Acoustic Propagation in a Medium with Spatially Distributed Relaxation Processes and a Possible Explanation of a Frequency Power Law Attenuation

A possible medium is considered with a large number of diverse but possibly similar relaxation processes. The equations for single spatially located relaxation processes of general type are developed and extended to a large spatially extended system. Each process is characterized by a relaxation time and a relaxation strength and is excited by the local pressure in an incident acoustic wave. A constant frequency model is initially assumed with complex amplitudes associated with acoustic pressure and relaxation responses. A smearing process results in the collective relaxation responses being regarded as a continuous function of relaxation times. An expression for entropy perturbation is developed in terms of the relaxation responses. The equations of fluid mechanics then lead to an expression for the pressure perturbation in terms of the dilatation and the relaxation processes. The extended result yields a time-domain wave equation for propagation through an inhomogeneous medium with distributed relaxation processes. Expressions for frequency-dependent attenuation and phase velocity are derived in which relaxation is characterized by a single function giving strength as a continuous function of relaxation time. Possible choices for this function are discussed, and it is shown that some choices lead to an attenuation varying nearly as a power of the frequency over any fixed and possibly large range of frequencies. A model set of parameters leads to good agreement with attenuation and phase velocity measurements for a suspension of human red blood cells in saline solution, as communicated to the authors by Treeby and analogous to those reported by Treeby, Zhang, Thomas, and Cox.