  A family of compressible exact periodic solutions is reported for the first
time in an open cavity flow setup. These are found using a novel framework
which permits the computation of such solutions in an arbitrary complex
geometry. The periodic orbits arise from a synchronised concatenation of
convective and acoustic events which strongly depend on the Mach number. This
flow-acoustic interaction furnishes the periodic solutions with a remarkable
stability and it is found to completely dominate the system's dynamics and the
sound directivity. The periodic orbits, which could be called `exact Rossiter
modes', collapse with a family of equilibrium solutions at a subcritical Hopf
bifurcation, occurring in the quasi-incompressible regime. This shows
compressibility has a destabilising effect in cavity flows, which we analyse in
detail. By establishing a connection with previous 2D and 3D stability studies
of cavity flows, we are able to isolate the effect of purely compressible
two-dimensional flow phenomena across Mach number. A linear stability analysis
of the equilibria provides insight into the compressible flow mechanisms
responsible for the instability. A close look at the adjoint modes suggests
that an eigenvalue merge occurs at a Mach number between 0.35 and 0.4, which
boosts the receptivity of the leading mode and determines the onset of the
unstable character of the system. The effect of the choice of base flow over
the transition dynamics is also discussed, where in the present case, the
frequencies associated to the leading eigenmodes show a strong connection with
the frequencies of the periodic orbits at the same Mach numbers.
