﻿Resonators exposed to Primary Plane Waves. 215 



respect to the line R joining the resonators, and if 6 be the 



angle between r and R ? r 1 — r 2 = R cos 0. 

 Thus 



r 2 . Mod 3 f = A 2 { 2 + 2 cos (Hi cos 0) } ; 



and on integration over angular space, 



f. sin e d0 = SttA: 2 h+ S11 ^ \ 



2ttt 2 C Mo 



Introducing the value of A 2 from (19), we have finally 



sin /cR\ 



8irk~ 2 (l 



•( 



2itt 2 \ Mod*ir.8in0d0 = - —^ ^1. . (24) 



Jo 1 I- 1 - . 2 * mkR 



if we suppose that Hi is large, but still so that R is small 

 compared with r, (24) reduces to %irk~ 2 or 2A, 2 /7r. The 

 energy dispersed is then the double of that which would be 

 dispersed by each resonator acting alone ; otherwise the 

 mutual reaction complicates the expression. 



The greatest interference naturally occurs when kR is 

 small. (24) then becomes 2PR 2 . '2X 2 /7r, or 16VR 2 , in 

 agreement with 4 Theory of Sound,' § 321. The whole 

 energy dispersed is then much less than if there were only 

 one resonator. 



It is of interest to trace the influence of distance more 

 closely. If we put &R — 27rm, so that Ii = m\, we may 

 write (24) 



2\ 2 

 S = — . F, (25) 



where S is the area of primary wave-front which carries the 

 same energy as is dispersed by the two resonators and 



F = 27rm + sin (27rm) 



2-7rm+ (2tt??i)- 1 + 2 sin (tirm)' ' ' ^ b > 



If 2m is an integer, the sine vanishes and 



F = - (27) 



l-|-(27r?n)- 2 ' ^~ iJ 



not differing much from unity even when 2m = 1 ; and 

 whenever 2m is great, F approaches unity. 



