  X-ray surface brightness fluctuations in the core ($650 \times 650$ kpc)
region of the Coma cluster observed with XMM-Newton and Chandra are analyzed
using a 2D power spectrum approach. The resulting 2D spectra are converted to
3D power spectra of gas density fluctuations. Our independent analyses of the
XMM-Newton and Chandra observations are in excellent agreement and provide the
most sensitive measurements of surface brightness and density fluctuations for
a hot cluster. We find that the characteristic amplitude of the volume filling
density fluctuations relative to the smooth underlying density distribution
varies from 7-10% on scales of $\sim$500 kpc down to $\sim$5% at scales $\sim$
30 kpc. On smaller spatial scales, projection effects smear the density
fluctuations by a large factor, precluding strong limits on the fluctuations in
3D. On the largest scales probed (hundreds of kpc), the dominant contributions
to the observed fluctuations most likely arise from perturbations of the
gravitational potential by the two most massive galaxies in Coma, NGC4874 and
NGC4889, and the low entropy gas brought to the cluster by an infalling group.
Other plausible sources of X-ray surface brightness fluctuations are discussed,
including turbulence, metal abundance variations, and unresolved sources.
Despite a variety of possible origins for density fluctuations, the gas in the
Coma cluster core is remarkably homogeneous on scales from $\sim$ 500 to
$\sim$30 kpc.
