Approximate Dispersion Relation for Wave‐Current Interactions

Abstract

An approximate dispersion relation for water waves on a depthdependent current is examined. For infinitely deep water, the approximate dispersion relation provides asymptotic results that are identical to established asymptotic formulas for high and low wave numbers. Further, the approximate dispersion relation provides a highly satisfactory representation to known, exact dispersion relations throughout wave number space. For water of finite depth, the approximate dispersion relation again provides asymptotic results that are identical to established asymptotic results for high wave numbers. For low wave numbers (long waves), the approximate dispersion relation gives results that are in error from established asymptotic formulas by an amount proportional to the standard deviation of the current about its mean value. This error is small for the conditions under which the approximate dispersion relation is derived. The goodness of the approximate dispersion relation allows the extension of the wave kinematic/wave action formulation to the analysis of wave‐current interactions for depth‐dependent current fields.