16 Lord Kelvin on the 



Hence, performing the operations, dz } we find 



J i 



A= — km cos m(x — vt) \— | cos m(.v - rt ) -hgl—^ + ~ L jj (160). 



§ 123. Remarking now that 2irjmv is the periodic time 

 of the wave, and denoting by W the total work per period, 

 done by the water on the negative side of the plane (,v) upon 

 the water on the positive side, we have 



^=\ dt.A=—.±Pmv = ±7rtf. (161). 



Jo mv 2 2 



§ 124. We are going to compare this with the total energy, 

 kinetic and potential, K + P, per wave-length. In the first 

 place we shall find separately the kinetic energy, K, and the 

 potential energy, P. We have (the density of the water 

 being taken as unity) 



K=iCd.vCdz(? + t) ■ . ■ (162); 



Jo „'l 



P = i ? fW? (163), 



where £\ denotes the surface, displacement. 

 By (140) and (152) we find 



gr = — mite- w(z - 1) cos m{x—vt) . . (164); 

 5=jnAe- m( * _1) sinm(# — vt) . . . (165); 



k 



f 1= -cos m(.v— ft) . . . (156) repeated. 



Hence, 



" 1 2/n 4 ^ 



Jo 



(166); 



p =^S=r F • • • • < iii7 >- 



where r 2 is eliminated by (157). 



§ 125. Tims we see that the kinetic energy per wave- 

 length, and the potential energy per wave-length, are each 

 equal to the work done per period by the water on the 

 negative side, upon the water on the positive side, of any 

 vertical plane perpendicular to the length and sides of the 

 canal. Thus we arrive at the remarkable and well-known 

 conclusion that in a regular procession of deep-sea waves, the 



