256 



DYNAMICAL THEOEY OF SOUND 



The condition (2) requires that d(f>/dr = for r a, the radius of 

 the cavity. Hence 



tan kaka ......................... (7) 



This is a transcendental equation to find k, and thence n (= kc). 

 The roots are obtained graphically (see Fig. 76) as the abscissae 

 of the intersections of the lines y = tan x, y=-x y the zero root 

 being of course excepted as irrelevant. We have, approximately, 

 &a=(ra+i)7r, where m= 1, 2, 3, .... More accurate values of 

 the first three roots are 



ka/ir = 1-4303, 2-4590, 3*4709 ............. (8) 



The numbers give the ratio of the diameter 2a of the cavity to 

 the wave-length. In the modes after the first there are internal 

 spherical nodes (i.e. surfaces of zero velocity) whose relative 

 positions are indicated by the roots of inferior rank. In the 

 higher modes the nodal surfaces tend, as we should expect, to 

 become equidistant, since the conditions, except near the centre, 

 approximate to those of plane waves. 



Equations of somewhat similar structure to (7) occur (as 

 we have seen) in various parts of our subject, as well as in 

 other branches of mathematical physics, and processes of 

 numerical solution have been de- 

 vised by Euler, Lord Rayleigh 

 and others. There is one method, 

 of very general application, which 

 is so elegant, and at the same 

 time so little known, that it may 

 be worth while to explain it. It 

 is given by Fourier in his Theorie 

 de la Chaleur (1822). Starting 

 with a rough approximation, say 

 = 0?!, to a particular root of (7), 

 we calculate in succession the 

 quantities x z , x s , 0? 4 , ... determined 

 by the relations 



# 2 = tan" 1 0?i , # 3 = tan" 1 # 2 

 The figure illustrates the manner in which these converge 

 towards the desired root as a limiting value, no matter from 



Fig. 77. 



tan" 1 



(9) 



