For these short waves, a considerable amount is deduced by using 

 asymptotic methods. These include the case of waves trapped between 

 caustics and the low-mode number cases where the caustics are too close 

 together to be treated individually. 



No exact solutions were found for the above cases, so corresponding 

 results for both short and long waves are deduced for the case where the 

 current only varies with depth. A number of exact linear solutions for 

 special velocity profiles are given and compared with approximations. 



Coastal Engineering Significance . These solutions can be used in 

 several other contexts when a change of reference frame can make the 

 waves stationary. 



44. PEREGRINE, D.H., and SMITH, R., "Nonlinear Effects upon Waves Near 

 Caustics," Philosophical Transactions of the Royal Society, London, 

 England, Series A, Vol. 292, No. 1392, Aug. 1979, pp. 341-370. 



Keywords . Averaged Lagrangian; Caustics; Current Refraction; Currents, 

 Large-Scale; Currents, Shearing; Dispersion Relation; Theory; Waves, 

 Nonlinear. 



Discussion . The usual linear theory of waves in a slowly varying medium 

 indicates that wave amplitudes are especially large at caustics. The 

 effects of nonlinearity are considered by taking a general averaged 

 Lagrangian for near-linear waves. In the first approximation to a 

 slowly varying medium this shows that caustics are of two types: an R 

 type, which has a singularity of the approximation at small amplitude 

 and hence is probably regular, and an S type, which has no singularity 

 but has solutions growing without bound near caustics and hence, if 

 there is a limit to wave growth, such as breaking, the waves may reach 

 it. 



The theory is discussed for straight and curved caustics and a 

 detailed example of water waves on currents is given. A wide set of 

 current distributions is considered. They have the form of any current 

 field which depends on position through a single coordinate, 



e.g., U(x) i + V(x) j 



The character of the caustics that can arise is examined, and it is 

 found that both types of caustics occur. Further properties of these 

 are illustrated by PEREGRINE and THOMAS (1979). 



A second approximation improving the representation of the "slow 

 variations" leads to uniform solutions involving Airy functions for 

 linear waves. Details of their use are given. For nonlinear waves, 

 corresponding solutions involve a Painleve transcendent function. 



49 



