Context. Umbral flashes (UFs) and running penumbral (RP) waves are believed to be closely related oscillatory phenomena of sunspots. Aims. We investigate the association of UFs and RP waves to see whether the latter are a visual pattern created by a common source with UFs or a trans-sunspot wave driven by UFs.Methods. Simultaneous, two-dimensional, dual-line observations in Ca ii 8542 Å and Hα 6563 Å, obtained with the Multichannel Subtractive Double Pass (MSDP) spectrograph mounted on the German VTT at Teide Observatory on Tenerife, are used for this study. High-cadence 8 s Doppler velocity images, spectrograms, and spectral-analysis results are used to study the characteristics and the relationship of UFs and RP waves. Results. Several UFs were observed that seem to fill the whole umbra. Doppler velocity variations with time indicate a shock behaviour for UFs, as well as for umbral and RP waves and a smooth continuous propagation of the latter from the umbra through the umbra-penumbra boundary out to the edge of the penumbra. Furthermore, the spectral analysis shows a decreasing oscillatory frequency as we move from the umbra outwards and a jump at the umbra-penumbra boundary that could possibly reflect, apart from a change in physical conditions, a drastic change of the magnetic field inclination with respect to the vertical. Conclusions. The results do not permit us to convincingly support one scenario over the other (i.e. visual pattern vs. trans-sunspot wave) for RP waves; however, they do provide important constraints for future models of sunspot oscillations and RP waves.