Newman 



in principle, on the grounds that its existence violates the preassumed 

 condition of a slowly varying wave system which is the basis for 

 Howe's work. 



V. THIRD-ORDER INTERACTIONS IN KELVIN WAVE SYSTEMS 



One of the fundamental properties of a linear boundary-value 

 problem is the principle of superposition; thus, for example, Kelvin's 

 ship-wave pattern, although originally derived for a single "pressure 

 point," is valid for any distribution of singularities and hence for 

 arbitrary ship hulls. But as soon as the assumption of linearity is 

 discarded, the possibilities for nonlinear interactions, among the 

 previously independent components of the solution, must all be 

 examined. In water-wave theory it was shown ten years ago by 

 Phillips (cf« Phillips [ 1966]) that for deep water gravity waves the 

 second-order interactions are relatively uninteresting, but when 

 third-order effects are included it is possible for "resonant" inter- 

 actions to occur. Thus two or three primary waves can interact, 

 over large scales of time and distance, so as to transfer a substantial 

 portion of their energy into a completely new wave system of a differ- 

 ing wavenumber. This striking result has been confirmed by others, 

 both theoretically and experimentally, and can be regarded as well 

 established. 



Motivated by the occurrence of third-order interactions in 

 ocean wave systems, and by the striking nonlinear effects obtained 

 for a special case of the ship- wave problem by Howe [ 1967, 1968] 

 as noted in the previous section, I have studied the third-order per- 

 turbation solution of the Kelvin wave problem. The details of this 

 investigation are "messy," to say the least, and will be presented In 

 a separate paper (Newman [ i97i]), but I shall briefly describe the 

 technique employed and the form of the results. First, as a pre- 

 liminary approach to this problem, we may examine the possibility 

 that, at any point in the Kelvin wave field, the transverse and diverg- 

 ing waves are such as to satisfy the criteria developed by Phillips 

 for resonance between two primary waves. It Is not difficult to show. 

 In fact, that the wavenumbers of the diverging and transverse waves 

 are not resonant, except possibly on or near the cusp line, where the 

 simple stationary phase results are Invalid, 



To develop a complete solution of the ship-wave problem valid 

 to third-order would be a formidable task; local effects near the hull, 

 and nonllnearltles associated with the boundary condition on the hull 

 would have to be Included, and the possibility of a breaking wave near 

 the bow would raise fundamental questions of validity of the solution. 

 Instead, we focus on nonllnearltles associated only with wave propa- 

 gation on the free surface, and taking place slowly over scales of 

 many wavelengths, so that local effects and hull nonllnearltles can 

 both be neglected. The first-order linearized velocity potential must 

 satisfy the familiar free -surface condition 



534 



