In coastal areas, wave-mud interaction is an important mechanism of wave attenuation. The present study on mud-induced wave attenuation is settled in a system composed of an inviscid water layer and a mud layer, in which the mud layer is modeled by a two-layered viscoelastic model. In the two-layered model, the upper layer is described by a Maxwell model, which is fluid-like; the lower layer is described by a Kelvin-Voigt model, which is heavier, and solid-like. Including the influence of the two-layered mud model, a new set of Boussinesq-type equations for shallow water waves is established. Degenerating to the case of a single-layered mud model, the Boussinesq-type equations are equivalent to the results in literatures. Applying the Boussinesq-type equations to one-dimensional waves, the attenuation of linear sinusoidal waves and a solitary wave are studied. For linear sinusoidal waves, the damping rates are calculated and are found consistent with literature results. The damping is dominated by the lower mud layer, which is much thicker. The influence of the upper mud layer is important near resonance. For a solitary wave, an evolution equation of the wave amplitude is obtained. The attenuation of a solitary wave is again dominated by the lower mud layer and the influence of the upper mud layer is small.
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