The problem of Compton scattering in an optically thick fluid flow in which bulk motion is the dominant source of photon heating is illustrated by analysing a radiation-dominated, plane-parallel shock of speed u with photon to electron ratio greatly exceeding ∼(m_p/m_e). In traversing the shock (of thickness ∼(c/u) Thomson optical depths), a typical photon experiences (c/u)2 scatterings, each one giving a secular fractional energy increase ∼(u/c)^2 and a total average increase of order unity. In a converging fluid flow, an exponentially small number of photons are accelerated to an exponentially large energy. Thus, a power-law spectrum will be transmitted at high frequencies. For a shock of Mach number M, bulk acceleration produces a spectral index α = (M^2 − ½)(M^2 + 6)(M^2 – 1)^(−2), which tends to unity for a strong shock. The applicability of these results to quasars and the microwave background is briefly discussed
