Long wave energy reflection induced between a submerged plate and a viscoelastic mud: Determination of wave damping

In this work, the long wave–submerged plate–viscoelastic fluid interaction is analyzed. The decay of the elevation of the free surface is determined through the damping, reflection, and transmission coefficients. The system is further divided into two layers: the first is a domain composed of water, further divided into four regions for the submerged plate presence, and the second includes a viscoelastic fluid that obeys a Maxwell rheological model. The hydrodynamics of the water is modeled by adopting the assumption of a potential flow. The viscoelastic mud motion is determined using Cauchy's equations. The damping coefficient is obtained by applying the kinematic boundary condition at the water–mud interface. The horizontal patching boundary conditions for the equality of pressure and mass flux between the different regions are fundamental in determining the reflection and transmission coefficients. The viscoelastic effects are identified to induce an increase in the magnitude of the free surface elevation and, consequently, the reflection coefficient. The maximum damping coefficient is identified to occur when the thickness of the mud is 20% of the total depth of the water. The reflection coefficient increases with increments of the damping coefficient, and when the plate's submergence depth is 25% of the water depth, large values of the reflection coefficient are obtained. The present solution is validated with a classical analytical solution reported in the specialized literature. The present solution can serve, as a first approximation, to identify the effects of the wave–plate–mud interaction on the reflection coefficient.