Periodicity makes galactic shocks unstable - I. Linear analysis
  We study the dynamical stability of stationary galactic spiral shocks. The
steady-state equilibrium flow contains a shock of the type derived by Roberts
in the tightly wound approximation. We find that boundary conditions are
critical in determining whether the solutions are stable or not. Shocks are
unstable if periodic boundary conditions are imposed. For intermediate
strengths of the spiral potential, the instability disappears if boundary
conditions are imposed such that the upstream flow is left unperturbed as in
the classic analysis of D'yakov and Kontorovich. This reconciles apparently
contradictory findings of previous authors regarding the stability of spiral
shocks. This also shows that the instability is distinct from the
Kelvin-Helmholtz instability, confirming the findings of Kim et al. We suggest
that instability is a general characteristics of periodic shocks, regardless of
the presence of shear, and provide a physical picture as to why this is the
case. For strong spiral potentials, high post-shock shear makes the system
unstable also to parasitic Kelvin-Helmholtz instability regardless of the
boundary conditions. Our analysis is performed in the context of a simplified
problem that, while preserving all the important characteristics of the
original problem, strips it from unnecessary complications, and assumes that
the gas is isothermal, non self-gravitating, non-magnetised.
