On Replacements Reducing One-Dimensional Systems of Shallow-Water Equations to the Wave Equation with Sound Speed $c^2=x$

We obtain point transformations for three one-dimensional systems: shallow-water equations on a flat and a sloping bottom and the system of linear equations obtained by formal linearization of shallow-water equations on a sloping bottom. The passage of these systems to the Carrier–Greenspan parametrization is also obtained. For linear shallow-water equations on a sloping bottom, we obtain the solution in the form of a traveling wave with variable velocity. We establish the relationship between the resulting solution and the solution of the two-dimensional wave equation.