Decay of free-gravity waves

It is generally believed that the major effects of viscosity are associated with vorticity. This belief is not always well founded; major effects of viscosity can be obtained from purely irrotational analysis of flows of viscous fluids. Here we illustrate this point by comparing irrotational solutions with Lamb's (1932) exact solution of the problem of the decay of free-gravity waves. Excellent agreements, even in fluids 107 more viscous than water, are achieved for the decay rates n(k) for all wavenumbers k, excluding a small interval around a critical value kc, where progressive waves change to monotonic decay.

Introduction

Lamb (1932, §348, §349) performed an analysis of the effect of viscosity on free-gravity waves. He computed the decay rate by a DM, using the irrotational flow only. He also constructed an ES for this problem, which satisfies both the normal and shear stress conditions at the interface.

Joseph and Wang (2004) studied Lamb's problem by using the theory of VPF and obtained a dispersion relation that gives rise to both the decay rate and the wave velocity. They also used VCVPF to obtain another dispersion relation. Because VCVPF is an irrotational theory, the shear stress cannot be made to vanish. However, the shear stress in the energy balance can be eliminated in the mean by the selection of an irrotational pressure that depends on viscosity.