Long Wave Propagation in Canals with Spatially Varying Cross-Sections and Currents

Two aspects of wave propagation in inhomogeneous media moving with a spatially varying speed are considered. Firstly, we analyze the mutual transformation of co- and counter-current propagating waves. In sub-critical flows with the current velocity U less than the local wave speed relative to the medium c, this process is known as a reflection. In super-critical flows with \(U>c\) , the process manifests as a simultaneous amplification of positive- and negative-energy waves. The most interesting phenomena in such a case are the wave transformation in currents that transit from sub- to super-critical regime and vice versa. Secondly, we find and examine such inhomogeneous flows that allow an independent propagation of co- and counter-current waves, so-called reflectionless flows. In the latter case, wave energy can be transmitted most efficiently in space which can have both positive and negative effects depending on the particular situation.