Whitham, 1962; Hayes, 1973; Whitham, Ch. 15, 1974). A discussion of the 

 problem is given in Peregrine and Thomas (Sec. 5, 1979). This means 

 that a ray approach is not directly available. 



One effect of wave dissipation, whether it is by bed shear, inter- 

 action with turbulence, or breaking, is for momentum to be transferred 

 from the wave motion to current motion. This is particularly important 

 in the surf zone where such wave-induced currents include longshore 

 currents and rip currents and cause wave setup. 



An effect of wave motion on flowing currents, described in Section 

 II, 3, is to change the bottom shear stress. This change can lead to 

 significant changes in the magnitude of the current depending on how the 

 current is maintained. Numerical examples are given by Chrlstof fersen, 

 Skovgaard, and Jonsson (1981). 



These last two effects can only be modeled by simultaneously solving 

 for both the current and wave fields; see Noda (1974), Birkemeier and 

 Dalrymple (1975), Dalrymple (1980), and Ebersole and Dalrymple (1980). 



11. Caustics and Focusing . 



The concept of a caustic arises out of ray theory. In a family of 

 rays, successive rays may cross. In the limit of a full, infinitely 

 dense, set of rays the successive crossing points define curves by their 

 envelopes. These curves are known as caustics; straight-line examples 

 are shown in Figure 7. However, in a smoothly varying refractive 

 medium, a single caustic does not initiate by itself. Rather two 

 caustics initiate from a cusp, which is like a focus (Fig. 8). 



Ray theory gives singular amplitudes at caustics and their cusps; 

 this is a shortcoming of that theory. Better approximations have been 

 known since the work of Airy (1838) in optics and are discussed in the 

 water wave literature (e.g., Pierson, 1951; Chao, 1971). The wave 

 behavior near a caustic is described by the Airy function, and all the 

 major wave field properties can be found directly from ray theory 

 solutions and the Airy function. Peregrine and Smith (1979) give some 

 appropriate formulas. Similar results hold at cusps of caustics where 

 the Pearcey (1946) function can be used (see also Peregrine and Smith, 

 1979). McKee (1974, 1975, 1977) has also studied linear caustics on 

 currents (see Smith, 1981). 



Caustics of two types occur in refraction patterns of rays (this is 

 based on depth refraction patterns because of the lack of real examples 

 of current refraction): (a) numerous crossings of a few rays 

 corresponding to weak focusing and forming weak caustics (sometimes 

 called "spaghetti" diagrams); (b) clear, well-defined focuses and 

 caustics caused by major topographic features, such as the edge of a 

 dredged channel or a lens-shaped shoal. 



48 



