Abstract: In coastal area, the interactions between waves andmuddy seabed lead to energy dissipation. We use Maxwell model as theconstitutive equation of non-Newtonian mud and develope a new set ofBoussinesq-type equations to describe wave propagation under wave-mudinteraction. Based on these equations, the damping rate for one-dimensionallinear long waves is analytically obtained. The present result is consistentwith previous literatures when the model degenerates to Newtonian case. Whendimensionless relaxation time λ is large, extremums of the dampingrate appear quasi-periodically as the mud depth increases. These extremepoints correspond to the peaks of modal damping rates.