Bow Waves and other Non-Linear Ship Wave Problems 



[1967] 



T+\ 



exp 



L^^^o i-u^ -lUvT^VTTx L ^^-'o i-u2 J( 



(28) 



The separation of Eq, (27) can now be accomplished, pro- 

 vided that we select a given body shape h.(x). 



We limit ourselves here to the case of the completely blunt 

 shape of Fig. 4d. The forebody length i^ is of order e, such that 

 at the limit e -*" the bow degenerates at first order into a point 

 singularity at the origin. Any shape with the same length scale of the 

 forebody will yield the same first order body condition. 



With (Fig. 4e) 



H;(x) = -^r e'^^h,(e)de 



77772 i\ ^ .TTiTF i'k-i/i <^9) 



(2Tr) (2ir) 



we obtain fronn the separation of Eq. (27) 



^'*<^^ = Tif Lx " ^^"^^ xtitfJ iFocy Mi^ ^ 'm*(\) 



(30) 



where the last term, representing eigensolutions , results from the 

 application of Liouville's theorem, C|. being arbitrary. 



Equation (3 0) cannot be inverted exactly, because of the 

 integral appearing in M*(X.) , but the inversion can be carried out 

 for large \ by expanding M*(\) . After carrying out this process 

 (see Dagan and Tulin [ 1970]) we arrive at the following expression 

 for g,(6) in the vicinity of the origin 



n 



where d|, are related in a unique manner to C|. . 



617 



