suitable multiples of the momentum conservation equation. This is 

 illustrated by considering the energy of a system of particles in a 

 moving system. 



Stokes' second-order wave solution is given, and particular atten- 

 tion is given to a term proportional to amplitude squared which can 

 appear in any of three places: (a) in Bernoulli's equation, (b) in the 

 velocity potential, and (c) as a change in mean level. 



The conservation equations for total mass, momentum and energy are 

 derived to second order for waves on a current. The equation for 

 "conservation of waves" (a consistency equation for the existence of a 

 phase function) is then added to these equations. Three examples are 

 considered in detail to demonstrate that currents and depths cannot in 

 general be specified in advance since they depend on wave conditions. 



The four unsteady equations for the case of unidirectional waves on 

 initially undisturbed flow are a hyperbolic system. It is shown to have 

 four characteristics with velocities equal to the waves' group velocity 

 and the long wave velocity in both directions. 



The paper was stimulated by LONGUET-HIGGINS and STEWART (1960, 

 1961) and broadens their discussion of radiation stress. 



Coastal Engineering Significance. This paper shows some of the diffi- 

 culties of dealing with the refraction of finite-amplitude water waves. 

 Water depths and currents may not be specified in advance but should 

 come from initial and boundary conditions. Unlike linear wave theory, 

 wave energy does not travel with a single group velocity. The group 

 velocity splits into two distinct velocities for perturbation (see also 

 Hayes, 1973 and PEREGRINE and THOMAS, 1979). 



60. WHITHAM, G.B., Linear and Non-Linear Waves, Wiley-Interscience, New 

 York, 1974. 



Keywords. Waves, Nonlinear. 



Discussion. This book draws together much of the research of the 

 previous 25 years. It is divided into two large sections. The first 

 section, "Hyperbolic Waves," deals with waves described by hyperbolic 

 equations, such as shallow-water waves, sound waves, and shock waves. 

 Five chapters discuss unidirectional propagation and are primarily con- 

 cerned with nonlinear effects. The remaining four chapters of this 

 section discuss propagation in two or three dimensions including geomet- 

 rical optics approximations for linear waves and work on shock waves. 



The second section discusses dispersive waves with particular 

 emphasis on water waves. There are discussion and derivation of the 



60 



