Effect of Viscosity on Long Gravity Waves

Abstract

The effect of viscosity is examined on long gravity waves of small but finite amplitude. The reductive perturbation method combined with the usual boundary layer theory reveals that the inviscid Korteweg-de Vries equation is not affected by the viscosity if O(α⁻⁵)<R, where R is the Reynolds number and α(<<1) the wavenumber. For O(α⁻¹)<R≤O(α⁻⁵), the effect of viscosity modifies the Korteweg-de Vries equation and yields new types of equation. On the other hand, for R<O(α⁻¹), the complex phase velocity becomes purely imaginary and the free surface is found to be governed by a nonlinear diffusion equation which was first obtained by Nakaya.