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:—
Also a little consideration will show that—
since λ is nearly equal, by assumption, to λ′. Hence we have—
Accordingly—
Let us write 2π / k instead of λ, and 2π / k′ instead of λ′; then we have—
And since k and k′, v and v′ are nearly equal, we may write the above expression as a differential coefficient; thus—
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—
But λ = 2π / k , hence—
Hence if we differentiate with respect to v, we have—
Again, k = 2π / λ = 2πC / v² ; therefore—
Hence, dividing the expression for d(vk) / dv by that for d(k) / dv , we have—
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—
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 nth root of the wave-length, or let vⁿ = Cλ; and let λ = 2π / k as before.
Then—
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 mglθ, 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 mglθ 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 = ml².
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 | } × { | displacement per unit of |
| the axis of rotation | 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—
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 | } × { | inductance in | |
| microfarads | 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
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
But V′ = √2₁/⁴ λ , and V = nλ , 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
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
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 = 2r 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—
[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—
For the ice prism the refracting angle was 50°, and the deviation 50°; accordingly for ice we have—
See “Cantor Lectures,” Society of Arts, December 17, 1900. J. A Fleming on “Electric Oscillations and Electric Waves.”