﻿Thus 



Resonators exposed to Primary Plane Waves. 219 

 2<7rr .Mod 2 ^= — , (43) 



IT 



which expresses the width of primary wave-front carrying 

 the same energy as is dispersed by the linear resonator 

 tuned to maximum resonance. 



A subject which naturally presents itself for treatment is 

 the effect of a distribution of point resonators over the whole 

 plane of the primary wave-front. Such a distribution may 

 be either regular or haphazard. A regular distribution, <?. y. 

 in square order, has the advantage that all the resonators 

 are similarly situated. The whole energy dispersed is then 

 expressed by (29), but the interpretation presents difficulties 

 in general. But even this would not cover all that it is 

 desirable to know. Unless the side of the square (h) is 

 smaller than X, the waves directly reflected back are accom- 

 panied by lateral "■ spectra " whose directions may be very 

 various. When /><X, it seems that these are got rid of. 

 For then not only the infinite lines forming sides of the 

 squares which may be drawn through the points, but a for- 

 tiori lines draw T n obliquely, such as those forming the 

 diagonals, are too close to give spectra. The whole of the 

 effect is then represented by the specular reflexion. 



In some respects a haphazard distribution forms a more 

 practical problem, especially in connexion with resonating 

 vapours. But a precise calculation of the averages then 

 involved is probably not easy. 



If we suppose that the scale (b) of the regular structure is 

 very small compared with X, we can proceed further in the 

 calculation of the regularly reflected wave. Let Q be one 

 of the resonators, the point in the plane of the resonators 

 opposite to P, at which ty is required; OP = </', OQ=//, 

 PQ = ?\ Then if m be the number of resonators per unit 

 area, 



f 00 e~ if 

 \jr = 2 77 ma I y dy — 



Jo r 



or since ydy = rd> 



ty = 2mna I e~ lk, 'd 



The integral, as written, is not convergent: but as in the 

 theory of diffraction we may omit the iutegral at the upper 

 limit, if we exclude the case of a nearly circular boundary. 



