Salve sen 

 INTRODUCTION 



The determination of the wave resistance of ship hulls is one of the most 

 important and challenging problems the naval architect has to contend with. In 

 the last century, many outstanding scholars in the field have been engaged in 

 this subject; moreover, the last several years have brought a renewed interest 

 in the theory of ship waves. Due to the complexity of the problem, however, we 

 are by no means able to predict theoretically the wave resistance of a ship with 

 sufficient accuracy. 



The three methods in use for obtaining the wave resistance are (a) model 

 testing, subtracting the estimated viscous drag from the measured total drag 

 force, (b) wave survey behind the model, assuming small waves and using linear 

 wave theory, and (c) theoretical analysis, using linear theory and representing 

 the hull by some singularity distribution. The correlations between these three 

 techniques have been far from satisfactory. In many cases, the differences are 

 as large as 40 to 50 percent (1). Fairly good agreements have been achieved 

 only by introducing artificial correction factors; however, very little is known 

 in general about how these factors are related to physical parameters. 



The author believes that a main part of the discrepancies between analytical 

 and experimental results could be due to the neglect of nonlinear effects at the 

 free surface and that the viscous effect is probably not as important as often 

 stated. This nonlinearity is investigated here, and in particular its effect on the 

 wave resistance. 



The first nonlinear wave theory for irrotational flow of an ideal fluid was 

 derived by G. G. Stokes (2). Applying this theory to the problem of flow past 

 submerged two-dimensional bodies, one can easily relate the wave resistance of 

 the body to the far downstream wave height. On the other hand, no relationship 

 between the shape of the body and the wave elevation or the wave resistance can 

 be obtained from the Stokes wave. Appendix A gives a detailed discussion of 

 "Stokes Waves and their Application to Wave Resistance Problems." 



Only two investigators have applied a consistent second-order wave theory 

 to the problem of free-surface effects on the flow past bodies: M. Bessho (3) 

 and E. O. Tuck (4). Both restricted themselves to the simplified two-dimensional 

 case of a submerged circular cylinder. Bessho in his fine work derived correctly 

 the complex potential but unfortunately obtained the forces incorrectly such that 

 the most important higher-order term was not included. His final result and 

 many of his conclusions are therefore incorrect. Tuck, on the other hand, cor- 

 rectly obtained the wave resistance and the lift for the circular cylinder, and he 

 also correctly stated the very opposite conclusion of Bessho, namely, that for a 

 circular cylinder "it is more important to correct for nonlinearity at the free 

 surface than for the fact that the boundary condition is not satisfied exactly by 

 the first approximation on the body surface." This excellent paper by Tuck ap- 

 pears to be the only work to date in which the effect of nonlinearity at the free 

 surface has been treated correctly to the second order. 



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