Dagan and Tulin 



the influence of the bow shape on the breaking wave inception, and 

 therefore serves in selecting shapes which retard the phenomenon. 

 Its application to three-dimensional bows Is left for future studies. 



III. THE HIGH FROUDE NUMBER APPROXIMATION (THE JET 

 MODEL) 



(a) The Outer Expansion 



As Fr increases the breaking wave develops. Energy is 

 dissipated there and the momentum loss is associated with an induced 

 bow drag. To treat analytically this free-surface nonlinear problem 

 we have to adopt a flow model which: (i) permits an ideal fluid 

 representation of the phenomenon and (ii) lends itself to the lineari- 

 zation of the free- surface condition. 



We adopt here the jet model, well known from planing theories, 

 and neglect the returning jet flow, such that the jet monnentum loss 

 is equivalent to that of the breaking wave (Fig. 4a). Moreover we 

 expand the exact equations of flow in an e = l/Fr^^ small parameter 

 expansion, consistent with the usual linearized ship wave theory. 



The expansion, and the associated model, are probably valid 

 for sufficiently high Fr^, and we only may hope, like in other 

 asymptotic solutions, that the results are sufficiently accurate even 

 for moderate Fr_. 



We consider now an outer expansion with the outer variables 

 made dimensionless by referring them to U' and U /g (see Nota- 

 tion). Using a procedure followed in similar problems in the past 

 (Tulin [ 1965] , Wu [ 1967] ) we carry out the expansion in an auxiliary 

 t, plane (Fig. 4c) related to the complex potential plane (Fig. 4b) 

 through the transformation 



il= 1 - -JL (15) 



We assume now that the jet thickness t and the distance to 

 the stagnation point b are both o(l), such that under the outer 

 process e -* , C, = 0(i) they coalesce with the origin of the I, 

 plane (Fig, 4e). 



Under these conditions we obtain by a systematical expansion 

 of the complex velocity 



w(C) = 1 + 5,(e)w,(C) + 52(e)w2(;) + ... (i6) 



after taking into account that by definition the body profile has the 



614 



