﻿Resonators exposed to Primary Plane Waves. 211 



The potential o£ the symmetrical waves issuing from the 

 resonator may be taken to be 



a 



■f=<— = "(l -ikr +....). ... (6) 



Since the resonator is supposed to be an ideal resonator, 

 concentrated in a point, r is to be treated as infinitesimal 

 in considering the conditions to be there satisfied. The first 

 of these is that no work shall be done at the resonator, and 

 it requires that total pressure and total radial velocity shall 

 be in quadrature. The total pressure is proportional to 

 d(d> + yjr)/dt^ or to i(<f> + yfr), and the total radial velocity is 

 d(<j) + ^)/dr. Thus {(f>T^r) and d(<f> + yjr)/dr must be in the 

 same (or opposite) phases, in other words their ratio must 

 be real. Now, with sufficient approximation, 



a 



«*(»'+*)_ 



ifj-T y — a 



r dr. 



9 5 



r 





so that 



ik — real. . 



a 







I£ we write 











a = Ae ia , 1/a = A" 1 e" i ", 





. 



then 



A= — k~ l sin a.. . 







(J) 



(8) 



(9) 



So far « is arbitrary, since we have used no other condition 

 than that no work is being done at the resonator. For 

 instance, (9) applies when the source of disturbance is 

 merely the presence at the origin of a small quantity of gas 

 of varied character. The peculiar action of a resonator is 

 to make A a maximum, so that sin a = + 1, say — 1. Then 



xmd 



As in (o), 



A = l/*, a=-i\L; (10) 



[p-ikr 



Airr 2 Mod 2 f = Itt/F = \ 2 /tt, . . . (12) 



and the whole energy dispersed corresponds to an area o£ 

 primary wave-front equal to X 2 /7t. 



The condition of resonance implies a definite relation 

 between ((f> + yp) and d(<j> + yjr)/dr. If we introduce the 



P2 



