The Analytical Structure of Acoustic and Elastic Material Properties

TL;DR

This paper analyzes the analytical structure of acoustic and elastic material transfer functions, clarifying their dispersion relations using passivity and Herglotz-Nevanlinna functions, with implications for material property modeling.

Contribution

It establishes the order of dispersion relations for various acoustic and elastic properties based on passivity and complex analysis, extending understanding beyond electromagnetism.

Findings

Dispersion relations of order 1 apply to density and refractive index.

Order 3 dispersion relations apply to stiffness and bulk modulus.

Order 2 dispersion relation applies to wavenumber.

Abstract

In this paper, we take an in-depth look at the analytical structure of the material transfer functions which govern acoustic and elastic response. These include wavenumber ($\kappa$) in such media and refractive index ($n$), density ($\boldsymbol{\rho}$) and its inverse, stiffness ($\boldsymbol{C}$) and compliance ($\boldsymbol{D}$) tensors as well as the Bulk modulus ($B$), and finally the broader generalization of these properties which is now known as the Willis tensor ($\boldsymbol{L}$). Our goal is to clarify the appropriate dispersion relations applicable to these properties from the perspective of passivity. Under some mild assumptions, causality ensures that these properties are analytical in the upper half but deriving dispersion relations for them requires one to know how they behave in the limit $|\omega|\rightarrow\infty$. Unlike electromagnetism, such a determination cannot be made on physical grounds since in that limit the continuum approximation breaks down. Instead, we can exploit the properties of the Herglotz-Nevanlinna (H-N) functions along with their tensorial counterparts which characterize the transfer functions of certain passive systems and for which the appropriate dispersion relation is known. Our aim, therefore, is to clarify the relationship that these transfer functions have with Herglotz functions, which in turn determines the appropriate dispersion relation for them. Our analysis shows that based upon passivity alone, dispersion relations of \emph{minimum} order 1 apply to the Fourier transforms of $\boldsymbol{D},\boldsymbol{\rho}, n'$, and the inverse of $B$, order 3 apply to $\boldsymbol{C},B$, and the inverse of $\boldsymbol{\rho}$, and order 2 applies to $\kappa$.