FOOTNOTES
[1] The wave-velocity in the case of waves on deep water varies as
√gλ/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′ = √2₁/⁴ 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
1/ϵ2π
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 = 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—
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.”