Wave Resistance 



nonlinear free- surface effects is emphasized, especially at low Froude 

 numbers. 



The second-order theory for thin surface ships was outlined by Wehausen 

 (1963) and subsequently has been studied extensively by Eggers (1966), Yim 

 (1968), and Wehausen (1968). In principle, the numerical calculations of wave 

 resistance are tractable, but as yet these have not been completed. 



Salvesen (1968) and Newman (1968) have considered certain aspects of the 

 third- order theory, from different viewpoints. Salvesen showed that third- 

 order effects were negligible for the submerged two-dimensional cylinder, thus 

 validating the second- order truncation. Newman showed that no resonant non- 

 linear energy exchange mechanisms exist in the third- order solution of the 

 three-dimensional free-wave distribution, as opposed to the opposite conclu- 

 sion reached in the case of wind- generated wave spectra by Phillips and others. 



The above-mentioned work is all based upon the Eulerian description (with 

 the notable exception of Wehausen (1968) who has formulated the thin-ship 

 problem in Lagrangian coordinates) of the fluid motion and the asymptotic ex- 

 pansion in terms of a suitable, small parameter such as the ship's beam or the 

 wave elevation. The leading- order contribution to the asymptotic expansion is 

 the classical linear solution associated with Kelvin's ship-wave pattern and 

 Michell's wave-resistance integral. Second- and third-order contributions can 

 be regarded as approximations to the nonlinear effects, although solutions ob- 

 tained in this manner are, at each successive order of approximation, associ- 

 ated with linear boundary value problems. An entirely different and inherently 

 nonlinear approach to wave problems is that developed by Whitham and others, 

 in which the wave system is assumed to be of large amplitude but slowly varying 

 in amplitude and wave number, so that it is in essence a perturbation of the 

 exact nonlinear solution for purely periodic unidirectional wave motion. (Vari- 

 ous papers describing and using this method are contained in the Proceedings of 

 the Royal Society of London, Series A, Vol. 299, No. 28, 1967.) This method has 

 been applied to the study of ship waves by Howe (1967, 1968) with striking non- 

 linearities along a "cusp" line where, in effect, a shock wave is formed. So far, 

 however, this study has been restricted to a system of diverging waves only, 

 since the assumption of slowly varying wave numbers does not permit the simul- 

 taneous existence (and interaction) of diverging and transverse wave systems. 

 The interrelationship between the Whitham technique and the Eulerian perturba- 

 tion approach has been examined in a recent paper by Hoogstraaten (1968). 



Gadd (1968) has proposed a second- order correction for the hull boundary 

 condition which is similar to the earlier technique of Guilloton, A computer 

 program has been written which incorporates this correction and which is 

 claimed to give realistic estimates of the wave resistance of fairly fine hull 

 forms. (The free- surface nonlinear ities were neglected after initial investiga- 

 tions showed that they required excessive computer time.) 



VISCOUS EFFECTS 



One only need observe the flow in the wake behind a ship's stern to realize 

 that viscous effects, and especially separation, affect the ship's waves and 



1550 



