The dispersion relation at large frequencies for linear surface gravity waves on a Maxwell fluid exhibiting a spatial decay

Abstract

The phase velocity of linear surface gravity waves on Maxwell fluids exhibiting a temporal decay and waves exhibiting a spatial decay have similar phase velocity in the limit of long waves where both the dimensionless wave number and frequency is small. For dimensionless wave numbers and frequencies of the order of unity in magnitude the phase velocity curves of these two types of waves is completely different. This is the case even for waves on Newtonian fluids. It is shown that both of these two phase velocity curves for any Maxwell fluids with non-zero relaxation time resume the value of the phase velocity of a Rayleigh wave on an incompressible Hookean solid in the limit of infinite wave number and frequency.