Bow Waves and other Non-Linear Ship Wave Problems 



REPLY TO DISCUSSION 



Gedeon Dagan 



Teohnion-Israel Institute of Technology 

 Haifa, Israel 



The authors have indeed applied Taylor's instability criterion 

 to a flow field which is different from that considered originally by 

 Taylor: The free- surface is curved, the basic flow is not uniform 

 and the instability is local. 



For a plane free -surface Taylor [ 1950] has shown that any 

 disturbance, of arbitrary wave length, is unstable for 9p/8y < 0; 

 moreover, the rate of growth of the disturbance amplitude becomes 

 large for small wave lengths. It is reasonable, therefore, to assume 

 that for wave lengths which are much smaller than the radius of the 

 curvature of the free-surface, Taylor's criterion is locally valid. 



The radius of curvature of the free-surface (r') may be 

 quite large. At instability V'Vgr' = 1, i.e. r'/T' ^ V'VgT'. For 

 the marginal Froude number characterizing instability, Fr^. s 1.5, 

 it was found that V /V^ = 0,7. Hence, in this case, r'/T'^ = 

 (V'^/Vi,^) Fr^= 1. For a draft T ' of order of a few meters , dis - 

 turbances of wave lengths much smaller than r' are physically 

 conceivable. 



Although the example suggested by Prof, van Wijngaarden, 

 of a collapsing spherical bubble, shows undoubtedly that Taylor's 

 criterion cannot be applied indiscriminately, its resemblance with 

 the case of a steady free-surface is questionable. The inspection of 

 Plesset and Mitchell [ 1956] article reveals that only for R/Rq "^ 

 (R being the actual radius of the bubble and Rq its initial radius) 

 the disturbances are unstable, although the pressure gradient is 

 positive. The two cases are quite different. 



We agree in principle with Prof, van Wijngaarden's criticism 

 concerning the need for a more rigorous treatment of the local 

 stability of a steady curved free-surface. Unfortunately, the basic 

 flow itself has been determined only approximately and further 

 refinements of the stability criterion do not seem justified at this 

 stage. The aim of the computations presented in the paper was 

 limited to offering a model of the free- surface breaking and the order 

 of magnitude of the marginal Froude number. 



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