Extended Data Fig. 2: Geometric effects on hop length modulations.
From: Emergence of wind ripples controlled by mechanics of grain–bed impacts
(a) Cotangent of impact angles \(\cot \theta (\ell )\) for Mars (red), Earth (blue), Titan (green) and Venus (gold) conditions, measured from DEM simulations. \(\cot \theta (\ell )\) is rescaled by its average for clarity of presentation. For Mars, Earth, Titan and Venus \(\langle \cot \theta \rangle = (1.4,1.5,2.8,3.7)\), respectively. (b) The Fourier transform of geometric hop length modulations (for small surface perturbations \(\hat{Z}\)) are defined as \(\Delta \hat{\ell }=\cot \theta (\ell )({e}^{-ik\ell }-1)\hat{Z}\) (Eq. (6)). Plotted in (b) is the (negative) phase and magnitude of \(\Delta \hat{\ell }/\hat{Z}\) versus rescaled hop length ℓ/λ for λ = 200d in Earth conditions (〈ℓ〉 ≈ 100d). Also shown over the magnitude is the envelop \(2\cot \theta (\ell )\).
