265-266] ARBITRARY INITIAL CONDITIONS. 483 



266. Iii the case of simple-harmonic motion, the time-factor 

 being e i<rf , the equation (4) of Art. 265 takes the form 



(V a + #)0 = ........................ (1), 



where k a/c .............................. (2). 



It appears on comparison with Art. 258 (7) that 2-7T/& is the wave- 

 length of plane waves of the same period (27r/cr). 



There is little excuse for trespassing further on the domain of 

 Acoustics; but we may briefly notice the solutions of (1) which 

 are appropriate when the boundary- conditions have reference to 

 spherical surfaces, as this will introduce us to some results of 

 analysis which will be of service in the next Chapter. 



In the case of symmetry with respect to the origin, we have 

 by Art. 263 (5), or by direct transformation of (1), 



the solution of which is 



, sin AT cos&r 



When the motion is finite at the origin we must have B = 0. 



1. This may be applied to the radial vibrations of air contained in a 

 spherical cavity. The condition that d<f)/dr = at the surface r = a gives 



tan ka = ka .................................... (i), 



which determines the frequencies of the normal modes. The roots of this 

 equation, which presents itself in various physical problems, can be cal- 

 culated without much difficulty, either by means of a series*, or by a 

 method devised by Fourier f. The values obtained by Schwerdt for the 

 first few roots are 



#a/ir = l-4303, 2-4590, 3-4709, 4*4774, 5-4818, 6'4844 ......... (ii), 



approximating to the form m + ^, where m is integral. These numbers give 

 the ratio (2<x/X) of the diameter of the sphere to the wave-length. Taking 

 the reciprocals we find 



A/2a=-6992, -4067, '2881, -2233, -1824, -1542 ............... (iii). 



In the case of the second and higher roots of (i) the roots of lower order give 



* Euler, . Introductio in Analysin Infinitorum, Lausanna, 1748, t. ii., p. 319; 

 Rayleigh, Theory of Sound, Art. 207. 



t Theorie analytique de la Chaleur, Paris, 1822, Art. 286. 

 $ Quoted by Verdet, Optique Physique, t. i., p. 266. 



312 



