Even where waves are too short to affect or be affected by bottom 

 friction, they can propagate through the turbulence generated by bottom 

 friction of a current. In other cases, the turbulence associated with 

 wind stress, which is in part transmitted to currents by wave breaking, 

 may be more relevant. There is dissipation of waves in turbulence, but 

 there is insufficient information to confidently assess its magnitude. 

 Skoda (1972) describes experiments on waves propagating through 

 turbulence, and Peregrine (Sec. V, 1976) reviews some of the few other 

 papers on the topic. Experiments are difficult to interpret, as Skoda 

 indicates, since turbulence may scatter the waves as well as increase 

 dissipation (Phillips, 1959), and since it is difficult to generate 

 turbulence, or waves, without setting up some small mean currents. Even 

 if these currents are measured, their effect can easily be as great as 

 the dissipation being sought. 



7. General Solution Methods . 



Only in simpler cases such as those described in Section II, 5, can 

 explicit analytical solutions to refraction problems be found. In some 

 respects, wave-current refraction is similar to wave-depth refraction 

 and in practice the two are combined, i.e., in the dispersion equa- 

 tion (5) both u and d are functions of position. 



The natural way to approach any large-scale problem is to use ray 

 methods following the wave. For steady currents, the most significant 

 difference from Stillwater refraction is that the wave orthogonal 

 direction, i.e., the direction of k, differs from the ray direction, u + 

 C , and hence an extra variable, the ray direction, must be accounted 

 for. No extra equations are required, but this extra variable and the 

 directional dependence of the dispersion equation (5) interfere with 

 some simplifications that are possible when u = 0. C is parallel to 

 k. 



In the process of calculating rays both a and k are also found, and 

 singularities of the method such as focuses or caustics may be identi- 

 fied. If wave conditions are required only at a point, it may be more 

 efficient to trace the rays backward from the point as is done for depth 

 refraction. 



Once rays have been found, the conservation of wave action between 

 rays gives the wave amplitude. In depth refraction, it is usually 

 efficient to use a differential equation as first described by Munk 

 and Arthur (1952); see also Skovgaard, Jonsson, and Bertelsen (1975). 

 A similar equation can be derived for the more general case involving 

 currents (Skovgaard and Jonsson, 1976). For the steady flow case, 

 Christoffersen and Jonsson (1981) find an integral of the energy 

 equation along the streamlines of the current. This cannot replace 

 the equations necessary to obtain k and a, but it may be useful in 

 some circumstances for finding wave amplitude and water depth. 



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