Sound wave propagation through multiphase materials

Abstract

Theories for infinitesimal, planar sound wave propagation in a dilute suspension of rigid particles in a viscous fluid has been investigated by generally three approaches: (1) wave scattering, (2) hydrodynamic, and (3) ad hoc approaches specific to particular systems. Here, a hydrodynamic development that uses spatially averaged continuum balance and constitutive equations for multiphase materials is presented. Two alternative approaches derive equivalent results (i.e., a bicubic polynomial equation) for the complex propagation constant χ = − (a + ik), where a is the spatial attenuation coefficient and k is the wavenumber. One approach uses linear momentum equations for each individual phase, while the other uses an overall linear momentum equation and a relative momentum equation, obtained by taking the sum and difference of the individual momentum equations for the continuous and particulate phases. The advantage of this latter aproach is that it expresses the results in the form of a generalized Fick's law, and expresses the diffusion of particles, as well as the supply terms such as barophoresis, thermophoresis, and pycnophoresis, explicitly. Furthermore, an Einstein relation can be obtained by simply defining a diffusion coefficient, and interpreting this coefficient in the linear momentum supply term that is proportional to the relative velocity as a Stokes viscous drag force.