Numerical simulations of wave propagation in a sunspot umbra. Panel a: vertical velocity at z = 1.35 Mm for all the horizontal positions. Panel b: vertical velocity (red line, left axis) and temperature fluctuations (blue line, right axis) at X = −1.1 Mm (location indicated by the vertical green line in panel a). Panel c: average phase difference between velocity and temperature oscillations for waves with a frequency between 5 and 6 mHz as a function of atmospheric height. The red line represents the phase shift for standing waves from a simulation with a resonant cavity. The blue line corresponds to propagating waves from a simulation without a transition region. Dotted (dashed) lines indicate a phase shift of ±π/2 (0).