singular Perturbation Problems in Ship Hydrodynamios 



the result is not yet of much use, since we do not know either function, 

 A| or a|. It is worth noting, however, that the inner expansion can 

 be rewritten: 



<|)(x,y,z;e) ~ Ux + <^|(x,0,z;€) . 



Thus, to two terms the inner expansion is determined entirely by the 

 far-field solution, the latter being evaluated on the centerplane. In 

 other words , in the near-field view, the fluid velocity (to this degree 

 of approximation) is caused entirely by remote effects. 



Solution of the j^g problem; This is much more straight- 

 forward, and the results are more interesting. We may expect that 

 $2 = 0(€^), since we still have the nonhomogeneous body condition 

 to satisfy. In this case, then. 



$2(x,Y,z;c) = A2(x,z;€) + B2(x,z;c)|Y 



:ondition require; 

 three-term inner expansion is: 



2 



and the body condition requires that BgCxjZjc) = € UHx(x,z). The 



2 I I 



!^x,y,z;c) ~ Ux + q',(x,z;c) + A2(x,z;c) + € UHx(x,z) |Y (. 



0(1) 0(e) 0(€^) 0(€^) 



The two-term outer expansion of this three-term inner expansion is: 



4>(x,y,z;e) ~ Ux + (3'f(x,z;€) + Uhj^(x,z;e) (y | . 

 0(1) 0(c) 0(€) 



The three-ternn inner expansion of the two-term outer expansion is, 

 from (2-13): 



4>(x,y,z;e) ~ Ux + a,(x,z;€) + -^ |y |o-, (x,z;e). 



(The ttg in (2-13) is not carried over to the above expansion, since 

 it originates in the third term of the outer expansion.) These two 

 match if: 



(r,(x,z;e) = 2Uhx(x,z;€) = 0(c). (2-22) 



Thus, finally, we have found (r|(x,z;e), the source density In 

 the first far -field approximation, as a function of the body geometry. 

 It is the familiar result from thin- ship theory. In addition, we can 



687 



