detail by Tayfun, Dalrymple, and Yang (1976) In the particular case of 

 one-dimensional variations in currents and bathymetry. This case has a 

 simple analytical solution. 



There appear to be no published accounts of attempts to calculate 

 the refraction of wave fields for currents of only moderate complexity 

 (e.g., flow around a headland or out of an estuary) except for the early 

 work of Arthur (1950) on rip currents which preceded the full 

 understanding of th'e relevant equations. In fact, rip currents appear 

 to have too small a scale to be adequately dealt with by refraction 

 theory. Rather more complex computations are described in Noda (1974), 

 Birkemeier and Dalrymple (1975), and Ebersole and Dalrymple (1980) which 

 are concerned with nearshore wave-generated currents. It is difficult 

 to interpret and gain physical understanding from complex models before 

 models of intermediate complexity are understood. 



A different set of one-dimensional examples is that of currents 

 caused by wave motions. Examples are the tides, certain currents 

 produced by internal waves, and currents induced by surface waves much 

 longer than the waves riding upon them. The surface velocities due to 

 traveling waves can usually be described as functions of x - Ct, where C 

 is the phase velocity of these long waves. By considering the motion in 

 a reference frame moving with the long wave at velocity C, the current 

 field becomes steady and analysis of the shorter wave motion becomes 

 similar to that of the above examples. However, the character of the 

 current field, usually periodic, and the initial conditions for the 

 waves, created on a current of -C, are different. 



Longuet-Higgins and Stewart (1960) discuss short waves on swell and 

 waves on tidal currents. These are both cases where the "currents" are 

 also surface gravity waves, and in such cases, the vertical acceleration 

 of the surface should also be taken into account. As shown in Peregrine 

 (Section IIF, 1976), this does not make the analysis any more 

 complicated. The major effect of these long gravity waves on shorter 

 gravity waves is to cause the short waves to be steeper on the crests of 

 long waves and gentler in the troughs. Tide-induced variation of this 

 type initiated some of the earliest analysis of waves on currents (Unna, 

 1941, 1942). 



The interaction of short waves with swell led Longuet-Higgins (1969) 

 to suggest a possible interaction with the longer waves causing short 

 waves to grow. Hasselmann (1971) identified more possibilities for 

 energy transfer but disagreed with Longuet-Higgins on the magnitude of 

 any growth. Further work by Garrett and Smith (1976) introduces the 

 possibility of even more interaction when a wind is blowing. 



The behavior of water waves on the current field due to internal 

 waves has been studied more intensively than any other aspect of wave- 

 current interaction. This is largely because of the desire to 

 understand how internal waves in the ocean are generated. Interactions 

 with surface water waves are a possible growth mechanism for internal 



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