A Study On Visco-elastic Seabed Mud And Its Influence On The Attenuation Of Free-surface Water Waves

It has been observed that seabed mud can dissipate the energy of surfacewater waves effectively. As an important mechanism of wave attenuation, thewave-mud interaction has been extensively studied by coastal engineers fordecades.Among varies models for mud rheology, linear viscoelastic models arecommonly used in analytical studies on wave-mud interaction. However, mostof these studies focus on linear sinusoidal waves. In the very limited studies onthe interaction between the weakly nonlinear waves and seabed mud, theBoussinesq-type equations are powerful tools. The limitation of currentBoussinesq-type equations is that the mud is modeled by Newtonian fluid,which cannot cover the complexity of mud rheology. Hence, it is necessary toextend the Boussinesq-type equations to linear viscoelastic models, which is thegoal of present work.In the present work, we choose the Maxwell model and the Kelvin-Voigtmodel as mud models, which are typical models for viscoelastic fluid andviscoelastic solid, respectively. Basing on a two-layer system composed ofinviscid water and viscoelastic mud, and combining the perturbation analysis,the Fourier expansion and the Laplace transformation, we set upBoussinesq-type equations for surface waves over the Maxwell model and theKelvin-Voigt model, respectively. When the Maxwell model and theKelvin-Voigt model degenerating to the Newtonian model, our Boussinesq-typeequations can degenerate to the existing Boussinesq-type equations in theliterature.Applying the Boussinesq-type equations to one-dimensional waves, thedamping rate of linear sinusoidal waves and the evolution equation for theamplitude of a solitary wave are obtained. For the Maxwell model, the dampingrate of linear sinusoidal waves is dominated by low-order modes, and shearwave induced by solitary waves propagates along the vertical direction in themud layer. For the Kelvin-Voigt model, the first mode dominates the mud motion. Hence the damping rate of linear sinusoidal waves has only one obviouspeak and the velocity profile of the mud layer tends to that of the first modeafter the solitary wave crest. Modal analysis shows that for the Maxwell model,all the modes decay simultaneously, while for the Kelvin-Voigt model, the firstmode decays slowest and takes domination.We also introduce a Fractional-ordered Maxwell model, which can overcome thedefect of traditional viscoelastic models when fitting the experimental data of some realmud. Basing on this fractional-order Maxwell model, we obtain the damping rate of alinear sinusoidal wave. The influence of model parameters on the damping rate is alsodiscussed.