Restrictions on wave equations for passive media.

Most derivations of acoustic wave equations involve ensuring that causality is satisfied. Here, the consequences of also requiring that the medium should be passive are explored. 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. These models are described as a system of springs and dampers with positive parameters and belong to the important class of completely monotone systems. 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.