235-236] STANDING WAVES : UNIFORM DEPTH. 425 



condition to be satisfied at the surface is, in the case of infinitely 

 small motions, 



and the kinematical surface-condition is 



dt 

 Hence, for z = 0, we must have 



d 2 (f> d6 



+- 



or, in the case of simple- harmonic motion, 



if the time-factor be e* (<rt+e) . The proof is the same as in 

 Art. 216. 



The equation of continuity, V' 2 <f> = 0, and the condition of zero 

 vertical motion at the depth z = h, are both satisfied by 



< = <j cosh k (z 4- h) ..................... (5), 



where < x is a function of x, y y provided 



The form of ^ and the admissible values of k are determined by 

 this equation, and by the condition that 



at the vertical walls. The corresponding values of the ' speed ' (a) 

 of the oscillations are then given by the surface-condition (4), viz. 

 we have 



o- 2 = gk tanh kit ........................ (8). 



From (2) and (5) we obtain 





(9). 



The conditions (6) and (7) are of the same form as in the case 

 oltamall depth, and we could therefore at once write down the 



