domain D , where D, is the near field including S , and D„ is outside of D n . Then 

 2 1 s z 1 



at the interface S of D and D , we need to have 



(4) 



'in " "^n 



The outer potential <j>„ in D is assumed to satisfy the linear free surface condition 



*2tt + S *2z ' ° (5) 



3 4 



For such <j>„ we know the time-dependent Green function. 



Now we will construct a Lagrangian for the previously described problem, con- 

 sidering the Lagrangian that Luke used 



J JJ (Y V *l V *l~*lt) dzdxdt + J J gf dxdt "J J (*!" I * 2 ) <}) 2n dzdt (6) 



J = 



D l ° S 1F ° S J 



where (j> , ())_ and h vary with time, and t is a sufficiently large time after the body 

 has either exited from or come to rest in the water so that we can safely assume that 

 the variation &$ , 6(f> , n 6$ vanishes at t = T, It will be shown later that the 

 use of a convolution integral necessitates only the initial condition without the 

 condition at t = T. 

 Since 



.h(x,t) r h(x,t) 



_ds + <Kz=h) h (7) 



Lj M , -J 



we have 



