  We extend Guderley's problem of finding a self-similar scaling solution for a
converging cylindrical or spherical shock wave from the ideal gas case to the
case of flows with an arbitrary equation of state closure model, giving
necessary conditions for the existence of a solution. The necessary condition
is a thermodynamic one, namely that the adiabatic bulk modulus, $K_S$, of the
fluid be of the form $pf(\rho)$ where $p$ is pressure, $\rho$ is mass density,
and $f$ is an arbitrary function. Although this condition has appeared in the
literature before, here we give a more rigorous and extensive treatment. Of
particular interest is our novel analysis of the governing ordinary
differential equations (ODEs), which shows that, in general, the Guderley
problem is always an eigenvalue problem. The need for an eigenvalue arises from
basic shock stability principles -- an interesting connection to the existing
literature on the relationship between self-similarity of the second kind and
stability. We also investigate a special case, usually neglected by previous
authors, where assuming constant shock velocity yields a reduction to ODEs for
every material, but those ODEs never have a bounded, differentiable solution.
This theoretical work is motivated by the need for more realistic test problems
in the verification of inviscid compressible flow codes that simulate flows in
a variety of non-ideal gas materials.
