﻿218 Lord Rayleigh on the Mutual Influence oj 



If now we put Ji — 1, 



S= -log (1 — <?-») 



= -log^2sin|Wit(*-w-) + 2tnw. . . (3G 



Thus 



r 



ka 



(37) 



^ = i_ A{- lo S ( 2 sin ¥) +i*( ffi -< r ) + 2 ™ 



If AR = 2>?i7r, or R = mX, where m is an integer, the 

 logarithm becomes infinite and a tends to vanish *. 



When R is very small, a is also very small, tending to 



a = R-r-21og(Z;R) (38) 



The longitudinal density of the now approximately linear 

 source may be considered to be a/R,and this tends to vanish. 

 The multiplication of resonators ultimately annuls the effect 

 at a distance. It must be remembered that the tuning of 

 each resonator is supposed to be as for itself alone. 



In connexion with this we may consider for a moment the 

 problem in two dimensions of a linear resonator parallel to 

 the primary waves, which responds symmetrically. As 

 before, we may take at the resonator 



= 1, cty/dr = 0. 

 As regards i/r, the potential of the waves diverging in two 

 dimensions, we must use different forms when r is small 

 (compared with X) and when r is large f. When r is small 



#»=(7i- log Y')l 1 ~"2 r+ i r ^~ ■■■■J 



+ ^--^ T2 (l + i)+ F ^ (1+ * + *>- — ; (39) 



and when r is large, 



By the same argument as for a point resonator we find, as 

 the condition that no work is done at ? , = 0, that the imaginary 

 part of l/<2 is —iir/2. For maximum resonance 



a = 2t/w, (41) 



so that at a distance ijr approximates to 



_ y/\ e _ i(kr _ kn) , 42 ) 



* Phil. Mag. vol. xiv. p. 60 (1907) ; Scientific Papers, vol. v. p. 409. 

 t ' Theory of Sound,' §341. 



