  We examine a model for the variable free-free and neutral hydrogen absorption
inferred towards the cores of some compact radio galaxies in which a spatially
fluctuating medium drifts in front of the source. We relate the
absorption-induced intensity fluctuations to the statistics of the underlying
opacity fluctuations. We investigate models in which the absorbing medium
consists of either discrete clouds or a power-law spectrum of opacity
fluctuations. We examine the variability characteristics of a medium comprised
of Gaussian-shaped clouds in which the neutral and ionized matter are
co-located, and in which the clouds comprise spherical constant-density neutral
cores enveloped by ionized sheaths. The cross-power spectrum indicates the
spatial relationship between neutral and ionized matter, and distinguishes the
two models, with power in the Gaussian model declining as a featureless
power-law, but that in the ionized sheath model oscillating between positive
and negative values. We show how comparison of the HI and free-free power
spectra reveals information on the ionization and neutral fractions of the
medium. The background source acts as a low-pass filter of the underlying
opacity power spectrum, which limits temporal fluctuations to frequencies
$\omega < \dot{\theta}_v / \theta_{\rm src}$, where $\dot{\theta}_v$ is the
angular drift speed of the matter in front of the source, and it quenches the
observability of opacity structures on scales smaller than the source size
$\theta_{\rm src}$. For drift speeds of $\sim 10^3\,$km s$^{-1}$ and source
brightness temperatures $\sim 10^{12}\,$K, this limitation confines temporal
opacity fluctuations to timescales of order several months to decades.
