Bessho 



planing boats or for submerged thin wings, [6,24] The second is 

 based on Gausz's principle, which converts the boundary value prob- 

 lem to a variational problem. This method is shown to be an exten- 

 sion of Riabouchinsky's principle of minimum virtual mass. [3,24] 



The latter principle is based on the stationary character of 

 the Lagrangian and has recently been used by Luke, in a more general 

 form, to study water wave dispersion problems , [3,12,13] We 

 also have analogous principles for light and sound wave diffraction 

 and for the radiation of energy due to the heaving, swaying, and 

 rolling oscillations of ships. [25,27,28,29,30] 



The variational principles emphasize the dynamical meaning of 

 the boundary value problems and permit us to solve them approxi- 

 mately by the Rayleigh-Ritz- Gale rkin procedure, [6,28,29] How- 

 ever, when we try to apply these principles to our problem, there 

 are two difficulties: 



The first is that our system is not conservative because of 

 the trailing wave. This may be bypassed by introducing an artificial 

 model, as in Fig. 2, or by introducing a reversed flow for the 

 linearized case. 



The second difficulty is for the surface-piercing body, in which 

 case the wave profile is not known, a priori, even in the linearized 

 case. This difficulty may be avoided by introducing homogeneous 

 solutions [ 27] which appear in the case of a surface pressure distri- 

 bution. [ 26] 



Finally, although a variational method does not necessarily 

 represent a new method of analysis, it does suggest new methods of 

 approximation. For this reajson, it may be useful, especially for 

 engineering purposes. 



REFERENCES 



1, Lamb, H. , Hydrodynamics , 6th Ed. , Cambridge University 



Press, 1932. 



2. Wehausen, J. V. andLaitone, E, V. , "Surface Waves," Handbuch 



der Physik Bd. 9, Springer and Co. , l9bU. 



3. Gilbarg, D. , "Jets and Cavities, " Handbuch der Physik Bd. 9, 



Springer and Co. , i960. 



4, Milne -Thomson, L. M, , Theoretical Hydrodynamics , 4th ed, , 



Macmiillan and Co, , 1962. 



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