  We consider the wave equation for sound in a moving fluid with a fourth-order
anomalous dispersion relation. The velocity of the fluid is a linear function
of position, giving two points in the flow where the fluid velocity matches the
group velocity of low-frequency waves. We find the exact solution for wave
propagation in the flow. The scattering shows amplification of classical waves,
leading to spontaneous emission when the waves are quantized. In the
dispersionless limit the system corresponds to a 1+1-dimensional black-hole or
white-hole binary and there is a thermal spectrum of Hawking radiation from
each horizon. Dispersion changes the scattering coefficients so that the
quantum emission is no longer thermal. The scattering coefficients were
previously obtained by Busch and Parentani in a study of dispersive fields in
de Sitter space [Phys. Rev. D 86, 104033 (2012)]. Our results give further
details of the wave propagation in this exactly solvable case, where our focus
is on laboratory systems.
