Viscous effects on transient long-wave propagation

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.