Pien 



stream lines. But if the body is piercing the undisturbed free surface, why 

 should just this distribution be selected for calculation of wave resistance ? 



Intuitively, I would prefer the combined source-dipole layer on the surface 

 used in Green's theorem, as this has minimal, i.e., zero-inner kinetic energy. 

 In any case we have to formulate proper restrictions for the flow within a ship's 

 waterplane area to keep variation of resistance calculated in reasonable limits. 



DISCUSSION 



J. N. Newman 



David Taylor Model Basin 



Washington, D.C. 



There has been considerable discussion this afternoon concerning the rela- 

 tive importance of nonlinearities in the free surface condition, and now Dr. Pien 

 has advanced the suggestion that the linear free surface condition is valid for 

 "fat" ships if they are ships of low wave resistance, or that fat ships of mini- 

 mum wave resistance are equivalent to thin ships, as far as the free surface 

 condition is concerned. This may in fact be a valid analogy from the engineer- 

 ing standpoint, but I hope that it will not be confused with a rigorous mathemati- 

 cal development. 



A necessary condition for the linearized free surface assumption is that the 

 elevation of the free surface is everywhere small, compared to the wave length 

 of a characteristic wave. This is clearly true for a thin ship since (in an ideal 

 fluid) the fluid disturbance and free surface elevation can be made arbitrarily 

 small by making the ship sufficiently thin. The free waves or far-field disturb- 

 ance associated with a ship of minimum resistance will also be small because 

 wave resistance implies wave energy radiation, but the free surface disturbance 

 near the ship will not necessarily be small since this is a local disturbance and 

 is essentially independent of the wave resistance of the ship. An obvious exam- 

 ple is the waveless (infinite draft) ship discussed by Dr. Yim; for this case 

 there will be no free waves and the linearized free surface condition will clearly 

 be justified in the far-field, but close to the ship there will be a local disturb- 

 ance which can be made arbitrarily large simply by increasing the singularity 

 strength. In other words, a ship of low wave resistance will satisfy the linear 

 free surface condition over most of the free surface, but not necessarily close 

 to the ship. 



Please let me emphasize that my objection is based upon the rationality of 

 the theory, and not upon practical considerations. For practical purposes I 



1140 



