394 SURFACE WAVES. [CHAP. IX 



Hence the circulation in a circuit moving with the fluid, if once 

 zero, is always zero. 



If (f&amp;gt; be the velocity-potential, the equations of motion have 

 now the integral 



k 2 ............... (4), 



this being, in fact, the form assumed by Art. 21 (4) when we write 

 n = gy-p(cx + &amp;lt;l&amp;gt;) ..................... (5), 



in accordance with (1) above. 



To calculate, in the first place, the effect of a simple-harmonic 

 distribution of pressure we assume 



y sin kx, \ 



f .................. W 



The equation (4) becomes, on neglecting as usual the square 

 of k/3, 



2 = ... - gy + fi^y (&c 2 cos kx + pc sin kx) ......... (7). 



This gives for the variable part of the pressure at the upper 

 surface (i|r = 0) 



= (3 [(kd 1 a) cos kx + pc sin kx} ............ (8), 



P 



which is equal to the real part of 



If we equate the coefficient to P, we may say that to the pressure 



7 = ^** ( 9 &amp;gt; 



corresponds the surface-form 



y = kd ,_ P _ i c e ikx (io). 



Hence taking the real parts, we find that the surface-pressure 



A/n 



(11) 



P 



produces the wave-form 



p (kc* g) cos kx /AC sin kz 



