singular Perturbation Problems in Ship Hydrodynamics 



d 9 9 ^/ -i\ 



97 ' "9^ ' a? ^ °^^ ^- 



To a first approximation, the potential function satisfies the Laplace 

 equation in two dimensions: 



and the linear free-surface condition 



^^ - v,^~ on z = 0. (3-22) 



With the assumptions made above, the two terms here are of the 

 same order of magnitude. (If we did not assume high frequency, we 

 would obtain just the rigid-wall boundary condition, ^2 ~ ^•) This 

 condition implies that we shall be solving a gravity-wave problem in 

 two dimensions. At infinity, we know from the far-field solution 

 that the appropriate condition is an outgoing-wave requirement. All 

 that remains is to put the body boundary condition, (3-21), into the 

 appropriate form. 



Let 9/9N denote differentiation in the direction normal to 

 the body contour in a cross section. Then, from (3-19) and (3-21), 



^^^i+d;)'/2^ d+d^'/^ 



(l+d^)'/2 (l+d2)'/2 



The last simplification involves an error which is 0(e ) higher 

 order than the retained terms. To the same approximation, we can 

 write (see (2-72')): 



n, = n • k ~ ; n^ ~ - xn_. 



(l+d^^)'/2 ' ^ 



Thus, the boundary condition is: 



|r~S^3^^5^5' °^ z = d(x,y). (3-27) 



As in the infinite -fluid problem (cf. (2-73)), we can define 

 normalized potential functions, (^j(x,y,z): 



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