This paper, authored by Maurice A. Biot, develops a theory for the propagation of elastic waves in a porous elastic solid saturated with a compressible viscous fluid. The focus is on materials where the densities of the fluid and solid are comparable, such as water-saturated rock. The study is restricted to the low-frequency range where Poiseuille flow is valid. The material is described by four nondimensional parameters and a characteristic frequency. There are two rotational waves and one dilatational wave. The physical interpretation is clarified by first considering the case of frictionless fluid and then extending to viscous fluids. Phase velocity dispersion curves and attenuation coefficients for the three types of waves are plotted as functions of frequency for various combinations of the characteristic parameters. The results show that the phase velocity of rotational waves increases slightly with frequency, while the absorption coefficient is proportional to the square of the frequency. The waves can be categorized into two types: the first kind, which are true waves with negligible dispersion, and the second kind, which are highly attenuated and behave like diffusion processes. The attenuation of the second kind waves can vary widely depending on the material's composition.This paper, authored by Maurice A. Biot, develops a theory for the propagation of elastic waves in a porous elastic solid saturated with a compressible viscous fluid. The focus is on materials where the densities of the fluid and solid are comparable, such as water-saturated rock. The study is restricted to the low-frequency range where Poiseuille flow is valid. The material is described by four nondimensional parameters and a characteristic frequency. There are two rotational waves and one dilatational wave. The physical interpretation is clarified by first considering the case of frictionless fluid and then extending to viscous fluids. Phase velocity dispersion curves and attenuation coefficients for the three types of waves are plotted as functions of frequency for various combinations of the characteristic parameters. The results show that the phase velocity of rotational waves increases slightly with frequency, while the absorption coefficient is proportional to the square of the frequency. The waves can be categorized into two types: the first kind, which are true waves with negligible dispersion, and the second kind, which are highly attenuated and behave like diffusion processes. The attenuation of the second kind waves can vary widely depending on the material's composition.