  Consider the dynamics of a gas bubble in an inviscid, compressible liquid
with surface tension. Kinematic and dynamic boundary conditions couple the
bubble surface deformation dynamics with the dynamics of waves in the fluid.
This system has a spherical equilibrium state, resulting from the balance of
the pressure at infinity and the gas pressure within the bubble. We study the
linearized dynamics about this equilibrium state in a center of mass frame: 1)
We prove that the velocity potential and bubble surface perturbation satisfy
point-wise in space exponential time-decay estimates. 2) The time-decay rate is
governed by scattering resonances, eigenvalues of a non-selfadjoint spectral
problem. These are pole singularities in the lower half plane of the analytic
continuation of a resolvent operator from the upper half plane, across the real
axis into the lower half plane. 3) The time-decay estimates are a consequence
of resonance mode expansions for the velocity potential and bubble surface
perturbations. 4) For small compressibility (Mach number, a ratio of bubble
wall velocity to sound speed, \epsilon), this is a singular perturbation of the
incompressible limit. The scattering resonances which govern the anomalously
slow time-decay, are {\it Rayleigh resonances}. Asymptotics, supported by
high-precision numerical studies, indicate that the Rayleigh resonances which
are closest to the real axis satisfy | \frac{\Im \lambda_\star(\epsilon)}{\Re
\lambda_\star(\epsilon)} | = {\cal O} (\exp(-\kappa\ \We\ \epsilon^{-2})),
\kappa>0. Here, \We denotes the Weber number, a dimensionless ratio comparing
inertia and surface tension. 5) To obtain the above results we prove a general
result, of independent interest, estimating the Neumann to Dirichlet map for
the wave equation, exterior to a sphere.
