be used--a problem in itself in some cases. The equations given by 

 Stiassnie and Peregrine (1979) are a good starting point, but there is 

 no simplification to a ray theory such as there is for linear waves. 



Simple examples, depending on a single coordinate, and using 

 accurate periodic wave solutions, are given by Stiassnie and Peregrine 

 (1980) and Ryrie and Peregrine (1982). These are for waves incident on 

 a beach with zero mass flow toward the beach; they show that current and 

 depth variations are quite small in the absence of dissipation. This 

 suggests that for some engineering purposes an inconsistent 

 approximation neglecting current and depth changes due to finite- 

 amplitude irrotational effects might be adequate. However, a wider 

 class of problems should be examined before reaching such a conclusion, 

 since nonlinear effects from bed friction can be important; for example, 

 in the surge-tide interaction (Sec. II, 6). 



The major differences from linear theory are, as may be expected, 

 for the steepest waves. An advantage of using accurate nonlinear 

 solutions in a refraction calculation is that where the steepest possi- 

 ble progressing wave is predicted, it can reasonably be assumed that 

 real waves will break within a few wavelengths of that position. See 

 Stiassnie and Peregrine (1980) for more details. Sakai, et al. (1981) 

 present experimental information on the effect of nonlinearity for waves 

 breaking on an opposing current with three different bed slopes. 



For deep water, with currents defined independently of the waves, 

 Peregrine and Thomas (1979) give the results of refraction calculations 

 for the two current distributions u(x)i and Vj . The major interesting 

 results concern the neighborhood of caustics, as discussed in Section 

 II, 11. Although it is not strictly consistent to define the current 

 (e.g., see Mclntyre, 1981), in these cases no significant error is 

 likely. 



Both Peregrine and Thomas (1979) and Ryrie and Peregrine (1982) show 

 that for waves nearly perpendicular to the gradient of the medium, e.g., 

 for waves whose direction is at glancing incidence to a beach, the 

 refraction differs from linear theory in a qualitative manner. This is 

 discussed in more detail in Peregrine (unpublished, 1982) where it is 

 shown that diffraction becomes the dominant influence, suggesting that 

 linear ray theory may be unreliable even for quite gentle waves in some 

 circumstances . 



It is difficult to extend refraction theory for nonlinear waves to 

 complex realistic examples. The above-mentioned cases are all dependent 

 on a single coordinate so that much of the necessary work can be done 

 analytically. For nonlinear waves, there is no direct equivalent of 

 group velocity. The differential equations have several characteristic 

 velocities. For one -dimensional problems, there are two velocities 

 corresponding to the single group velocity of linear waves plus another 

 long wave velocity corresponding to depth and current changes (see 



47 



