Dagan and Tulin 



U' Velocity at infinity 



z' = x' + iy' Complex variable; z = z'g/U'^ z = 2.^ /l^) 



Z = z'/T' 



P Angle at bow 



6| , 62* ^1 Gauge functions 



C = T'g/U'^ Small parameter 



^ = 4 + i^ Auxiliary variable; % - t,/^ 



T|' Free- surface elevation; "n = "n'g/U' ; T| = r|'/T'; 



N = t|7T' 



n = in(i/\v) Logarithm of complex velocity = t + i9 



I. INTRODUCTION 



The conventional linearized theory of ship waves is based on 

 a first-order perturbation expansion in which the length Froude num- 

 ber is of order one, while the beam Froude number (thin ships) 

 and/or the draft Froude number (slender of flat ships) tend to infinity. 

 While the theory is in fair agreement with laboratory results in the 

 case of schematical fine shapes (e.g. Weinblum et al. [1952]), it is 

 of a qualitative value at best in the case of actuarTiuIis. To improve 

 the accuracy of the linearized solutions, second order nonlinear 

 effects have been considered, either in the free -surface condition 

 or in the body condition (e.g. Tuck [1965], Eggers [ 1966] ) . 



A different nonlinear effect, overlooked until recently for the 

 case of displacement ships, is that associated with the bow bluntness. 

 It is well known from the theory of inviscid flow past airfoils or 

 slender bodies (Van Dyke [ 1957]) that the linearized solution is 

 singular near a blunt nose in the stagnation region. The singularity 

 may be removed by an inner expansion in which the length scale is 

 a local one associated with the nose bluntness. 



In the case of a free -surface flow with gravity the phenomenon 

 is more complex. The pressure rise in the stagnation region is 

 associated with the free-surface rise and the formation of a breaking 

 wave or spray and the existence of a genuine bow drag. The inner 

 expansion of the Bernoulli equation shows that the inertial nonlinear 

 terms become more important than the free gravity term, for 

 sufficiently high local Froude numbers. 



The bow nonlinear effects have been recognized a long time 

 ago in the case of planing plates (Wagner [ 1932] ), but they have been 

 always associated with a relatively high Fr, , such that the lift /buoy- 

 ancy ratio is of order one. Here we are prinaarily interested in the 

 case of displacement ships which move at a small Frj^ and the hull 



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