# Dispersion of waves in two and three-dimensional periodic media

## TL;DR

This paper develops a new asymptotic method to analyze wave dispersion in 2D and 3D periodic media with small inclusions, providing explicit formulas for effective wave speed and dispersion relations.

## Contribution

A novel approach to derive complete asymptotic expansions of dispersion relations for waves in periodic media with small inclusions, applicable to various boundary conditions.

## Key findings

- Explicit asymptotic formulas for dispersion relations.
- Effective wave speed as a function of frequency and material properties.
- Illustrations of parameter dependence through graphs.

## Abstract

We consider the propagation of acoustic time-harmonic waves in a homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius $a$ of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two and three-dimensional cases as $a \to 0$ and evaluate explicitly several first terms. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Neumann, Dirichlet and transmission boundary conditions are considered. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs.