The imprint of the analogue Hawking effect in subcritical flows

We study the propagation of low frequency shallow water waves on a one dimensional flow of varying depth. When taking into account dispersive effects, the linear propagation of long wavelength modes on uneven bottoms excites new solutions of the dispersion relation which possess a much shorter wavelength. The peculiarity is that one of these new solutions has a negative energy. When the flow becomes supercritical, this mode has been shown to be responsible for the (classical) analog of the Hawking effect. For subcritical flows, the production of this mode has been observed numerically and experimentally, but the precise physics governing the scattering remained unclear. In this work, we provide an analytic treatment of this effect in subcritical flows. We analyze the scattering of low frequency waves using a new perturbative series, derived from a generalization of the Bremmer series. We show that the production of short wavelength modes is governed by a complex value of the position: a complex turning point. Using this method, we investigate various flow profiles, and derive the main characteristics of the induced spectrum.