Stability and waves on a two-dimensional inviscid shear flow

Abstract

The stability of an inviscid parallel flow with a free surface is considered. The numerical solution technique is a spectral method. The eigenvalues are found by converting the resulting quadratic eigenvalue problem into a linear eigenvalue problem, using the QR method. The results are compared to an iterative scheme, and have been found to match to the first five decimal places. The bottom of the liquid layer is treated as either a rigid boundary or in contact with an infinite layer of initially still liquid. A linear and an exponential velocity profile is examined. The case with the infinite layer of liquid below the shear layer has been found to be less unstable than the case with a rigid bottom for very long waves, more unstable for an intermediate interval of wavenumber, and about the same for very short waves.