185-186] WAVES ON AN OPEN SHEET OF WATER. 303 



If we put f=-f% ........................... (9), 



so that f denotes the equilibrium-elevation corresponding to the 

 potential H, these may be written 



du d ,<* t\ dv d ... t\ 



In the case of simple-harmonic motion, these take the forms 



whence, substituting in the equation of continuity (1), we obtain 



OV + frK-vtf ...................... (12), 



if Vi>=:d 2 /dtf + d?/dy*, .................... (13), 



arid k- = <r*/gh, as before. The condition to be satisfied at a 

 vertical boundary is now 



186. The equation (3) of Art. 185 is identical in form with 

 that which presents itself in the theory of the vibrations of a 

 uniformly stretched membrane. A still closer analogy, when 

 regard is had to the boundary conditions, is furnished by the theory 

 of cylindrical waves of sound*. Indeed many of the results 

 obtained in this latter theory can be at once transferred to our 

 present subject. 



Thus, to find the free oscillations of a sheet of water bounded 

 by vertical walls, we require a solution of 



O ........................ (1), 



subject to the boundary condition 



dg/dn = ........................... (2). 



Just as in Art. 175 it will be found that such a solution is possible 

 only for certain values of k, and thus the periods (27r/kc) of the 

 several normal modes are determined. 



* Lord Kayleigh, Theory of Sound, Art. 339. 



