Bow Waves and other Non-Linear Ship Wave Problems 



'///////////////A-/////. 



X=xVT' 



N =ri' /T' 



N (X)= Fr^ N| (X)+ Fr* N^ {X) ' 



Fig. 3. The free- surface shape in front of a rectangular body 



The dimensionless pressure gradient component normal to 

 the free- surface is proportional to 



- 1 + Fr 



4ri/dN, Y z d'^N,] 



T V dX dX I Hv2 ' 2 ,^2 / 



dX' 

 dX" 



(14) 



where the first two terms of the expansion contribute to order Fr^^ 

 in the pressure gradient. Taylor's marginal stability is reached for 

 the value of Fr which renders Eq. (14) equal to zero. This value 

 has been found to be Fr_ = 1. 5 and the point of instability at X = 0,3, 



Although the expression for the pressure gradient can hardly 

 be expected to converge rapidly at such a high Fr_ , the result is of 

 the order of magnitude of that found by Baba and would seem to con- 

 firm the mechanism of free-surface disruption assumed by us. 



The effect is nonlinear since only when taking into account the 

 second order term does the steepening of the free-surface depend 

 strongly on Fr . There is no bow drag in the small Fr^ limit. 



The present method suggests a possible way for determining 



613 



