An exact solution to the inverse problem of steady free-surface flow over topography (Q6593237)

The author presents an exact solution to the inverse problem for steady flow over topography that supplies the bottom shape given knowledge of the surface profile. The exact solution is straightforward to compute, and it is shown that it can be used to recover known topographical data, as was demonstrated in the case of a hydraulic fall. The inverse problem is formulated as a Stieltjes integral equation which is solved exactly using a Fourier transform. The solution requires the analytic continuation of two real functions representing the surface speed and the angle between the surface velocity vector and the horizontal. Some example surface profiles and their corresponding bottom topographies are discussed. Although the solution requires the prescription of the surface as a function of the velocity potential, it is shown to closely resemble the corresponding profile in physical space, even for quite large surface displacements, while significant discrepancy occurs at the bottom. Inference of the bottom profile from discrete surface data is accomplished by way of polynomial interpolation and rational approximation in the complex plane for the sample case of a hydraulic fall.