Ship Waves and Wave Resistance 



surface. The Jacobian is equal to 1, and the one-to-one correspondence holds. 

 Since f(x,y) is small, the first-order potential in (f.r,,/;) space again satisfies 

 the Laplace equation. Thus, the first-order problem is essentially the same as 

 that of Michell's solution except the interpretation of the free surface. The 

 higher order potential, in general, satisfies the Poisson's equation in the trans- 

 formed space. Thus, the effect due to the change of ship singularity domain 

 comes from near the ship bottom rather than the free surface. 



If we investigate the influence of the Froude number (or the wavelength) on 

 the higher order potential, we find that we have to be careful to analyze the or- 

 der of magnitudes. The inverse of the wavenumber is equal to the square of the 

 Froude number, and this is a smaller quantity than the beam/length ratio for the 

 usual merchant ship. However, it does not seem to affect the consistency of the 

 development except in the case of a high-speed, small-draft ship where the 

 second-order potential is simpler. This case was also discussed by Wehausen 

 (3). 



I certainly believe that Dr. Eggers' contribution will be a very significant 

 one in the development of more accurate ship wave theory. 



DISCUSSION 



Lawrence W. Ward 



Webb Institute of Naval Architecture 



Glen Cove, New York 



It is a great pleasure for me to have the opportunity to comment on this 

 paper, as I had the rewarding experience of spending the past academic year at 

 the Institut fur Schiffbau der Universitat Hamburg under a National Science 

 Foundation Fellowship, and I therefore know something of the ingenuity and ef- 

 fort which my friend and colleague Dr. Eggers expended to produce these re- 

 sults. Development of the computer program for the linearized case including 

 the total wave field for shiplike bodies in an ideal fluid provides us with a mini- 

 mum test for our various wave analysis methods, which are based on the same 

 assumptions, that must be met more-or-less exactly, before we can expect any 

 hope of even approximate validity of these methods for an actual model in a real 

 fluid. I will comment further on this point in respect to the "X-Y" method in a 

 moment. The carrying on of a consistent next approximation beyond the linear- 

 ized theory, following the lead of Wehausen, promises to be even more interest- 

 ing. It is not obvious to me that the effect of viscosity alone must explain the 

 major portion of the discrepancy between the Michell theory and experiment — 

 this would be equivalent to saying that the linearized theory would agree with 

 tests run on the actual model in an ideal fluid; we must however await numeri- 

 cal results from the author to clarify this picture. 



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