Causality analysis of waves and wave equations obeying attenuation

TL;DR

This paper investigates the causality conditions of attenuated waves, showing that the Kramers-Kronig relation alone is insufficient for causality, and derives wave equations that satisfy causality and attenuation laws, with applications to thermoacoustic tomography.

Contribution

It provides a detailed causality analysis of wave equations with attenuation, including a new generalized thermo-viscous wave equation suitable for thermoacoustic imaging.

Findings

Kramers-Kronig relation is necessary but not sufficient for wave causality.

Derived wave equations obeying both attenuation and causality.

Proposed a generalized thermo-viscous wave equation for thermoacoustic tomography.

Abstract

In this paper we show that the standard causality condition for attenuated waves, i.e. the Kramers-Kronig relation that relates the attenuation law and the phase speed of the wave, is necessary but not sufficient for causality of a wave. By causality of a wave we understand the property that its wave front speed is bounded. Although this condition is not new, the consequences for wave attenuation have not been analysed sufficiently well. We derive the wave equation (for a homogeneous and isotropic medium) obeying attenuation and causality and with a generalization of the Paley-Wiener-Schwartz Theorem (cf. Theorem 7.4.3. in \cite{Ho03}), we perform a causality analysis of waves obeying the frequency power attenuation law. Afterwards the causality behaviour of Szabo's wave equation (cf. \cite{Szabo94}) and the thermo-viscous wave equation are investigated. Finally, we present a generalization of the thermo-viscous wave equation that obeys causality and the frequency power law (for powers in $(1,2]$ and) for small frequencies, which we propose for Thermoacoustic Tomography.