Starting point are the linearized one-dimensional Saint-Venant equations for an open channel with a rectangular cross section (vertical side walls). These equations describe the evolution of a wave with free surface elevation  and horizontal flow velocity  with  the horizontal coordinate along the channel axis and  the time:
where  is the gravity of Earth (taken as a constant),  is the mean water depth,  is the channel width and  and  are denoting partial derivatives with respect to space and time. The slow variation of width  and depth  with distance  along the channel axis is brought into account by denoting them as  and  where  is a small parameter:  The above two equations can be combined into one wave equation for the surface elevation:
In the Liouville–Green method, the approach is to convert the above wave equation with non-homogeneous coefficients into a homogeneous one (neglecting some small remainders in terms of ).