Case (a) can be dealt with adequately by using the parabolic 

 approximation to simplify analysis; see for instance Radder (1979) and 

 Booij (1981). Results of calculations for weak focuses using this 

 approximation are given by Buckley (1975). For coastal engineering 

 purposes, Buckley's results require an assessment in the context of 

 water wave refraction. 



Significant reflection from caustics (case b) seems to be partic- 

 ularly important in navigation; see for instance Pier son (1972) and 

 Smith ( 1976) . The former mentions waves reflected from a caustic formed 

 by a submarine canyon as the possible cause of shipwreck, the latter 

 waves from a caustic formed at the side of the Agulhas Current, giving 

 rise to frequent ship damage. 



Strong caustics can easily be calculated on the basis of linear 

 theory from refraction results, as indicated above. However, a caustic 

 or focus region is also a region of relatively large amplitudes so that 

 nonlinear effects might be important. 



Nonlinear behavior of water waves near caustics has been discussed 

 by Peregrine and Thomas (1979) for waves on currents, by Peregrine 

 (1981a) for circular caustics, and in a more general context by 

 Peregrine and Smith (1979). These works show that there are two types 

 of caustics, R and S. At R-type caustics, which are the only type of 

 water wave-depth refraction, waves are unlikely to break unless there is 

 too much convergence of wave energy before the caustic region is 

 reached. Peregrine (1981a) gives a "caustic parameter" which can be 

 used in estimating the likelihood of breaking, and the case of a perfect 

 focus is also analyzed. 



S-type caustics have nonlinear solutions which suggest that waves 

 will normally break at such caustics unless their incident amplitude is 

 very small. For water waves, some caustics on currents are type S; 

 these include the "stopping current" caustic. Peregrine and Smith 

 (1979) analyzed the different types of caustics for deepwater waves on 

 currents to indicate which are S- and R-type caustics. 



At R-type caustics, which are the most common for water waves, the 

 above-mentioned papers give an incomplete analysis since they do not 

 include interaction between incident and reflected waves. The 

 importance of this interaction is shown by Yue and Mei (1980) who 

 describe reflection of nonlinear water waves from a small-angle wedge. 

 They find that reflection leads to a "wave jump," i.e., there is a jump 

 in wave properties, amplitude, direction and length, along a line 

 starting at the apex of the wedge. Reflection at R-type caustics may 

 lead to similar jumps. Peregrine (1982) shows that caustics must be 

 considered from their initiation, i.e., at cusps, focuses or 

 discontinuities. It appears that nonlinear effects oppose the 

 convergence of wave energy, and even a refraction calculation for a 

 caustic cusp may have no singularity. If it does have a singularity 

 then one, or two, wave jumps form. These jumps partially reflect the 



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