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We consider the acoustic flow field of rotationally symmetric systems, like an annular combustor and the flow in a round duct, in absence of a mean azimuthal flow field. We focus on azimuthal instabilities, which manifest as either spinning (rotating) waves or standing waves, or a linear combination of the two. These instabilities are often excited by some level of background noise that makes the system randomly change between spinning and standing states, undergo amplitude variations and changes to the azimuthal orientation of the solution. To account for this random change, we make use of a novel ansatz to track as a function of time the amplitude, orientation, nature (standing/spinning) and temporal phase of these instabilities. To capture the effect of the background noise, we apply stochastic averaging on the governing equations and obtain a novel differential equation. The equation allows to study the effects of acoustic sources and sinks on the statistics of the solution, and of explicit and spontaneous symmetry breaking and noise intensity. We focus on this last effect and show how the noise intensity affects the system preference for spinning and/or standing states. We find for example that, when present, background noise pushes the system away from spinning states and towards standing states, consistently with experiments and numerical simulations.