170-172] ENERGY. 279 



In a system of waves travelling in one direction only we have 



so that the expressions (1) and (2) are equal ; or the total energy 

 is half potential, and half kinetic. 



This result may be obtained in a more general manner, as 

 follows*. Any progressive wave may be conceived as having been 

 originated by the splitting up, into two waves travelling in opposite 

 directions, of an initial disturbance in which the particle-velocity 

 was everywhere zero, and the energy therefore wholly potential. 

 It appears from Art. 168 that the two derived waves are symme 

 trical in every respect, so that each must contain half the original 

 store of energy. Since, however, the elevation at corresponding 

 points is for the derived waves exactly half that of the original 

 disturbance, the potential energy of each will by (1) be one-fourth 

 of the original store. The remaining (kinetic) part of the energy 

 of each derived wave must therefore also be one-fourth of the 

 original quantity. 



172. If in any case of waves travelling in one direction only, 

 without change of form, we impress on the whole mass a velocity 

 equal and opposite to that of propagation, the motion becomes 

 steady, whilst the forces acting on any particle remain the same as 

 before. With the help of this artifice, the laws of wave-propa 

 gation can be investigated with great easef. Thus, in the present 

 case we shall have, by Art. 23 (4), at the free surface, 



(1), 



where q is the velocity. If the slope of the wave-profile be 

 everywhere gradual, and the depth h small compared with the 

 length of a wave, the horizontal velocity may be taken to be 

 uniform throughout the depth, and approximately equal to q. 

 Hence the equation of continuity is 



q(h + 7)) = ch, 



* Lord Rayleigh, &quot;On Waves,&quot; PhiL Ma0. t April, 1876, 

 t Lord Kayleigh, I. c&amp;gt; 



