  We present a generic mechanism for the thermal damping of compressive waves
in the interstellar medium (ISM), occurring due to radiative cooling. We solve
for the dispersion relation of magnetosonic waves in a two-fluid (ion-neutral)
system in which density- and temperature-dependent heating and cooling
mechanisms are present. We use this dispersion relation, in addition to an
analytic approximation for the nonlinear turbulent cascade, to model
dissipation of weak magnetosonic turbulence. We show that in some ISM
conditions, the cutoff wavelength for magnetosonic turbulence becomes tens to
hundreds of times larger when the thermal damping is added to the regular
ion-neutral damping. We also run numerical simulations which confirm that this
effect has a dramatic impact on cascade of compressive wave modes.
