to a Travelling Disturbance. 391 



be taken in the rear of the disturbing body. Owing to the 

 extension of the wave-train the space in front of this point 

 gains energy to the amount cE per unit time, whilst the 

 energy transmitted at the fixed point is UE, by the dynamical 

 property of group-velocity. The difference is equal to the work 

 done by the force (R) which generates the waves. Hence 



E= £=S E =lol^C .... (32) 



G I C[C— U) 



If U>c, the fixed point must be taken in advance of the 

 disturbing body, and c— TJ is replaced by U — c. 



5. The application to waves in superposed fluids has an 

 interest, owing to its bearing on the phenomenon of " dead- 

 water " *. 



Suppose we have a stratum of depth h and density p 

 resting on a liquid of greater density p' whose depth is prac- 

 tically infinite. The problem of free waves in this case was 

 solved long ago by Stokes f- If the origin be taken in the 

 upper surface, with the axis of y drawn vertically upwards, 

 we may write 



</> = ( A cosh ky + B sinh ky) cos kx cos art, . (33) 



cf>' = Ce k v cos kx cos at, (34) 



the two formulae relating to the upper and lower fluids 

 respectively. 



If tj, 7/ denote vertical displacements of a particle, we 

 have 



drj _ B</> drj _ "dcfr' 



dt ~dy ' dt ~dy ' 



whence 



7)= (A s'mh ky -\-B cosh, ky) cos kx sin at, . (36) 



v ' = --Ce'v cos kx sin at .... (37) 

 a 



The variation of pressure about a particle is given by 



In order that the pressure at the upper surface may be 

 constant we must have, therefore, 



4a-B = 0. ...... (39) 



gk 



* Ekman, /. c. infra. 



f " On the Theory of Oscillatory Waves," Camb. Trans, vol. viii. 

 (1847) ; Math, and Phys. Papers, vol. i. p. 212. 



(35) 



