Abstract <p>The behavior of compressional waves and the mechanical interactions between fluid and solid phases are fundamental aspects of rock physics and rock mechanics. However, a critical question pertains to the roles of creep and stress relaxation in wave propagation within fluid-saturated viscoelastic porous media. In this study, the effects of viscoelasticity on the propagation of pore pressure and stress in such media are theoretically examined, utilizing the Kelvin–Voigt and Maxwell models. The dispersion and attenuation characteristics of both fast and slow compressional waves are analyzed with taking into account the viscous energy dissipation within the solid phase. The findings reveal that the solid viscosity-dependent characteristic frequency is an intrinsic physical property of Newtonian fluid-saturated viscoelastic porous media. The results demonstrate that both wave modes are significantly influenced by creep and stress relaxation mechanisms. Furthermore, wave dissipation is primarily attributed to the viscoelastic nature of the medium and the viscous damping associated with pore-fluid motion.</p>