VARIATIONAL APPROACHES TO STEADY SHIP 

 WAVE PROBLEMS 



Masatoshi Bessho 



The Defense Academy 

 Yokcsukay Japan 



INTRODUCTION 



Although there have been many fruitful engineering applica- 

 tions of the theory of the wave-making resistance of ships, it is still 

 not possible to completely explain the wave resistance of the usual 

 surface-piercing ships. The so-called order theory gives us insight 

 into the structure and composition of our approximate theory; however, 

 we do not yet have a consistent and practical theory which is univer- 

 sally acceptable. 



The author has speculated on what would be the best approxi- 

 mation to our boundary value problem. In this connection, is there 

 a useful principle which corresponds to the Rayleigh-Ritz principle 

 in the theory of elasticity? The present paper will provide a partial 

 answer. 



Our first aim is to introduce a variational principle which 

 corresponds to the linearized boundary value problem. This is 

 accomplished by introducing Flax's expression from wing theory. [6] 



Our second aim is to find an alternate expression which will 

 enable us to treat blunt bodies, since Flax's method is useful only 

 for thin wings. Gauss' variational expression [ 24,25] for the 

 boundary problem of a harmonic function is introduced for this pur- 

 pose. This is shown to be equivalent to extremizing the Lagrangian 

 or kinetic potential. The resulting dynamical interpretation of the 

 boundary value problem is similar to the approaches of many other 

 authors who have studied free surface problems by using the 

 Lagrangian [3,12,13,14], 



I. FLAX'S VARIATIONAL PRINCIPLE 



The variational principle introduced by A. H. Flax in wing 

 theory [ 6] may be directly applied to our problem. Those unfamiliar 

 with this principle are directed to Appendix A. 



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