﻿210 Lord Rayleigh on the Mutual Influence of 



inadmissible for transverse vibrations. Nevertheless they 

 mav afford suggestions. 



The action of a simple resonator under the influence of 

 suitably tuned primary aerial waves was considered in 

 'Theory of Sound/ §319 (1878). The primary waves were 

 supposed to issue from a simple source at a finite distance c 

 from the resonator. With suppression of the time-factor, 

 and at a distance r from their source, they are represented * 

 by the potential 



*=v> (1) 



in which 7c=2ttI\, and \ is the wave-length ; and it appeared 

 that the potential of the secondary waves diverging from the 

 resonator is 



p -iJcc p -ikr' 



+ -TE-V' (2) 



so that 



W 2 Mod 2 <t|r = 47r/Fe 2 (3) 



The left-hand member of (3) may be considered to represent 

 the energy dispersed. At the distance of the resonator 



Mod 2 4> = 1/e 2 . 



If we inquire what area S of primary wave-front propa- 

 gates the same energy as is dispersed by the resonator, we 

 have 



SA- = 4tt/Fc 2 , 



or S = 4tt//- 2 = X 2 /tt (4) 



Equation (4) applies of course to plane primary waves, 

 and is then a particular case of a more general theorem 

 established by Lamb f . 



It will be convenient for our present purpose to start 

 de novo with plane primary waves, still supposing that the 

 resonator is simple, so that we are concerned only with 

 symmetrical terms, of zero order in spherical harmonics. 



Taking the place of the resonator as origin and the direction 

 of propagation as initial line, we may represent the primary 

 potential by 



^ _ gifo-cos*^ i + ;£ rC os<9-pVcos 2 <9+ (5) 



* A slight change of notation is introduced. 



t Camb. Trans, vol. xviii. p. 348 (1899) ; Proc. Math. Soc. vol. xxxii. 

 p. 11 (1900). The resonator is no longer limited to be simple. See also 

 Kayleigh, Phil. Mag. vol. iii. p. 97 (1902) ; Scientific Papers, vol. v. p. 8. 



