STABILITY OF LINEAR SHEAR FLOWS IN SHALLOW-WATER

The stability or instability of various linear shear flows in shallow water is considered. The linearized equations for waves on the surface of each flow are solved exactly in terms of known special functions. For unbounded shear flows, the exact reflection and transmission coeff icients R and T for waves incident on the flow, are found. They are sh own to satisfy the relation \R\(2) = 1 + \T\(2), which proves that ove r-reflection occurs at all wavenumbers. For flow bounded by a rigid wa ll, R is found. The poles of R yield the eigenvalue equation from whic h the unstable modes can be found. For flow in a channel, with two rig id walls, the eigenvalue equation for the modes is obtained. The resul ts are compared with previous numerical results.