Theory of Short and Long Waves 21 



P = \gQA 2 s\n 2 {xx — at) 



and for one wave length P = \gqA 2 l. 



A water particle has in its orbit an angular velocity Ae xz 2n\T and the 

 kinetic energy of a volume element dxdz is represented by 



p An 2 ... , , 

 | -= A 2 e^dxdz . 



In order to obtain the energy for the whole water column, this expression 

 must be integrated for dz from — oo to 0. Considering (11.11), the kinetic 

 energy per unit area will be K = \ ggA 2 that per wave length \ QgA 2 X. Con- 

 sequently, at any time the total energy per unit area in the case of a progressive 

 wave will be 



E = \ Qg A 2 . (11.14) 



The energy at any instant is always half potential, and half kinetic. The 

 energy of a progressive wave system of amplitude A, therefore, is equal to 

 the work required to raise up a waterlayer of thickness A through a height \A. 

 One has to remember that, in considering a part of the ocean surface, its 



Fig. 10. Streamlines and orbits in a short wave travelling to the right. 



wave energy depends not only on the wave height, but also on the wave 

 length. The long waves being mostly at the same time the highest ones, have 

 more energy than short waves. " 



In each progressive wave there is a transport of energy in the direction 

 of propagation of the wave. If v is the velocity of this energy transport, 

 then vE is the amount of energy propagated across vertical planes which 

 are a unit width apart; for surface waves, according to Lamb (1932, p. 383), 

 this is 



vE = ±QgA 2 cs'm 2 x(x-ct) . (II. 14a) 



