Abstract: Based on the ZWT viscoelastic constitutive relation, the governing equations which reflect high-strain-rate effects, for viscoelastic spherical waves are established, including five partial differential equations to solve five unknown variables such as v, r, , r and . By using the characteristics method, the problem is reduced to solve five ordinary differential equations along three sets of characteristics. The numerical results by the characteristics method show that the degree of attenuation and dispersion for viscoelastic spherical waves are more remarkable than those of linear elastic spherical waves. The tangential tensile stress/strain induced by the surface extension of spherical wave front plays a significant role in tensile failure of medium, while the constitutive viscosity can relieve the tensile stress/strain. An analytic expression for the attenuation of strongdiscontinuous viscoelastic spherical wave front is further derived, which analytically illustrates that the more remarkable attenuation of strongdiscontinuous viscoelastic spherical wave front is resulted from the combined effects of both the geometrical extension effect and the constitutive viscosity effect.