304 Prof. K. Pearson, On plane waves of the [May 25, 



Similarly: 

 v = B cos (pz — qt + a) — ^ t sin 3 (pz — qt + a) 



p(p + 2m) m 2 vA 2 B 



24 (q + 2n) 



p(p — 2m) m 2 vA 2 B . 



tsm (p + 2mz — q + 2nt+a) 



24 (q - 2n) * sin ^ ~ 2m z ~ q ~ 2n % + ^' 



, 2 2 2 ^ ^M" 



where 5 — k p = -^— — 4- <— - — . 



4 b 



Let ns write m = — - , n=—-k, 



A, A. 



2tt 2tt . , 



therefore F = « 2 + 4<ir' 



1 X' 

 A 2 B z 



(A*_ jr\ 



[4X 2 + 3A/V ' 



/ A 2 R z 



In our solution above we have endeavoured to find waves 

 which might be propagated through an isotropic elastic medium in 

 such fashion that although anomalous waves might arise there 

 should not be any such waves of the same arguments as the 

 principal waves. Let us see at what results we have arrived. 



If two waves consisting of vibrations in planes at right angles 

 be propagated through an isotropic elastic medium, then 



(i) These waves will interfere with each other, that is to say, 

 either will produce anomalous waves in the plane of the other. 

 This production of anomalous waves exists whatever may be the 

 relation between the wave lengths, or between the velocities of 

 propagation of the principal waves. 



(ii) The velocities of propagation of the principal waves are 

 altered by their coexistence, and in a manner which depends not 

 only on the wave lengths but on the intensities. 



(iii) If the two principal waves have the same velocities of 

 propagation then it is necessary that 



A_B 

 X~X" 



or the amplitudes must be in the ratio of the wave lengths. If 

 this condition be not fulfilled, there must be anomalous waves of 



