Eigenfrequency of the adiabatic solution (green lines) based on the integral formulation (63) compared to its integrated approximation (83) (orange lines), which is based on the approximate description of the flow profile v,ℳ and acoustic structure Y₀. The estimate of the oscillation frequency $2\pi /{\tau }_{\mathrm{adv}}^{\mathrm{ns}}$ neglecting the phase of 𝒬 is shown as a dashed blue line. For reference, the eigenfrequency of the flow with cooling using (α, β) = (3/2, 5/2) is also shown (purple lines with crosses).