394 SURFACE WAVES. [CHAP. IX 



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

 zero, is always zero. 



If (j> be the velocity-potential, the equations of motion have 

 now the integral 



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



in accordance with (1) above. 



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

 distribution of pressure we assume 



<f>/c = x + fte^y sin kx, \ 



'' > (b). 



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

 of k&, 



n\ 



" = . . . ay + B^y (kc 2 cos kx + ac sin kx) (7). 



P 



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

 surface (^ = 0) 



= j3 }(&c 2 g) cos kx + /z,c sin kx} (8), 



which is equal to the real part of 



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



7 = p tt * W 



corresponds the surface-form 



9-w^=ijf < 10 >- 



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



^=Pcoskx (11) 



produces the wave-form 



_ r> (1* ~~ 9} cos hx fie sin kx 



