Long waves in water over a visco-elastic muddy seabed

Abstract

The propagation of surface waves over a flat muddy seabed are studied. Mud is first considered as a Newtonian fluid. Water and mud equations are derived in order to obtain governing equation for surface and interface waves. By the method of multiple scales. nonlinear evolution equations are derived for the harmonic amplitudes. These equations are numerically solved for a finite number of harmonics to show the behavior of surface and interface motions. A drift current in mud is found at the second order., meaning that surface waves induce mud motion. Equation governing the total wave energy variation is derived and used to verify the accuracy of numerical solutions. The model is extended to viscoelastic mud, using the results of four experiments carried on different types of muds. These muds range from very elastic to rather viscous, allowing us to compare hie differences in behavior. Surface and interface variations. mud drift current and energy variations are plotted and compared to the results with Newtonian muds. A sloping muddy seabed is then considered. Mud is modeled as viscoelastic to avoid a constant static current that would happen with Newtonian mud. By the method of multiple scales and by introducing a space-dependent wavenumber, mud drift current and energy variation equations are derived again, taking into account the effect of the slope. An analytic stud v of the surface variations demonstrates that surface and interface waves, as well as drift current, exponentially decay to reach a zero value at the shore.