On calculating the lengths of water waves

Abstract

A discussion is given of the physical approximations used in obtaining water wave dispersion relations, which relate wave length and height, period, water depth and current. Several known explicit approximations for the wave length are presented, all of which ignore effects of wave height and current. These are compared and are shown to model the usual linear dispersion relation rather more accurately than it describes the physical problem. A simple approximation is obtained: in terms of wave period T, depth d and gravitational acceleration g, which is exact in the limits of short and long waves, and in the intermediate range has an accuracy always bettern than 1.7%. Explicit approximations which include the effects of current are presented, plus an algorithm based on Newton's method which converges to engineering accuracy in one evaluation, and requires the specification of a value of current, which is a useful reminder that one is obtaining an approximate solution to an approximate problem, and no great effort should go into refining methods or solutions.