Multiple scattering of acoustic waves with application to the index of refraction of fourth sound

Abstract

A multiple-scattering model which previously explained the conducting properties of fluid-saturated porous fused glass beads is applied to the acoustic index of refraction $n$ of an ideal fluid in a rigid porous frame. The model consists of ellipsoidal grains coated with an effective medium consisting of fluid and other fluid-coated grains; this nesting is continued ad infinitum in order to insure connectedness of the pore space to very low values of porosity. For the relevant cases considered here, the result is ${n}^{2}={P}^{\ensuremath{-}\ensuremath{\beta}}$, where $P$ is the porosity and $\ensuremath{\beta}$ depends on the aspect ratio of the grains. By assuming that the scatterers used in fourth-sound experiments can be characterized by long filaments or needles, randomly oriented ($\ensuremath{\beta}=\frac{2}{3}$), we have achieved excellent agreement with experimental values of ${n}^{2}$; the data also seem to agree with an earlier theory valid for low concentration of aligned needles (${n}^{2}=2\ensuremath{-}P$), when extrapolated to high concentrations of scatterers (\ensuremath{\gtrsim}50%). The present theory contains neither approximation. We also resolve a controversy over the relationship between $n$ and the hydrodynamic drag parameter $\ensuremath{\lambda}$; the correct result is ${n}^{2}={(1\ensuremath{-}\ensuremath{\lambda})}^{\ensuremath{-}1}$.