incident waves and a completely different wave field from that predicted 

 by linear theory can arise. A sketch of the lines parallel to k. at such 

 a focus, with jumps, is given in Figure 9. Additional details, 

 including the waves reflected, are given in Peregrine (1982). 



The nonlinear results can be interpreted as a tendency to smooth the 

 wave field; hence, for the case of weak focuses, it may be justifiable 

 to smooth linear computations so that they do not arise. Further work 

 on this topic is desirable. 



12. Wave Breaking . 



There is still much that needs to be understood about the breaking 

 of both wind-generated waves on deep water and waves approaching a 

 beach. Currents affect the likelihood of breaking, producing, for 

 example, the less steep waves over rip currents already mentioned. 



Refraction of water waves can cause breaking whether that refraction 

 is due to variation of currents or depth. Breaking may be due to a 

 convergence of wave action or to a slowing down of wave action transport 

 when the flux is constant. Both phenomena can occur on simple shear 

 flows (Isaacs, Fig.l, 1948; Jonsson and Skovgaard, 1978). 



The focusing described in Section II, 11 is another description of 

 the convergence of wave action by current. As indicated, nonlinear 

 effects can delay or eliminate any singularity, but if Peregrine's 

 (1981a) caustic parameter is small enough, the waves will break. 

 Features of wave breaking for this case, which is essentially three 

 dimensional, have not been studied. 



A slowing down of wave action transport occurs as waves approach 

 beaches; the total group velocity decreases, causing waves to steepen 

 and break. Although a slowing down also occurs for waves on opposing 

 currents (see discussion of Fig. 2, Sec. 11, 2), the wave breaking that 

 occurs on opposing currents usually has a different character from that 

 on beaches. The waves are generally more complex and the surface motion 

 appears to be more oscillatory than translatory. In part, this is 

 because water depth is important on beaches and is less so on opposing 

 currents. Also the general refraction pattern is often quite simple on 

 beaches and is probably complex on currents. 



In both beach and current cases, linear theory predicts reflection 

 if waves are gentle enough, but the details are very different. On a 

 beach, linear theory predicts reflection, and most of the reflection 

 occurs within one wavelength of the shore. In practice, waves often 

 break before this point. 



Against a current reflection can be assumed to occur in the first 

 "wavelength" of the Airy function, but the Airy function modulates the 

 wave in this case (see Peregrine (eq. 2.109, 1976) for the approximate 

 linear solution). Thus reflection takes place over a number of 



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