Maurice A. Biot's 1956 paper "Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range" presents a comprehensive theory for the propagation of elastic waves in a porous solid saturated with a compressible viscous fluid. The paper focuses on the low-frequency range where Poiseuille flow is valid. The material is described by four nondimensional parameters and a characteristic frequency. The analysis reveals two dilatational waves and one rotational wave. The paper discusses the physical interpretation of these waves, considering both frictionless and viscous fluid cases. It also explores the effects of fluid viscosity on wave propagation, including phase velocity dispersion and attenuation coefficients. The paper introduces the concept of dynamic compatibility, where the fluid and solid move in unison, leading to reduced dissipation. The results are illustrated with plots of phase velocity and attenuation coefficients for different wave types and parameter combinations. The paper also discusses the transition to higher frequencies in a subsequent part. The analysis is based on the assumption of statistical isotropy and the use of Lagrangian mechanics to derive dynamic equations. The paper concludes with numerical discussions of attenuated waves, showing how phase velocity and attenuation vary with frequency. The results are presented in terms of nondimensional parameters and characteristic frequencies, providing a foundation for understanding wave propagation in porous media.Maurice A. Biot's 1956 paper "Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range" presents a comprehensive theory for the propagation of elastic waves in a porous solid saturated with a compressible viscous fluid. The paper focuses on the low-frequency range where Poiseuille flow is valid. The material is described by four nondimensional parameters and a characteristic frequency. The analysis reveals two dilatational waves and one rotational wave. The paper discusses the physical interpretation of these waves, considering both frictionless and viscous fluid cases. It also explores the effects of fluid viscosity on wave propagation, including phase velocity dispersion and attenuation coefficients. The paper introduces the concept of dynamic compatibility, where the fluid and solid move in unison, leading to reduced dissipation. The results are illustrated with plots of phase velocity and attenuation coefficients for different wave types and parameter combinations. The paper also discusses the transition to higher frequencies in a subsequent part. The analysis is based on the assumption of statistical isotropy and the use of Lagrangian mechanics to derive dynamic equations. The paper concludes with numerical discussions of attenuated waves, showing how phase velocity and attenuation vary with frequency. The results are presented in terms of nondimensional parameters and characteristic frequencies, providing a foundation for understanding wave propagation in porous media.