Reflectionless internal wave propagation in an exchange flow of shallow two-layer medium in a channel with variable cross-section

The propagation of long linear internal waves of arbitrary shape in a steady flow of shallow two-layer fluid in a narrow channel of variable cross-section is considered. The flow velocities in the layers are oppositely directed. It is shown that there are three classes of flows in which waves can propagate without reflection in both directions, and the properties of such flows are examined. A detailed analysis has demonstrated that only one of these classes contains regular flows of unlimited extent, while the flows belonging to the other two classes are always limited (on one or both sides) by a singularity. Possible applications of the results obtained and prospects for further research are discussed.