Abstract: In this study, the homogeneous viscoelastic beam with a variable cross-section is used to model the lamprey, an anguilliform (eel-like) swimmer, and to research the propagation characteristics of the curvature wave driven by an active bending moment wave along the fish body. The results show that as long as the excitation frequency is higher than the structure fundamental frequency of the fish body, there will be a phase lag between the two waves, which increases from the head to the tail of the fish body. The increasing phase lag indicates that there exists a speed difference between propagations of the active bending moment and the body curvature, that is, the speed of the later is lower than that of the former. This features are consistent with the experimental results published. The dimensional analysis indicates the ratio of the speed of bending curvature wave to that of the active bending moment wave is associated with the dimensionless excitation frequency, wavelength and damping coefficient, but independent of the swimming Reynolds number. For anguilliform (eel-like) swimmers, the wave speed ratio decreases with increasing frequency or wave length of active bending moment, and it rises if the damping coefficient becomes larger. In addition, we also carried out a small perturbation analysis to linearize the equations, and found an integrated similarity parameter which includes the dimensionless frequency, wave length and damping coefficient. This parameter can uniformly describe the dependence of the speed ratio on the excitation and material parameters.