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In the linear approximation, we study a one-dimensional problem of thereflectionless wave propagation on a surface of a shallow duct with thespatially varying water depth, duct width, and current. We show that bothglobal and bounded exact solutions describing reflectionless propagation inopposite directions of long waves of arbitrary shape exist for the particularvariations of these parameters. A general analysis of the problem isillustrated by a few solutions constructed for the specific cases of spatialprofiles of the flow velocity. The results obtained can be of interest tomitigate the possible impact of waves on ships, marine engineeringconstructions, and human activity in the coastal zones.

Reflectionless wave propagation on shallow water with variable bathymetry and current