Bow Waves and other Non-Linear Ship Wave Problems 



(b) . The Inner Expansion and Its Matching with the Outer Sol ution 



We stretch now the coordinates and adopt the following inner 

 variables , 



J = ^/e; w = w; z = z/c; t = t/€; b = b/e (36) 



and expand the function Q, = in(l/w) = t + 10 in a perturbation 

 series 



?i = So + A|(€)n| + ... (37) 



For the body of Fig. 4d we obtain from the inner expansion of the 

 exact equations the boundary conditions for ^q specified in Fig, 4f, 

 which represent a nonlinear free-surface flow without gravity. The 

 conditions at infinity are provided by the matching with the outer 

 expansion. Only in the case of the straight bow of Fig, 4e are the 

 inner conditions so simple. In a general case we have to solve an 

 integral equation for 6 [ Wu, 1967] or to start with a given 0(?). 



The solution of Wq is readily found in the form 





\if^ +b 



(if^ - b 



^/ir 



-P( / 1^ ) <3«' 



where the exponential represents the eigensolutions of the problem, 

 3qj being arbitrary constants. 



Expanding Wq for large t, we obtain 



~ ^ I/O fj \ ~ \/9 /~ 2 i doi 



Wo= i - 2[(tA)'^ - Pb/Tr]^ + 2[(tA)^ - PbA] 1+ ' +...(39) 

 z = r 1±M dl=l- A[ (t/^f^ - pbA] + 2[ (t A)'/" - (3b A]' in I 



Wo 



r . ~ ~ ~ 1/2 



.lin; - ai +2do|; +... (40) 



Before proceeding to the matching we rule out the eigen- 

 solutions appearing in Eqs. (38), (3 9) and (40) because they lead 

 either to an infinite velocity in the jet or to an infinite jet thickness, 

 depending on whether doi are positive or negative. The matching 

 of Wq and z (Eqs. (39) and (40)) with w and z (Eqs. (34) and 

 (35)) now gives 



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