An alternative to using wave action is to consider the total energy 

 of the system, which is conserved in the absence of dissipation. For 

 steady flows and wave conditions, total energy is a relatively 

 straightforward quantity to use as Is shown by Jonsson (1978a). He also 

 found that under irrotational flow conditions total energy flux is pro- 

 portional to wave action flux, B. 



B = (u + C„)A (25) 



One feature of this method of solution which is rarely pointed out 

 is that the current field should satisfy the nonlinear shallow-water 

 wave equations. In that approximation, the horizontal flow is uniform 

 with depth. This causes no problem in most cases, but there are some 

 detailed difficulties in reconciling the general equations with the 

 deepwater limit. (For linear waves, the deepwater limit is no problem.) 

 This point is mentioned again in Section II, 10 on nonlinear effects. 

 As noted previously, vertical velocity profile effects do have 

 influence, but except for unidirectional problems, no refraction 

 examples have been examined. 



5 . Refraction of Waves by Currents — Simple Examples . 



For those cases in which refraction problems can be reduced to 

 problems depending on one coordinate only, being uniform with respect to 

 other coordinates, it is possible to find analytical solutions. For 

 example, the consistency condition (eq. 14) reduces to Snell's law 



k^ = constant (26) 



if there is no variation in the v.^ direction. An interesting range of 

 problems can be solved in this manner. 



One example, which has similarities with waves obliquely incident on 

 a beach, is a current V(x)j, where j is a unit vector in the positive y 

 direction. That is, consider a current perpendicular to the direction 

 of variation x. Any type of shear of this unidirectional current is 

 possible. A relatively detailed discussion is given in Peregrine (Sec. 

 HE, 1976) where different axes are chosen. 



In this case, it is relatively simple to understand what is 

 happening simply by considering how the current acts to convect the 

 waves. For simplicity, a horizontal bed is assumed. If waves propagate 

 onto steadily stronger currents, wave direction turns toward the slowest 

 part of the current when the waves have any component of propagation in 

 the current direction (compared with depth refraction). See curves for 

 60° and 240° in Figure 6. On the other hand, if waves have a component 



33 



