﻿Resonators exposed to Primary Plane Waves. 213 



disturbed by numerous small obstacles composed of a medium 

 of different quality. There is then no difficulty in supposing 

 the obstacles so small that their mutual reaction may be 

 neglected, even although the average distance of immediate 

 neighbours is much less than a wave-length. When the 

 obstacles are small enough, the whole energy dispersed may 

 be trifling, but it is well to observe that there must be some. 

 No medium can be fully transparent in all directions to 

 plane waves which is not itself quite uniform. Partial 

 exceptions may occur, e.g. when the want of uniformity is 

 a stratification in plane strata. The dispersal then becomes 

 a regular reflexion, and this may vanish in certain cases, 

 even though the changes of quality are sudden (black in 

 Newton's rings) *. But such transparency is limited to 

 certain directions of propagation. 



To return to resonators : when the} 7 may be close to- 

 gether, we have to consider their mutual reaction. For 

 simplicity we will suppose that they all lie on the same 

 primary wave-front, so that as before in the neighbourhood 

 of each resonator we may take 



</> = 1, d<f>/dr = (14) 



Further, we suppose that all the resonators are similarly 

 situated as regards their neighbours, e. g., that they lie 

 at the angular points of a regular polygon. The waves 

 diverging from each have then the same expression, and 

 altogether 



*=«jV + V- + --- ' • • • (15) 



where r l9 r 2i . . . are the distances of the point where yjr is 

 measured from the various resonators, and a is a coefficient 

 to be determined. The whole potential is <^ + ^, and it 

 suffices to consider the state of things at the first resonator. 

 With sufficient approximation 



+ ^ = l + -(l-^r^ + aS^p-..., . • (1(5) 



R being the distance of any other resonator from the first, 

 while (as before) 



d(<t> + ^ ) _ __ a ^ ^ ^ ^ 17 j 



dr r{ 2 ' 



We have now to distinguish two cases. In the first, 

 which is the more important, the tuning of the resonators is 



* See Tree. Roy. Soc. vol. 80 a, p. 207 (1912). 



