Singing Sand Dune

What is modelled

Wind lifts sand grains only above a threshold friction speed that grows with the grain size and with moisture. Grains hopping over the crest (saltation) fall onto the steep lee face and set off avalanches. The grains in an avalanche shear against each other at a rate of order the square root of g over the grain diameter, which gives a tone. The dry sand above a firmer base behaves as a one-dimensional resonator: sound travels slowly near the loose surface and faster at depth, so the layer has a ladder of resonant frequencies. When the avalanche tone sits on one of them the dune booms.

Equations

• Threshold u*t = 0.1 √((ρs - ρa) g d / ρa), raised for moisture by the Belly fit.
• Wind profile U(z) = (u*/κ) ln(z/z0), z0 = d/30.
• Bagnold sand flux q = 1.8 (ρa/g) √(d/D) u*³, D = 0.25 mm.
• Avalanche tone f = k √(g/d) with an empirical k near 0.5.
• Wave speed c = c1 z1/6 (contact stiffness grows with pressure), travel time T = (6/5) h5/6/c1, modes fn = (2n - 1)/(4T).

Checks run in the page

• The travel time integral against the closed form.
• A finite-difference wave simulation of the layer against its fourth resonance.
• The log-law wind profile integrated numerically.
• The half-power width of the resonance against f / Q.

Honest simplifications

Why dunes boom is still debated; this page uses one published picture (avalanche shear rate plus a waveguide in the dry layer) with single constants for each part. The wave-speed law, the avalanche constant k and the moisture damping are assumptions, not measurements of a particular dune, the dune is one-dimensional, the base is rigid and the sound is not radiated. Real booming dunes also vary the pitch with the slope and the layer. The 3D cathedral is an artistic wind harp, and its sliders carry no units.

• Choose Fine grains: the avalanche tone rises with the inverse square root of the grain size.
• Choose After rain (damp): the threshold rises and the resonance is damped, so the dune is silent.
• Choose Thick dry layer: the resonances move down, and the tone and a mode may no longer line up.