Recent Research on Ship Waves 



where P denotes the right-hand side of (15), A solution correspond- 

 ing to (16) is readily obtained, and in the limit e -* we find that 

 the only modification is to replace Eq. (17) for the roots of the 

 denominator of W by the new equation 



^123 + (^ + ^ 2y ^^"^ ®J^^ " ^' ^^^^ 



j = l 



Thus the singularities in (16) become slightly complex, and for e > 

 the integrals in (16) are proper. 



Finally, the behavior of (16) as e -♦ must be examined. 

 Here the algebraic details are critical, since cancellation occurs 

 between many of the leading-order terms. In view of the numerous 

 possibilities for error, the following surprising result must be re- 

 garded as tentative, and I would hope that it will be verified inde- 

 pendently by others who are willing to tackle the algebra involved. 



As e — * 0, Eq. (16) predicts waves on the cusp line of the 

 same form (0 = 35°) as the first-order solution, but with an ampli- 

 tude which tends to infinity logarithmically in e. Thus there can 

 be no steady-state solution of the third-order initial value problem, 

 as posed in Eq, (18) and, presumably, in the more general case of 

 a "steady" moving disturbance initiated from a state of rest. Ulti- 

 mately, as in the analogous case for ocean waves, the logarithmic 

 growth rate will be modified by further nonlinearities , but, never- 

 theless, we must conclude that significant amounts of energy can be 

 exchanged, through nonlinear processes, in the region of the cusp 

 line, among adjacent wavenumbers. 



VI. CONCLUSIONS AND RECOMMENDATIONS 



The caption above is the standard one for theses and reports. 

 In this paper we have obviously raised more questions than we have 

 answered. The observations of the UNCATENA show that Kelvin's 

 wave patterns are confirmed, even for a highly nonlinear near-field, 

 provided the observation point is sufficiently far downstream (much 

 further than is possible in a conventional towing tank). But are the 

 differences noted in the photographs of the full-scale vessel and the 

 1 /24th- scale model due to differences in photographic conditions 

 and experimental errors, or are the transverse waves (and very short 

 diverging waves) of substantially larger amplitude on the model scale? 

 Here we would emphatically recommend further experiments in which 

 the wave heights can be measured quantitatively, both for the full- 

 scale vessel and for Its model. This task can be simplified if only 



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