Effective acoustic response and spectral representation

Abstract

The problem of finding an effective or average acoustic properties of inhomogeneous materials has been treated extensively in connexion with problems of porous media, rock physics and composite materials. Recently, there has been a renewed interest due to the potential applications of acoustic metamaterials. There are different techniques for finding effective acoustic properties. These range from simple averaging techniques, to variational methods, to coherent phase approximations. There is not a unique method for finding effective properties. In the case of dielectric materials, there are at least thirteen procedures reported. In this work, we present the acoustic version of the spectral representation of effective media, first developed to find dielectric effective properties. This method has the advantage that it separates the geometric contribution from the physical property to be calculated, in our case the acoustic impedance. The method is based on a Green's function solution of the acoustic wave equation and finding the effective properties is done by calculating the poles of the so called spectral function. Furthermore, we show that any effective medium model that can be described in the spectral representation satisfies the Kramers-Kronig relations. Numerical examples and comparisons between the spectral representation and other existing procedures will be discussed.