To obtain Equation (17), we used a convolution expression of Equation (16), where we 

 can use the initial condition <j) = <J> = 0. 



Since 5<j)- , 6<f>„, and 6<j) are arbitrary, we obtain the corresponding equations 

 for a time-dependent linear free surface boundary value problem that has a unique 



i • 3 

 solution. 



If we lift out S so that D occupies the entire fluid domain, then the last 

 integral of Equation (15) disappears. The resulting equation appears much simpler 

 than that obtained by Murray due to a simple difference in the treatment of the 

 free surface condition. Equation (15) does not give the wave height as a natural 

 boundary condition, whereas Murray's corresponding equation does. However, from h = 

 (j> /g the wave height can be obtained. 



If we consider eigen solutions that satisfy only the Laplace equation; the 

 linear free surface condition, Equation (5); and the radiation condition in D U D , 

 then <J>, which satisfies the body boundary condition in Equation (11), can be derived 

 from Equations (15) and (17) by using 



(18) 



When we know a functional whose minimum value is attained by the solution, we 



g 

 can find the solution numerically by such methods as the finite-element technique 



9 

 or singularity method. 



For example 



N 



* * 21 m ± *i 

 i=l 



(19) 



where <b is the Green function, which is available for this problem for a source 

 l 



distribution on the body surface. The source distribution m. will be obtained from 

 soltuion of the simultaneous equations, 



^- = , i=l,2,...N (20) 



