Submerged Two-Dimensional Bodies 



in which 3/Bn represents differentiation along the normal to the surface wall. 

 For two-dimensional bodies with a sharp trailing edge the Kutta-Jukowski con- 

 dition specifying that the trailing edge is the stagnation point should also be 

 satisfied. 



The depth of the water is assumed to be infinite, which results in the condi- 

 tion 



lim (grad $) = Ui . /rj\ 



y -» - 00 ^ ' 



One additional condition is necessary to ensure uniqueness, namely the ab- 

 sence of waves far upstream. To make the list of boundary conditions complete, 

 we should also mention the absence of wave reflection far downstream. 



Introducing the complex variable z = x + iy and the complex velocity poten- 

 tial W(z) = a)(x,y) + iw(x,y) and assuming that a perturbation about the flow past 

 the body in an infinite fluid can be carried out, we write the expansion 



W(z) = (uz+WbJ + e^wp^ + wgj + e2^Wp^+WBj+ ... , (8) 



where e is some physical parameter which vanishes with the disturbance at the 

 free surface. The potentials wq^, wg^, etc., are analytic everjrwhere outside 

 the body and are chosen so that the terms in parentheses in Eq. (8) satisfy the 

 cylinder -wall condition exactly. On the other hand, wp^ and wpj are analytic 

 for all im z < b and are the first-order and second-order free -surface contri- 

 butions. 



Assuming that the free-surface disturbances are small, of order e , we 

 have the following expansion for the surface elevation: 



7j(x) = e77(^>(x) + e27](2)(x) + ... . (9) 



It also follows from the assumption of small disturbances at the free surface 

 that the body singularities wq^, wg^, etc., are of one order higher in e near the 

 free surface than near the body. The expansion near the free surface, therefore, 

 is of the form 



W(z) = Uz + ew( ^) + £^^(2) + ^ (10) 



with 





Substitution of the expansion given by Eq. (10) in the "exact" free-surface 

 conditions of Eqs. (4) and (5) and including only first-order terms gives the 

 first-order free-surface condition 



599 



