Equations for the self-consistent field in random medium

TL;DR

This paper derives an integral-differential equation for the self-consistent acoustic and electromagnetic fields in a medium with many small, randomly distributed bodies, considering different size and spacing conditions.

Contribution

It introduces a new integral-differential equation for the effective field in media with small bodies, extending Ramm's wave scattering theory to complex distributions.

Findings

Derived an integral-differential equation for the effective field.

Applicable to media with small bodies of arbitrary shape.

Considers different spacing regimes between bodies.

Abstract

An integral-differential equation is derived for the self-consistent (effective) field in the medium consisting of many small bodies randomly distributed in some region. Acoustic and electromagnetic fields are considered in such a medium. Each body has a characteristic dimension $a\ll\lambda$, where $\lambda$ is the wavelength in the free space. The minimal distance $d$ between any of the two bodies satisfies the condition $d\gg a$, but it may also satisfy the condition $d\ll\lambda$. Using Ramm's theory of wave scattering by small bodies of arbitrary shapes, the author derives an integral-differential equation for the self-consistent acoustic or electromagnetic fields in the above medium.