  We study the stability properties of hydrodynamic shocks with finite Mach
numbers. The linear analysis supplements previous analyses which took the
strong shock limit. We derive the linearised equations for a general specific
heat ratio as well as temperature and density power-law cooling functions,
corresponding to a range of conditions relevant to interstellar atomic and
molecular cooling processes. Boundary conditions corresponding to a return to
the upstream temperature ($R$ = 1) and to a cold wall ($R$ = 0) are
investigated. We find that for Mach number $M > 5$, the strong shock
overstability limits are not significantly modified. For $M < 3$, however,
shocks are considerably more stable for most cases. In general, as the shock
weakens, the critical values of the temperature power-law index (below which
shocks are overstable) are reduced for the overtones more than for the
fundamental, which signifies a change in basic behaviour. In the $R$ = 0
scenario, however, we find that the overstability regime and growth rate of the
fundamental mode are increased when cooling is under local thermodynamic
equilibrium. We provide a possible explanation for the results in terms of a
stabilising influence provided downstream but a destabilising effect associated
with the shock front. We conclude that the regime of overstability for
interstellar atomic shocks is well represented by the strong shock limit unless
the upstream gas is hot. Although molecular shocks can be overstable to
overtones, the magnetic field provides a significant stabilising influence.
