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.
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