Solution of the homogeneous equation Y₀ associated with the fundamental mode ℓ = 1 (solid orange line) for r_sh/r_ns = 5. This solution is very well approximated analytically by Eq. (73) (dashed orange line with crosses). The radial profile of the amplitude of the forcing term in Eq. (63) (purple solid line) is compared to its analytical approximation (dashed purple line with crosses) using Eqs. (72) and (74). The Mach number profile in the adiabatic model (green solid line) is compared to its analytical approximation (Eq. (72), green dashed line with crosses). The Mach profile of the flow including cooling is shown for reference (dark green line).