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

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 $n^2=P^{-\beta }$, where $P$ is the porosity and $\beta$ 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 ($\beta =\frac{2}{3}$), 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 ($n^2=2-P$), when extrapolated to high concentrations of scatterers (≳50%). The present theory contains neither approximation. We also resolve a controversy over the relationship between $n$ and the hydrodynamic drag parameter $\lambda$; the correct result is $n^2={(1-\lambda )}^{-1\mathrm{}}$.