  We compute the analogue Hawking radiation for modes which posses a small wave
vector perpendicular to the horizon. For low frequencies, the resulting mass
term induces a total reflection. This generates an extra mode mixing that
occurs in the supersonic region, which cancels out the infrared divergence of
the near horizon spectrum. As a result, the amplitude of the undulation
(0-frequency wave with macroscopic amplitude) emitted in white hole flows now
saturates at the linear level, unlike what was recently found in the massless
case. In addition, we point out that the mass introduces a new type of
undulation which is produced in black hole flows, and which is well described
in the hydrodynamical regime.
