Hydrodynamic Flow from Fast Particles

We study the interaction of a fast moving particle in the Quark Gluon Plasma with linearized hydrodynamics. We derive the linearized hydrodynamic equations on top of an expanding fireball, and detail the solutions for a static medium. There are two modes far from the jet -- a sound mode and a diffusion mode. The diffusion mode is localized in a narrow wake behind the jet while the sound mode propagates at the Mach angle, $\cos(\theta_M) = c_s/c$. A general argument shows that the strength of the diffusion mode relative to the sound mode is directly proportional to the entropy produced by the jet-medium interaction. This argument does not rely on the linearized approximation and the assumption of local thermal equilibrium close to the jet. With this insight we calculate the spectrum of secondaries associated with the fast moving particle. If the energy loss is large and the jet-medium interaction does not produce significant entropy, the flow at the Mach angle can be observed in the associated spectrum. However, the shape of associated spectra is quite fragile and sensitive to many of the inputs of the calculation.