The propagation of a solitary wave over seabed mud of the Voigt model

Abstract

In shallow water, seabed mud can dissipate the energy of surface gravity waves effectively. In this paper, solitary wave attenuation induced by seabed mud is studied based on a two-layered system, in which the water is assumed to be inviscid and the mud layer is described by the Voigt model. A set of Boussinesq-type equations suitable for solitary waves over the mud of the Voigt model is established, by combining the perturbation analysis and the Laplace transformation. Degenerating into the case of Newtonian model, our Boussinesq-type equations are equivalent to those of Liu and Chan (2007), while the term indicating mud influence is greatly simplified. Based on the equations, the attenuation of solitary waves is studied. An evolution equation of wave amplitude is obtained and the development of mud velocity profiles is discussed. The modal analysis shows that the first mode always dominates mud dynamics. The results are also compared with those of the Maxwell model. solitary wave, seabed mud, wave-mud interaction, Boussinesq-type equations, Voigt model PACS number(s): 47.35.Fg, 47.35.Bb, 47.50.Cd Citation: Xia Y Z, Zhu K Q. The propagation of a solitary wave over seabed mud of the Voigt model.It has been observed that seabed mud can dissipate the energy of surface water waves effectively [1]. As an important mechanism of wave attenuation, the wave-mud interaction has been extensively studied by coastal engineers for decades.The problem of wave-mud interaction is commonly discussed in a two-layered system of water and a muddy seabed. Since water viscosity is very small compared with mud viscosity and makes little contribution to wave attenuation, the water can be treated as an inviscid fluid [2]. Since mud induced wave attenuation is significant only in shallow water, the water layer thickness is often chosen to be small compared with the wave length [2,3].The muddy seabed is a natural cohesive suspension composed of sediment grains and flocs containing minerals and organic matter. Kinds of mud models have been proposed, since the components of mud vary with sites and environments. Jain and Mehta [4] classified basic rheological models of mud according to mud density and the size of solid particles. They pointed out that when seabed sediments are composed of fine suit, the solid particles of the seabed can follow the fluid motion and the entire sediment suspension can be treated as a continuum. These singlephased models are further classified into viscoplastic solid, viscoelastic solid, viscoelastic fluid and viscous fluid according to mud density. For the heaviest mud, viscoplastic solid model is valid, such as the Bingham model [5,6]. For the lightest mud, the viscous fluid model is valid, such as the Newtonian model [2,7-9] and the power-law model [3]. For mud with moderate density, viscoelastic models are widely used, including viscoelastic fluid and viscoelastic solid. Furthermore, viscoelastic models are time-dependent models that describe the response of mud in periodic or transient motions. In these visc...