For linear and time-invariant systems, the concept of causality as a general tool implies that a function takes the value of zero for a range of its arguments (primitive causality condition). Based upon this, it is possible to show that the Fourier transform of such a function can be analytically continued into a part of complex plane. Using this property and the knowledge about the asymptotic behavior of functions in high frequencies, it is possible to write dispersion relations (Kramers-Kronig relations). Such dispersion relations here are interpreted as the relation between real and imaginary part of a transfer function in frequency domain. Therefore, derivation of dispersion relations from "causality" requires further knowledge of high frequency behavior of the transfer function. The behavior of a transfer function in the Laplace domain (complex frequency domain) constitutes the analytical structure of the function and the knowledge of this analytical structure is essential in arriving at the correct dispersion relations. 
&nbsp;In this talk, we take an in-depth look at the analytical structure of the material transfer functions which govern acoustic and elastic waves. These include wavenumber in such media, density and its inverse, Young's modulus and compliance, their tensorial counterparts, as well as the broader generalization of these properties which is now known as the Willis tensor. 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. For wavenumber, this analysis was done by Weaver and Pao some forty years ago. Here we present a more straightforward treatment of the same from the perspective of distribution theory. We simply proceed to evaluate the boundary value of the H-N function as the Fourier transform of the wavenumber and then take its inverse Fourier transform to arrive at its time domain representation. We show that this representation is a specific order derivative of a continuous function which allows us to immediately determine the appropriate dispersion relation without having to refer to additional assumptions and analysis that Weaver and Pao made. The authors hope that the present work along with the previous novel works in this field can pave the way for finding the exact dispersion relations of physical parameters of elastic systems without any concern of the function’s behavior at high frequencies.&nbsp;

The Analytical Structure of Acoustic and Elastic Material Properties

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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