  Hydrodynamic and acoustic scales separate as the Mach number decreases,
making the modelling of aeroacoustic phenomena singular in this flow regime.
The benchmark of the flow developing around an oscillating and vibrating
cylinder is one of the scarce configuration that is fully analytically
tractable, and is thus precious in the validation of new theories or solvers.
This work carefully derives the complete incompressible flow solution for this
case, extending the axisymmetric results of the literature to more complex
cylinder motion. High-order finite element method solution of the
incompressible Navier-Stokes equations provide a reference to validate the
analytical formulae derived here. Both analytical and numerical investigations
agree on an independence of the configuration to the Reynolds number. Up to the
critical Reynolds number, found to lie above $10^4$, the hydrodynamic solution
of this configuration is solely governed by the Stokes number. The exact
expression of the sound radiated in the far field is computed by accounting for
the cylinder scattering by means of tailored Green's functions.
