The Stability of Stationary Waves Produced By Flow Through a Channel with Wavy Sidewalls (Surface, Resonance, Stability).

Abstract

Experiments by Binnie showed that there were unsteady waves produced by flow through a channel with symmetric, wavy sidewalls, with waves propagating both upstream and downstream. However, the first order solution to this problem that was obtained by Yih is a set of steady waves. The steady solution is shown to be unstable to a pair of infinitesimal disturbance waves which satisfy the resonance conditions of Phillips. For the Froude number range used by Binnie, a pair of disturbances has been found such that one wave propagates upstream, one propagates downstream, and the amplitudes have an exponential growth. The Froude numbers outside the range of Binnie are also shown to be unstable, but only to waves propagating downstream. The steady waves produced by flow through an antisymmetric channel are shown to be unstable in the same manner. The steady waves produced by the flow of a two-layer fluid through a symmetric wavy-walled channel are also shown to be unstable to disturbances which satisfy the resonance conditions.