Theory of acoustic attenuation, dispersion, and pulse propagation in unconsolidated granular materials including marine sediments

A unified theory of sound propagation in saturated marine sediments is developed on the basis of a linear wave equation, which includes a new dissipation term representing internal losses arising from interparticle contacts. This loss mechanism, which shows a “memory” or hysteresis, is proposed as being responsible for the acoustic properties of sediments. To accommodate the memory, the loss term in the wave equation is formulated as a temporal convolution between the particle velocity and a material response function, $\mathrm{h(t),}$ which varies as $t^{\mathrm{-n}},$ where $\text{0<n<1.}$ The compressional wave that emerges from the analysis shows: (1) an attenuation that scales almost exactly as the first power of frequency (corresponding to a constant $Q$ ) over an unlimited number of decades; and (2) weak logarithmic dispersion. As well as being characteristic of the wave properties of actual marine sediments, the predicted attenuation and dispersion are consistent with the Kronig–Kramers relationships. The theory also leads to pulse propagation that is strictly causal, which, although a necessary requirement, is an issue that has been widely debated in the literature in the context of an attenuation that is proportional to frequency. Finally, the wave properties (phase speed and attenuation) are related to the mechanical properties (grain size, density, and porosity) of a sediment by combining the Hertz theory of particles in contact with a new, rough-surface, random-packing model of mineral grains in unconsolidated granular media. The resultant relationships between the acoustical and mechanical properties (e.g., sound speed and porosity) of marine sediments are shown to follow the trends of published experimental data sets very closely.