The stability of capillary-gravity waves on flow over a wavy bottom

Abstract

The stability of free surface waves on a fluid flowing over a wavy bottom is considered. The mechanism for instability is the three-wave resonance conditions of Phillips. Previous studies not including surface tension have shown that there are two sets of disturbance waves which exist. It has now been found that with the inclusion of surface tension, two types of disturbances still exist, but may be multiply-valued, resulting in as many as four disturbance wave sets. The regions in parameter space with different numbers of solutions are identified. The zero surface tension case can be proven to be always unstable, but not the nonzero case. Numerical evaluation of the growth rates shows that the nonzero case is always unstable, and the instability may be a result of either type of disturbance wave. Increasing the value of surface tension from a non-zero value has been found to lead to a stronger instability. Increasing the surface tension from zero, however, may weaken the instability.