We seek to clarify the nature of running penumbral (RP) waves: are they chromospheric trans-sunspot waves or a visual pattern of upward-propagating waves? Full Stokes spectropolarimetric time series of the photospheric Sii10827 Å line and the chromospheric Hei10830 Å multiplet were inverted using a Milne-Eddington code. Spatial pixels were paired between the outer umbral/inner penumbral photosphere and the penumbral chromosphere using inclinations retrieved by the inversion and the dual-height pairings of line-of-sight velocity time series were studied for signatures of wave propagation using a Fourier phase difference analysis. The dispersion relation for radiatively cooling acoustic waves, modified to incorporate an inclined propagation direction, fits well the observed phase differences between the pairs of photospheric and chromospheric pixels. We have thus demonstrated that RP waves are in effect low-β slow-mode waves propagating along the magnetic field.