  Materials with properties that are modulated in time are known to display
wave phenomena showing energy increasing with time, with the rate mediated by
the modulation. Until now there has been no accounting for material
dissipation, which clearly counteracts energy growth. This paper provides an
exact expression for the amplitude of elastic or acoustic waves propagating in
lossy materials with properties that are periodically modulated in time. It is
found that these materials can support a special propagation regime in which
waves travel at constant amplitude, with temporal modulation compensating for
the normal energy dissipation. We derive a general condition under which
amplification due to time-dependent properties offsets the material
dissipation. This identity relates band-gap properties associated with the
temporal modulation and the average of the viscosity coefficient, thereby
providing a simple recipe for the design of loss-compensated mechanical
metamaterials.
