Hawking radiation and classical tunneling

``Hawking radiation'' is most familiar as a quantum field phenomenon in curved space-times that contain an event horizon. Unruh pointed out that acoustic waves in \textit{classical} fluids with \textit{nonuniform} background flows can exhibit analogous behavior. [1] The ``event horizon'' in that case consists of the set of spatial points where the flow speed and sound speed are equal. A WKB analysis [2] of the acoustic wave equation reveals that tunneling occurs at the ``event horizon,'' but it is not of the standard type. We have recast the Unruh model into a self-adjoint form using a formulation of linearized MHD due to Brizard [3]. A self-adjoint formulation of the linearized wave equation allows the use of variational methods. These provide a systematic means to derive conservation laws and, after discretization, symplectic integration schemes. \\[4pt] [1] G Volovik, M Novello, and M Visser, \textit{Artificial Black Holes} (World, 2002). \\[0pt] [2] ER Tracy, AN Kaufman, AJ Brizard, and AS Richardson, \textit{Ray Tracing and Beyond: Phase Space Methods in Plasma Wave Theory} (Cambridge, 2014). \\[0pt] [3] AJ Brizard, ``Hermitian structure for linearized ideal MHD equations with equilibrium flows,'' PLA \textbf{168} (1992) 357.