708 



where an arbitrary integration-function of t has been absorbed In .^L Including in 

 it also the constant 10.//O on the surface, we have, to the same order of approximation. 



for a email disturbance of the free surface 



-(HI 



-t- qzi =.0 



, free surface. 



Now kinematically the velocity of the fluid normal to the surface must be the same as the 

 normal component of the surface velocity so that approximately (to first order terms) 



Differentiating the preceding equation with respect to t and substituting for -^t 

 we obtain for our second boundary condition 





(3 



Suppose ^ ( /l> Z. , c ) is the cylindrically syrmnetric velocity potential for the 

 case of a bottom only, i.e., no free surface. Then 



# = 



& 





via 



where "TTT- is 'the strength of an assumed simple source, i.e., a point source with uniform 

 radial flow. Let ^ ( A, , Z , L ) be the additional term required for a free surface to 



be satisfied also. Thus 



Inasmuch as ^ satisfies Laplace's equation subject to the condition of the fixed bottom, 

 it follows from equations (l) and (2), respectively, that 



V'$, =0 



s*. 



d' 



_— =0 , Z--0 



