﻿220 Lord Rayleigh on the Mutual Influence of 



Thus 9 



yfr= -jj— e - .... (44) 



and , ., 2 a 9 



Mod- t/t = — p — (45) 



The value of A 2 is given by (19). We find, with the same 

 limitation as above, 



^ cos /jR , i ,0 ° ,_, __ 



S -g- = 2wm\ eos£RdR=0, 



v sin /jR 



7,r> /"»ao 



t^ = 'Irnn 1 sin£R<£R=29rm/£. 



Thus A* = l/(k-+2mn/k)? 



and Mod 2 ^= ^^y vo (46) 



When the structure is very fine compared with X, Jc 2 in the 

 denominator may be omitted, and then Mod 2 i|r=l, that is 

 the regular reflexion becomes total. 



The above calculation is applicable in strictness only to 

 resonators arranged in regular order and very closely dis- 

 tributed. It seems not unlikely that a similar result, viz. a 

 nearly total specular reflexion, would ensue even when there 

 are only a few resonators to the square wave-length, and 

 these are in motion, after the manner of gaseous molecules ; 

 but this requires further examination. 



In the foregoing investigation we have been dealing 

 solely with forced vibrations, executed in synchronism with 

 primary waves incident upon the resonators, and it has not 

 been necessary to enter into details respecting the consti- 

 tution of the resonators. All that is required is a suitable 

 adjustment to one another of the virtual mass and spring. 

 But it is also of interest to consider free vibrations. These 

 are of necessity subject to damping, owing to the communi- 

 cation of energy to the medium, forthwith propagated away; 

 and tbeir persistence depends upon the nature of the reso- 

 nator as regards mass and spring, and not merely upon the 

 ratio of these quantities. 



Taking first the case of a single resonator, regarded as 

 bounded at the surface of a small sphere, we have to establish 

 the connexion between the motion of this surface and the 

 aerial pressure operative upon it as the result of vibration. 

 We suppose that the vibrations bave such a high degree of 



