  As more cooling flow clusters of galaxies with central radio sources are
observed with the Chandra and XMM-Newton X-ray Observatories, more examples of
"bubbles" (low-emission regions in the X-ray coincident with radio emission)
are being found. These bubbles are surrounded by bright shells of X-ray
emission, and no evidence of current strong shocks has yet been found. Using an
analytic approach and some simplifying assumptions, we derive expressions
relating the size and location of a bubble, as well as the density contrast
between the bubble and the ambient medium, with the shock history of the
bubble. These can be applied straightforwardly to new observations. We find
that existing observations are consistent with a mild shock occurring in the
past, and with the bulk of the cool material in the X-ray shells being cooled
at the cluster center and then pushed outward by the radio source. Strong
shocks are ruled out unless they occurred more than 1 Gyr ago. We also discuss
Rayleigh-Taylor instabilities as well as the case of a bubble expanding into an
older bubble produced from a previous cycle of radio activity.
