Levitah

Levitah holds a row of pearls still in mid-air, caught in a column of sound no one can see.

The Physics

Sound carries momentum, and a strong standing wave exerts a steady average force, the acoustic radiation force, on small objects. King computed it for spheres in 1934. Gor'kov (1962) wrote it, for a sphere much smaller than the wavelength in an ideal fluid, as the gradient of a potential U, with a compressibility factor f₁ and a density factor f₂.

For a small bead much denser and much less compressible than air (ρ_p ≫ ρ₀, κ_p ≪ κ₀), both factors are close to 1. U is then lowest where the pressure amplitude vanishes and the velocity amplitude peaks, so the bead collects near the pressure nodes, which are the velocity antinodes, half a wavelength apart. At 40 kHz in air the wavelength is about 8.6 mm, so the trapping sites are about 4.3 mm apart.

U = (4π/3) a³ [ f₁ · ½ κ₀ ⟨p²⟩ − f₂ · ¾ ρ₀ ⟨v²⟩ ], F = −∇U

• U: radiation potential (J)
• a: sphere radius, a ≪ wavelength (m)
• f₁ = 1 − κ_p/κ₀: compressibility contrast
• f₂ = 2(ρ_p − ρ₀)/(2ρ_p + ρ₀): density contrast
• κ₀, ρ₀: compressibility and density of the air
• ⟨p²⟩, ⟨v²⟩: time-averaged squared acoustic pressure and velocity

King, Proc. R. Soc. Lond. A 147, 212 (1934); Gor'kov, Sov. Phys. Dokl. 6, 773 (1962).

Honest caveat: It is not anti-gravity, and it does not work on any object: size and density limits apply. Dense, rigid beads collect near pressure nodes (velocity antinodes), not at velocity nodes; in a vertical levitator gravity holds each bead slightly below its node.

On the shirt

Front: Levitah, the pixel-art character, with “HELD BY SOUND”. Back: Gor'kov's potential, the force, the pressure-node rule, the 40 kHz spacing and the 1934 and 1962 sources. Printed on a unisex tee.