  Fluid instabilities arise in a variety of contexts and are often unwanted
results of engineering imperfections. In one particular model for a magnetized
target fusion reactor, a pressure wave is propagated in a cylindrical annulus
comprised of a dense fluid before impinging upon a plasma and imploding it.
Part of the success of the apparatus is a function of how axially-symmetric the
final pressure pulse is upon impacting the plasma. We study a simple model for
the implosion of the system to study how imperfections in the pressure imparted
on the outer circumference grow due to geometric focusing. Our methodology
entails linearizing the compressible Euler equations for mass and momentum
conservation about a cylindrically symmetric problem and analyzing the
perturbed profiles at different mode numbers. The linearized system gives rise
to singular shocks and through analyzing the perturbation profiles at various
times, we infer that high mode numbers are dampened through the propagation. We
also study the Linear Klein-Gordon equation in the context of stability of
linear cylindrical wave formation whereby highly oscillatory, bounded behaviour
is observed in a far field solution.
