where 



by 



90 



r^^ = x^ + / + (z + f )2 



[2] 

 r/ = x^ + y2 + (z - f)2 



m/r^ represents the velocity potential of the source In an un- 

 bounded fluid, 



-m/rg Is the potential of a sink of equal strength at the Image 

 point under the same conditions, and 



0^ is the potential due to wave motion. 



For extreme Proude numbers ^^ vanishes; thus the potential is given 



^ = f-^ [3] 



In the other limiting case, F ->• 0, the surface acts as a rigid cover; the ap- 

 propriate expression for <^ is then 



^ = ?-+?- [4] 



a term, say ^ ' due to waves, disappears again. The expression 



is legitimate as (1); it has been used by Dickmann. 



Using the surface condition 



d' 



i^.^-MU=0 [6] 



§2x2 ^ "o dZ '^ax 



(where fi is Rayleigh's friction, Reference 25, K =3^) 



O q2 



and Laplace's equation 



A0 = [?] 



Havelock gives the expression for the velocity potential 



^-ni _"L _V!„^ r+% P„^. f exp [-Wf-z) + i/C6;]dx [8] 



r, r. 



^«« 1 



sec'^QdG 



K - K sec + ifisec 9 



where Re stands for the real part. Dickmann 's form yields 



