singular Perturbation Problems in Ship Hydrodynamias 



The two-term outer expansion is now: 



w(z;(3) ~ U + W,(^;p), 



with W| given as above. Its inner expansion to one term is easily- 

 found: 



w(z;p) ~ U -i^ . 



We cannot rccilly say positively that these two terms are the same 

 order of magnitude, but it turns out that they must be if this expres- 

 sion is to match the two-term outer expansion of the one-term inner 

 expansion. The latter is obtained readily from Green's solution for 

 w{z(C,)) which was given in (5-1). It is: 



/ .a\ ~ TT J. ^U sin a 

 w(z;(3) ~ U + y . 



Then, obviously, we find that: 



Cg = - U sin a. 



We cannot determine the other constants, C| , from the solu- 

 tions so far obtained. It is necessary to solve for the second term 

 in the inner expansion, and Rispin carries this through. Then, he 

 matches the two-term outer expansion of the two-term inner expan- 

 sion with the two-term inner expansion of the two-term outer expan- 

 sion, finding that C| = - aU/Tr. Thus, C\ is proportional to the 

 rate at which fluid leaves in the jet; the C, term represents a sink, 

 in fact. (The Cj term represents a vortex. ) 



Rispin obtains estimates for h as well, but the results are 

 rather complicated, and it would add no perspicuity to the present 

 section to repeat them. The important point in principle is that it is 

 possible now to specify the value of h and not come to a contradic- 

 tion as a result. The far-field description has effectively provided 

 a height reference, because of the effect of gravity. This effect 

 does not change the first-order inner solution, but it does modify the 

 second-order term. (The velocity magnitude is not constant on the 

 free surface in the second approximation.) 



In the second-order term of the inner expansion, there is 

 another interesting phenomenon, namely, the apparent angle of 

 attack changes. This means, physically, that the- occurrence of 

 gravity waves modifies the inflow to the planing surface. In the 



783 



