| The water wave-seabed interactions are studied for small amplitude waves propagating in a water-soft-mud system. Based on systematic experiments, the seabed is divided into different layers with different rheology properties. The thickness of each layer is a part of the solutions to the interaction problem. The nonlinear viscous and elastic stresses due to strain-dependent viscosity and shear modulus are linearized by Lorentz's condition of equivalent work. The solutions of motions in the system are sought in terms of the surface wave slope, {dollar}alpha{dollar}, which is a small parameter. The viscous damping effect of the motions is assumed to be of the same order of magnitude as {dollar}alpha{dollar}. The surface water wave decays spatially in the direction of wave propagation. The solutions of the periodic leading-order wave motions, {dollar}O(alpha{dollar}), in the system are presented. The second-order solutions for both periodic wave motions and steady streaming are also obtained. It is found that surface water wave motion and steady streaming are enhanced by seabed movements. The fluidized depths of seabed increase with {dollar}alpha{dollar} and decrease with seabed density. The viscous dissipation in the system also increases because of the seabed fluidization. The analytical results are verified by experimental data for simple cases.; Alternatively, water waves may decay temporally due to viscosity. To {dollar}O(alpha{dollar}), solutions of motions of a temporally decaying wave are in the same form as that of a spatially decaying wave, except for a different decaying factor. However, the second-order motions of a temporally decaying wave may be much different from that of a spatially decaying wave. The mass transport of waves decaying temporally in a two-layer fluid system is investigated with analytical and numerical approaches. The mean motions are treated as an initial-boundary-value problem. Because of viscous damping and diffusion, strong transient interfacial second boundary layers of {dollar}O(epsilonsp{lcub}1/2{rcub}{dollar}), where {dollar}epsilon{dollar} is the dimensionless Stokes boundary layer thickness, are established adjacent to the interfaces, in contrast to the parabolic profiles of a spatially decaying wave. Within the second boundary layers, there may exist the steady streaming of {dollar}O(alphasp2epsilonsp{lcub}-1/2{rcub}){dollar}. |
