Small frequency approximation of (causal) dissipative pressure waves

TL;DR

This paper investigates small frequency approximations of causal dissipative pressure waves, demonstrating that under certain conditions, the causal Green function can be approximated by a noncausal, low-frequency, power-law Green function with minimal error.

Contribution

It introduces a method to approximate causal dissipative pressure waves with a noncausal, small-frequency Green function, expanding understanding of wave behavior in this regime.

Findings

Green function $G^c$ can be approximated by $G_M^{pl}$ in small frequency range

The noncausal wave contains arbitrarily fast partial waves, but their sum is small in $L^2$-sense

Approximation is valid for situations with appropriate parameters and relaxation time $ au_0$

Abstract

In this paper we discuss the problem of small frequency approximation of the causal dissipative pressure wave model proposed in \cite{KoScBo:11}. We show that for appropriate situations the Green function $G^c$ of the causal wave model can be approximated by a noncausal Green function $G_M^{pl}$ that has frequencies only in the small frequency range $[-M,M]$ ($M\leq 1/\tau_0$, $\tau_0$ relaxation time) and obeys a power law. For such cases, the noncausal wave $G^{pl}_M$ contains partial waves propagating arbitrarily fast but the sum of the noncausal waves is small in the $L^2-$sense.