Edge waves on a longshore shear flow

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Abstract
A theoretical analysis of the effect of a longshore mean shear flow on edge waves is performed in the framework of linear shallow water equations. A single equation describing edge waves as well as neutral shear waves is obtained. A numerical method of calculation is given for the dispersion relations and the wave pattern, accounting for any beach topography and any mean flow profile (remaining constant alongshore and with a straight shoreline). Numerical calculations are presented for a simple exponential flow profile and for a plane bottom. A Doppler shift in the frequencies and a variation in the offshore extension of the waves are found, depending on the maximum local Froude number of the current, F, defined as F = [V(x)/√gH(x)]max, where V(x) stands for the mean longshore current, H(x) for the depth, and g for the gravity. The maximum shift in frequency is for wave numbers of about Vx(0)2/gm and frequencies of about Vx(0), where Vx(0) is the shear at the shoreline and m is the beach slope. For instance, these maximum differences may reach about 40% for F=0.5. Waves of any wavelength can always propagate downstream, but they can propagate upstream only for F≤Fc∼0.7. For mean flows with F≥Fc only waves shorter or longer than some forbidden wavelengths can propagate against the current. An analytical dispersion relation of asymptotic general validity for short waves (corresponding to the gravity range in real beaches) is given. The numerical model as well as this analytical dispersion relation is tested by means of a nonplanar real topography.

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