A perturbation method to solve dispersion equations for water waves over dissipative media

Abstract

A simple and innovative method to solve explicitly the dispersion equation for water waves over dissipative media is presented. The roots of this equation are themselves complex and difficult to obtain by standard numerical methods. A general structure for the dispersion relation is given, the dimensionless wave number, x, depending on the value of a dissipation parameter, φ, which is a function of the dissipative medium. Based on the hypothesis that a small perturbation in the dissipation parameter, δ φ, produces a small variation in the dimensionless wave number, δ x, a simple approach is proposed to calculate explicitly in an iterative manner the complex roots. The new method improves upon the unreliability of standard numerical schemes in the calculation of the roots in dissipative media.