We present new data on running penumbral waves (RPWs) obtained by appl
iction of special filtering techniques to high-resolution CCD images.
We find for the first time that the waves decelerate from an initial v
elocity of 25 to around 10 km s(-1) at the outer edge and are azimutha
lly symmetric. They are more or less circular and are only seen in sym
metric, radial sectors of sunspot penumbrae. We did not see any outwar
d decrease in amplitude. If present, it is not large. There appears to
be a connection between umbral flashes and RPWs, but we cannot define
it with the present data. We present a geometric model for the waves,
which fits the observed deceleration. In this model, RPWs are spheric
al waves expanding from a subsurface region on the axis of the spot. A
good fit is obtained with a depth of 15,000 km and a vertical velocit
y of 4.5 km s(-1). This model predicts a lag in the wave arrival at su
ccessive heights. For the single spot for which we had good data in th
ree wavelengths, we found an appropriate lag between the waves at H al
pha +/- 0.6 Angstrom and H alpha. Therefore, we propose a spherical wa
ve geometric model for RPWs, although we have no physical model.