  We use images of high spatial and temporal resolution, obtained using both
ground- and space-based instrumentation, to investigate the role magnetic field
inclination angles play in the propagation characteristics of running penumbral
waves in the solar chromosphere. Analysis of a near-circular sunspot, close to
the center of the solar disk, reveals a smooth rise in oscillatory period as a
function of distance from the umbral barycenter. However, in one directional
quadrant, corresponding to the north direction, a pronounced kink in the
period-distance diagram is found. Utilizing a combination of the inversion of
magnetic Stokes vectors and force-free field extrapolations, we attribute this
behaviour to the cut-off frequency imposed by the magnetic field geometry in
this location. A rapid, localised inclination of the magnetic field lines in
the north direction results in a faster increase in the dominant periodicity
due to an accelerated reduction in the cut-off frequency. For the first time we
reveal how the spatial distribution of dominant wave periods, obtained with one
of the highest resolution solar instruments currently available, directly
reflects the magnetic geometry of the underlying sunspot, thus opening up a
wealth of possibilities in future magneto-hydrodynamic seismology studies. In
addition, the intrinsic relationships we find between the underlying magnetic
field geometries connecting the photosphere to the chromosphere, and the
characteristics of running penumbral waves observed in the upper chromosphere,
directly supports the interpretation that running penumbral wave phenomena are
the chromospheric signature of upwardly-propagating magneto-acoustic waves
generated in the photosphere.
