Multiple scattering by cylinders immersed in fluid: high order approximations for the effective wavenumbers

TL;DR

This paper develops higher order approximations for the effective wavenumber of acoustic waves in fluids with cylindrical scatterers, extending existing models to include corrections up to fourth order in scatterer density.

Contribution

It introduces a method to extend the Linton and Martin formula to higher orders and compares various self-consistent schemes for effective medium theories.

Findings

Explicit formulas for corrections up to O(n_0^4)

Linton and Martin formula as a closed self-consistent scheme

Comparison of different dynamic effective medium theories

Abstract

Acoustic wave propagation in a fluid with a random assortment of identical cylindrical scatterers is considered. While the leading order correction to the effective wavenumber of the coherent wave is well established at dilute areal density ($n_0 $) of scatterers, in this paper the higher order dependence of the coherent wavenumber on $n_0$ is developed in several directions. Starting from the quasi-crystalline approximation (QCA) a consistent method is described for continuing the Linton and Martin formula, which is second order in $n_0$, to higher orders. Explicit formulas are provided for corrections to the effective wavenumber up to O$(n_0^4)$. Then, using the QCA theory as a basis, generalized self consistent schemes are developed and compared with self consistent schemes using other dynamic effective medium theories. It is shown that the Linton and Martin formula provides a closed self-consistent scheme, unlike some other approaches.