Wave Equations for Regular and Irregular Water Wave Propagation

Abstract

The Boussinesq equations are often used to simulate irregular nonlinear wave propagation in finite water depth. The mild slope equation is also frequently used for modeling regular linear wave propagation as there is no water depth restriction on it. In this paper a different approach is presented to derive a new set of equations. The Stokes first-order solution is used as a first approximation of distribution for velocity potential function or velocity over water column, and subsequently a new distribution function is derived. The new distribution function is then used to derive a new set of equations. For regular wave the linear dispersion relationship can be satisfied completely over entire range of water depth for the new equations. The shoaling gradient coefficient of the new equations is in good agreement with the theoretical value from the Stokes theory without water depth restriction. In finite water depth the new equations reduce to the improved Boussinesq equations. In very deep water the linearized new equations reduce to the classical wave equation. The new equations can be used for a regular wave propagating over the entire range of water depth taking into account the nonlinear effects in shallow water, or used for irregular waves in finite water depth. Numerical solutions of the new equations are compared with experimental data of regular and irregular waves. Good agreement is achieved.