singular Perturbation Problems in Ship Hydrodynamics 



o-|(x,z) = 2Uhj^(x,z). 



(See (2-22).) This obviously had to come out this way, since we have 

 not yet introduced any effects of the presence of the free surface. 

 It should be noted that only the function A2(x,z) is not already deter- 

 mined. (Knowledge of o-|(x,z) allows us to express a^{x,z) ex- 

 plicitly, from (4-1) and (4-3).) 



A systematic treatment of the free surface- conditions leads 

 to the following: 



[A] = gZ, + m^^ 0(e) 



+ gZg + U^2, + UZ, *,„ + -^ (^f, + ^f,) + J ($2y) 0( e^ 

 + . . . , on z = 0; 



[B] = UZ,^ - $1^ 0(c) 



+ UZg^ - ^2, - Z,^,„ + ^,,Z,^ + ^2^Z2^ 0(6^) 



+ . . . . on z = 0. 



The lowest-order conditions in [A] and [ B] together require that: 



^, + K^, =0, on z = 0. 



'xx 'z 



We see that this is automatically satisfied by our #|(x,y,z) = A,(x,z) = 

 Q'l(x,z). (See (4-4).) The first term in the expansion for wave shape 

 in the near field is also determined: 



Z,(x,y) i - -^ C>| (x,0). 



This really says only that the free surface appears in the near field 

 to be raised (or lowered) by just the limiting value (as y -* 0) of 

 t,(x,y) in the far field. Again, a rather trivial result. 



2 



When we consider the € terms in the free-surface conditions, 



it is a different matter. The two conditions can be combined into the 

 following: 



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