  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.
