# Restrictions on wave equations for passive media

## TL;DR

This paper investigates the restrictions imposed by passivity on wave equations in media, showing that passivity limits the frequency dependence of attenuation and phase velocity, and providing criteria to evaluate the passivity of wave models.

## Contribution

It establishes passivity as a stricter condition than causality, deriving restrictions on relaxation moduli and their derivatives, and characterizing the class of completely monotone systems in acoustics.

## Key findings

- Attenuation must increase slower than linearly with frequency.
- Phase velocity must increase monotonically with frequency.
- Passivity criteria can distinguish between different wave equation models.

## Abstract

Most derivations of acoustic wave equations involve ensuring that causality is satisfied. Here we explore the consequences of also requiring that the medium should be passive. This is a stricter criterion than causality for a linear system and implies that there are restrictions on the relaxation modulus and its first few derivatives. The viscous and relaxation models of acoustics satisfy passivity and have restrictions on not only a few, but all derivatives of the relaxation modulus. This is the important class of completely monotone systems. It is the only class where the medium is modeled as a system of springs and dampers with positive parameters. It is shown here that the attenuation as a function of frequency for such media has to increase slower than a linear function. Likewise the phase velocity has to increase monotonically. This gives criteria on which one may judge whether a proposed wave equation is passive or not, as illustrated by comparing two different versions of the viscous wave equation.