Sideband damping of water waves over a soft bed

Abstract

The Benjamin & Feir sideband instability of gravity waves over a rigid bed is extended here to the case of wave propagation over a compliant bed. The bed is assumed elastic with inhomogeneous properties, following some vertical stratification profile. The elastic stiffness at the mudline is assumed very small, as is typically the case for upper-stratum gel-like marine mud, so that its effect is assumed comparable in magnitude with that of wave nonlinearity. It is then shown that through the sideband instability, the carrier gravity wave would lose energy to sideband oscillations. These sidebands will be shown to be contaminated with small-amplitude, but very short elastic shear waves. These shear waves will interact with the viscous boundary layer at the bed-water interface, and thus significantly enhance the viscous damping of wave energy in the boundary layer.