  The highly dynamic atmosphere above sunspots exhibits a wealth of
magnetohydrodynamic (MHD) waves. Recent studies suggest a coupled nature of the
most prominent phenomena: umbral flashes (UFs) and running penumbral waves
(RPWs). From an observational point of view, we perform a height-dependent
study of RPWs, compare their wave characteristics and aim to track down these
so far only chromospherically observed phenomena to photospheric layers to
prove the upward propagating field-guided nature of RPWs. We analyze a time
series (58\,min) of multi-wavelength observations of an isolated circular
sunspot (NOAA11823) taken at high spatial and temporal resolution in
spectroscopic mode with the Interferometric BIdimensional Spectro-polarimeter
(IBIS/DST). By means of a multi-layer intensity sampling, velocity comparisons,
wavelet power analysis and sectorial studies of time-slices, we retrieve the
power distribution, characteristic periodicities and propagation
characteristics of sunspot waves at photospheric and chromospheric levels.
Signatures of RPWs are found at photospheric layers. Those continuous
oscillations occur preferably at periods between 4-6\,min starting at the inner
penumbral boundary. The photospheric oscillations all have a slightly delayed,
more defined chromospheric counterpart with larger relative velocities (which
are linked to preceding UF events). In all layers the power of RPWs follows a
filamentary fine-structure and shows a typical ring-shaped power distribution
increasing in radius for larger wave periods. The analysis of time-slices
reveals apparent horizontal velocities for RPWs at photospheric layers of
$\approx50\,\rm{km/s}$ which decrease to $\approx30\,\rm{km/s}$ at
chromospheric heights. The observations strongly support the scenario of RPWs
being upward propagating slow-mode waves guided by the magnetic field lines.
