—— ——, model for illustrating, 13, 14
—— ——, various kinds of, 4
—— produced in a canal, 33
——, reflection of a, 55, 56
—— resistance, 68
Waves, 1
—— and ripples in the æther, 232
——, conditions of, for interference of, 49, 50
——, electric, 185
——, interference of, 48
—— in the air, 103
—— made by canal-boats, experiments on, 101, 102
—— made by ships, 57, 58
—— on a snow surface, 31
—— on clouds, 30
—— produced by high-speed ships, 79
——, refraction of, 52, 53
——, sea, 2
——, stationary, 155
—— train, 20
—— ——, velocity of, 20
——, transverse, 7
—— velocity, a rule for determining, 9
Wind, influence of, upon distances at which sounds are heard, 120,
121, 122
Wireless telegraphy across the Atlantic, 280
—— ——, apparatus for, 267
—— ——, explanation of, 273
—— ——, Marconi’s system of, 269
—— ——, method of conducting, 274
—— ——, transmitter for, 271
—— ——, utility of, 281
Worthington, Professor, photographs of splash of drop, 16, 17

Y

Yacht-design, object of, 96
Yachts entered for the America Cup race, pictures of, 94, 95
Young, Dr. Thomas, investigations of, on interference of light, 190

THE END.

PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BECCLES.

————————————————— End of Book —————————————————

Transcriber’s Note (continued)

Errors in punctuation and simple typos have been corrected without note. Variations in spelling, hyphenation, accents, etc., have been left as they appear in the original publication unless as stated in the following:

Page 56 – “sea-side” changed to “seaside” (At the seaside)
Page 56 – “sea-side” changed to “seaside” (study of seaside pools)
Page 136 – “sound ray” changed to “sound-ray” in Fig. 51 caption
Page 145 – “limelight” changed to “lime-light” (lime-light lantern)
Page 156 – “over-tones” changed to “overtones” (accompanied by the
harmonics or overtones)
Page 162 – “key-board” changed to “keyboard” (keyboard of a piano)
Page 176 – “aërial” changed to “aerial” (stationary aerial oscillation)
Page 177 – “horse-hair” changed to “horsehair” (bow made of horsehair)
Page 274 - “Full Stop — — — — — —” changed to
“Full Stop — ——— — ——— — ———”

————

Footnotes have been re-indexed using numbers and placed before the Index.

————

Large bracketing is not available to surround built-up fractions in displayed equations and expressions so ordinary bracketing and the solidus (/) is used to represent them instead. Thus the built-up x + 1 fraction ————— is rendered in this text in the conventional manner 2 as (x + 1)/2. If the sin of that expression is required it would be rendered unambiguously as sin((x + 1)/2). For consistency this also means that a simple sin function that would normally be displayed as ‘sin x’ will instead be rendered as ‘sin(x)’.

A large radical (√) symbol is not available either so in order to make the radicand expression unambiguous a similar approach to the above is used. x + 1 Thus the square root of the fraction ————— is rendered in this text as 2 √((x + 1)/2). Again this means that, for consistency, the simple square root of, say, the natural number 5, which would normally be displayed as ‘√5’ will instead be rendered as ‘√(5)’.

In other expressions and equations some additional bracketing may have been added to make the meaning clear and unambiguous or consistent with bracketing as used in immediately surrounding equations and expressions.