Viscous effects on transient long-wave propagation

Abstract

Using a perturbation approach and the Boussinesq approximation, we derive sets of depth-integrated continuity and momentum equations for transient long-wave propagation with viscous effects included. The fluid motion is assumed to be essentially irrotational, except in the bottom boundary layer. The resulting governing equations are differential–integral equations in terms of the depth-averaged horizontal velocity (or velocity evaluated at certain depth) and the free-surface displacement, in which the viscous terms are represented by convolution integrals. We show that the present theory recovers the well-known approximate damping rates for simple harmonic progressive waves and for a solitary wave. The relationship between the bottom stress and the depth-averaged velocity is discussed.