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



(b) Next let us suppose 



u = A cos — (z — Kt), v = B sin — - [z — Kt), 



A A 



which would correspond to elliptic vibrations. 

 We find 



. 27T, .. , A(3A* + B 2 )TT 3 V; . 27T. 



m = A cos — .(# — /rf) H — ^ rj - t sin — (£ — /c£) 



A- o A. K A. 



3 A^' Sm A ( ^^' 



w = B sin — - (z — id) : — 5 ts--t cos — - (z — Kt) 



A, o A /e A 



BiB*-A 2 )TT 3 v^ 6tt, 4N 



— = -j - 1 cos — - (£ — /en. 



3 A 3 k A v ' 



Thus it seems that in an elastic medium such as we are 

 considering a wave of elliptic vibrations is not a stable form of 

 wave motion. 



If we neglect the third terms l,i u and v (which in the case of 

 light would correspond to a ray outside the visible spectrum), we 



find |J(1 + Q 2 )+||(^-P)+^(1 + P 2 ) = (1 + PQ) 2 , 



, „ 3A 2 + B 2 7T 3 V; 



where 1 = ~ — ; - t. 



3 A k 



_ 3ff 2 + ,4 2 *> v . 

 H ~ 3" A 3 k L 



Hence, neglecting for a first approximation the coefficients of 

 f, we may say that if a wave of elliptic vibrations be started in an 

 isotropic elastic medium the vibrational ellipse will retain the 



same shape but rotate with constant angular velocity r-^ - 



about its centre in direction opposite to that of the vibrational dis- 

 placement. Further, the rate of rotation is obviously a function of 

 the wave length. 



If we consider terms of the order (amplitude) 5 , we find that the 

 axes of the ellipse themselves vary with the time and the rate of 

 rotation is no longer constant. 



If the wave be one of circular vibration A = B, P = Q and we 

 find that to the fourth power of the amplitude this form of wave 

 motion is stable. In this case the anomalous terms of one-third 

 the wave length disappear. Even when we consider terms of the 



