Low-frequency attenuation of sound in marine sediments

Data on low frequency attenuation in marine sediments is surveyed and evidence is cited for the attenuation coefficient at such frequencies being something other than directly proportional to frequency. The original model of Biot for propagation in porous media is examined and simplified versions are derived for the low frequency limit. Acoustic waves are governed by a wave equation of the standard form, but with an additional terms that is proportional to the third time derivative. The "slow wave" disturbances are governed by a simple diffusion equation, such as results from Darcy's law. In this limit, the model predicts for acoustic waves an attenuation coefficient that is proportional to the square of the frequency and inversely proportional to viscosity. It is shown that this model also implies that the attenuation for fixed frequency and porosity should be proportional to the square of the grain size. The commonly used hybrid model, commonly referred to as the Biot-Stoll model, is subsequently examined in the low frequency limit, and it is shown that this model always predicts an attenuation coefficient varying linearly with frequency in the low frequency limit, as the linear term will always dominate the quadratic term if the frequency is sufficiently low. An alternate to the Biot-Stoll model is proposed in which the nonviscous attenuation is regarded as caused by a continuous distribution of relaxation processes. The model is characterized by a function g(tau) that represents the relative contribution per unit range of relaxation time at a given value of relaxation time tau. It is demonstrated that there is a choice for this function that does lead to a prediction of a linear dependence on frequency. However, a more realistic prediction, in which g has vanishingly small values at lower relaxation times, yields a prediction in which the quadratic term dominates at low frequencies, but a linear dependence is evident for intermediate ranges of frequency.