The traditional acoustic attenuation coefficient is derived from an analogy of the attenuation of an electromagnetic wave propagating inside a non-ideal medium, featuring only the attenuation of wave propagation. Nonetheless, the particles inside viscous solids have mass, vibrating energy, viscosity, and inertia of motion, and they go through transient and damping attenuation processes. Based on the long-wavelength approximation, in this paper, we use the energy conservation law to analyze the effect of the viscosity of the medium on acoustic attenuation. We derive the acoustic attenuation coefficient by combinations of the dynamical equation of a solid in an acoustic field with conventional longitudinal wave propagation under a spring oscillator model. Considering the attenuation of propagating waves and the damping attenuation of particle vibration, we develop a frequency dispersion relation of phase velocity for the longitudinal wave propagating inside viscous solid media. We find that the acoustic impulse response and vibrational system function depends on the physical properties of the viscous solid media and their internal structure. Combined with system function, the impulse response can be an excellent tool to invert the physical properties of solids and their internal structures. We select a well-known rock sample for analysis, calculate the impulse response and vibrational system function, and reveal new physical insight into creating acoustic attenuation and frequency dispersion of phase velocity. The results showed that the newly developed acoustic attenuation coefficients enjoy a substantial improvement over the conventional acoustic attenuation coefficients reported in the literature, which is essential for industrial applications; so are the dispersion characteristics.
Keywords: acoustic attenuation characteristics; damping attenuation; dispersion characteristics; particle displacement; propagation attenuation.