Yim 



This is a function of only k^x and k^y. The case when x^ o, y ^o for a certain 

 k^ is exactly the same as the case when k^ -*o for certain fixed values of x and 

 y. For k^-* 0, or the case of infinite Froude number, 



t>y = on z = . 



Therefore, for any k 



0, (46) 



x^0 = onz = 

 y - . 



The above argument can also hold for a point (l + x,y,o) as x^o, y^o. 



Although we considered points only on z = , we notice from the potential 

 theory that physical quantities change continuously into the potential flow field 

 from the boundary. This indicates that every surface ship which is represented 

 by a centerplane source distribution has as strong an influence of the free sur- 

 face on the shape of the ship in a certain neighborhood of the free surface as in 

 the case of infinite Froude number. 



The influence of the free surface can be explained much more eloquently by 

 Green's formula for the velocity potential 4 which satisfies the Laplace equa- 

 tion (2) with the boundary conditions (3), (4) and 



<^„=|^---nV. (47) 



on 



(n is the normal vector at the ship hull surface into the fluid.) On a given ship 

 hull, 



<^=4^ r [ [0Cb",^,O G^Ccf,77,^,x,y,z) - 0^(^^T],O G(^-,r,,r.,x,y,z)] clS (48) 



where S includes the free surface Sp and the ship surface S^ (see Fig. 2). G is 

 the well known Green's function (see e.g., Stoker 1957) which is a harmonic 

 function for C <0 except at (x,y, z) where it has the singularity 



and G satisfies the boundary conditions (3), (4) and 



G = on r; = . 



The integral on the free surface Sp in (48) can be written by using (3) 



1088 



