  We extend the linear analysis of the drag instability in a 1D perpendicular
isothermal C-shock by Gu & Chen to 2D perpendicular and oblique C-shocks in the
typical environment of star-forming clouds. Simplified dispersion relations are
derived for the unstable modes. We find that the mode property of the drag
instability generally depends on the ratio of the transverse (normal to the
shock flow) to longitudinal (along the shock flow) wavenumber. For the
transversely large-scale mode, the growth rate and wave frequency of the drag
instability in a 2D shock resemble those in a 1D shock. For the transversely
small-scale mode, the drag instability is characterized by an unstable mode
coupled with an acoustic mode primarily along the transverse direction. When
the shock is perpendicular or less oblique, there exists a slowly propagating
mode, which can potentially grow into a nonlinear regime and contribute to the
maximum growth of the instability. In contrast, when the shock is more oblique,
this slowly propagating unstable mode disappears, and the maximum growth of the
drag instability is likely contributed from the transversely large-scale mode
(i.e., almost 1D mode). In all cases that we consider, the magnitude of the
density perturbations is significantly larger than that of the velocity and
magnetic field perturbations, implying that the density enhancement governs the
dynamics in the linear regime of the instability. A few issues in the linear
analysis, as well as the possible astrophysical implications, are also briefly
discussed.
