Equilibrium-range spectra of sand waves formed by flowing water

Based on a dimensional consideration, the ‘−3 power law’ on the spatial spectrum of sand waves formed by flowing water S ηη ( k ) is derived for a large wave-number equilibrium subrange, \[ Sηη(k)~ α k⁻³, \] where α is a constant depending on the angle of repose of sand particles and k denotes the wave-number. Likewise, the frequency spectrum is shown to have the ‘−3 power law’ range for higher frequencies as well as the ‘−2 power law’ range for frequencies near a spectral peak. These spectra are shown to agree with experimental data from various sources.