Undulations from amplified low frequency surface waves

Abstract

We study the linear scattering of gravity waves in longitudinal inhomogeneous stationary flows. When the flow becomes supercritical, it is known that counterflow propagating shallow waves are blocked and converted into deep waves. Here we show that in the zero-frequency limit, the reflected waves are amplified in such a way that the free surface develops an undulation, i.e., a zero-frequency wave of large amplitude with nodes located at specific places. This amplification involves negative energy waves and implies that flat surfaces are unstable against incoming perturbations of arbitrary small amplitude. The relation between this instability and black hole radiation (the Hawking effect) is established.