for a slowly varying wave train on deep water which includes both terms 

 due to the current variation and terms arising because the waves are not 

 infinitesimal (i.e., a near-linear approximation is used for the waves). 



Two derivations for such an equation are given, one systematic, the 

 other heuristic. The equation obtained is similar to the nonlinear 

 Schrodinger equation. The solutions of the equation are investigated 

 for the neighborhood of a caustic where it is expected that waves may be 

 at their steepest. A radiation condition is determined for matching 

 with simpler approximations. A stability analysis indicates that wave 

 trains in such circumstances do not suffer the usual modulational 

 instability of deepwater waves. 



The systematic perturbation approach also gives information on the 

 asymmetry of the wave profile. PEREGRINE and SMITH (1979) build on some 

 aspects of this work, but their work does not supersede this paper. 



Coastal Engineering Significance. Giant waves have long posed a threat 

 to navigation, and the interaction between waves and currents has been 

 suspected as a cause. This paper indicates that large stable waves are 

 possible on currents in deep water. The analysis does not provide 

 symmetric wave profiles, as is usually the case. 



54. STIASSNIE, M. , and PEREGRINE, D.H., "On Averaged Equations for 

 Finite-Amplitude Water Waves," Journal of Fluid Mechanics , 

 Cambridge, England, Vol. 94, No. 3, Oct. 1979, pp. 401-407. 



Keywords . Averaged Equations; Averaged Lagrangian; Conservation 

 Equations; Current Depth Refraction; Currents, Large-Scale; Flow, 

 Rotational; Momentum Equation; Theory; Wave Action; Waves, Finite- 

 Amplitude . 



Discussion. Equations for slowly varying wave trains in inhomogeneous, 

 moving media can be derived either from an averaged Lagrangian or from 

 averaging the equations of motion. In the former case a wave action 

 conservation equation is found for nondissipat ive flows. For surface 

 water waves such equations have been derived from averaged Lagrangians 

 for potential flow and directly from the equations of motion. 



In this paper the equations obtained from averaging the equations 

 of motion are manipulated into the same form as the equations derived 

 from an averaged Lagrangian. The motion on the scale of the waves is 

 assumed to be irrotational in both cases. It is found that wave action 

 is still conserved and that the only differences arise in the equations 

 which correspond to the consistency conditions of a pseudophase. One of 

 these is the irrotationality condition for the large-scale current. The 

 more complete equations permitting large-scale vorticity can be written 

 in various forms. Some include the vorticity explicitly; alternatively, 

 the equations clearly show the assumption of a shallow-water approxima- 

 tion for large-scale flows. 



56 



