178 Lord Rayleigh : Hydrodynamical Rotes. 



it follows that in the derived wave the potential and kinetic 

 energies are equal"*. 



The assumption that the displacement in each derived 

 wave, when separated, is similar to the original displacement 

 fails when the medium is dispersive. The equality of the 

 two kinds o£ energy in an infinite progressive train of simple 

 waves may, however, be established as follows. 



Consider first an infinite series of simple stationary waves, 

 of which the energy is at one moment wholly potential and 

 half a period later wholly kinetic. If t denote the time and 

 E the total energy, we may write 



K.E. = E sin 2 nt, P.E. = E cos 2 nt. 



Upon this superpose a similar system, displaced through a 

 quarter wave-length in space and through a quarter period 

 in time. For this, taken by itself, we should have 



K.E. = E cos 2 nt, P.E.=E sin 2 ^. 



And, the vibrations being conjugate, the potential and 

 kinetic energies of the combined motion may be found by 

 pimple addition of the components, and are accordingly 

 independent of the time, and each equal to E. Now the 

 resultant motion is a simple progressive train, of which 

 the potential and kinetic energies are thus seen to be 

 equal. 



A similar argument is applicable to prove the equality of 

 energies in the motion of a simple conical pendulum. 



It is to be observed that the conclusion is in general limited 

 to vibrations which are infinitely small. 



Waves moving into Shallower Water. 



The problem proposed is the passage of an infinite train of 

 simple infinitesimal waves from deep water into water which 

 shallows gradually in such a manner that there is no loss of 

 energy by reflexion or otherwise. At any stage the whole 

 energy, being the double of the potential energy, is pro- 

 portional per unit length to the square of the height ; and 

 for motion in two dimensions the only remaining question 

 for our purpose is what are to be regarded as corresponding 

 lengths along the direction of propagation. 



In the case of long waves, where the wave-length (X) is 

 long in comparison with the depth (I) of the water, corre- 

 sponding parts are as the velocities of propagation (V), or 

 * "On Waves," Phil. Mag. i. p. 2-57(1876); 'Scientific Paper?/ 

 i. p. 254. 



