Reflection of oblique waves by currents: analytical solutions and their application to numerical computations

Surface waves superimposed upon a larger-scale flow are blocked and reflect ed at the points where the group velocities balance the convection by the l arger-scale flow. In this study, we first extended the theory of Shyu & Phi llips (1990) to the situation when short deep-water gravity waves propagate obliquely upon a steady unidirectional irrotational current and are reflec ted by it. In this case, the uniformly valid solution and the WKBJ solution of the short waves were derived from the Laplace equation and the kinemati cal and dynamical boundary conditions. These solutions in terms of some par ameters (the expressions for which have also been deduced in this case) tak e the same forms as those derived by Shyu & Phillips, which by referring to Smith's (1975) theory can even be proved to be valid for gravity waves in an intermediate-depth region and near a curved moving caustic induced by an unsteady multidirectional irrotational current. In this general case, the expressions for certain parameters in these solutions cannot be obtained so that their values must be estimated in a numerical calculation. The algori thm for estimates of some of these parameters that are responsible for the amplitude of the reflected wave not being equal to that of the incident wav e in the vicinity of the caustic and therefore are crucial for the computer calculation of the ray solution to be continued after reflection, was illu strated through numerical tests. This algorithm can avoid the error magnifi cation phenomenon that occurred in the previous estimates of the reflected wave in the vicinity of the caustic using the action conservation principle directly. The forms of the solutions have also been utilized to clarify th e wave profiles near caustics in a general situation, which indicate that i n storm conditions freak waves characterized by a steeper forward face prec eded by a deep trough will probably occur in the caustic regions.