Many-body wave scattering problems in the case of small scatterers

TL;DR

This paper develops formulas and equations for wave scattering by many small particles in inhomogeneous media, considering various boundary conditions and the limit as particle size approaches zero, providing a comprehensive theoretical framework.

Contribution

It introduces new formulas and asymptotic equations for many-body wave scattering with small particles, including the limiting effective field as particle size diminishes.

Findings

Derived formulas for scattering by small particles with different boundary conditions.

Established equations for the effective field in the limit of vanishing particle size.

Analyzed wave scattering in inhomogeneous media with many small particles.

Abstract

Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The case of transmission (interface) boundary conditions is also studied in detail. The limiting case is considered, when the size $a$ of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived. The theory is based on a study of integral equations and asymptotics of their solutions as $a\to 0$. The case of wave scattering by many small particles embedded in an inhomogeneous medium is also studied.