Figure 8

Shape characterization with Fourier plane waves. (A) Real sand dune, with surface ripples. (B) Numerically generated (idealized) sand dune with ripples. The underlying dune is represented by a single low spatial frequency, high amplitude wave. The ripples were generated by the sum of two high frequency, low amplitude waves oriented at \(45^{\circ }\) and \(-45^{\circ }\) relative to the y-axis. The entire dune is the sum of all these waves. (C) The magnitude of the Fourier Transform (FT) of (B) detects these three waves. The location of the peaks gives the spatial frequency, the angle of the peaks in the \((k_{x},k_{y})\) plane provides the angle in the spatial (x, y) domain. These estimated Fourier components characterize the dune shape. The mirror image peaks are redundant information that result from the symmetry of the FT. The small ridges emanating from the peaks are a consequence of the finite nature of the numerical FT.