Acoustic Levitator

What is modelled

A flat emitter at the bottom radiates a wave upward; a rigid reflector at the top sends it back. Between them the two waves form a standing wave whose pressure is zero at nodes spaced half a wavelength apart. A small, dense bead (much smaller than the wavelength) feels a time-averaged radiation force that pushes it toward the pressure nodes, and gravity makes it sit slightly below each node. If the sound is too weak for the bead's weight, the bead falls out of the trap.

Equations

• Sound speed c = 331.3 √(1 + T / 273.15) m/s, wavelength λ = c / f, node spacing λ / 2.
• Cavity of round-trip amplitude r = 0.8: gain 1 / |1 - r e2ikL|, largest when the gap L is a whole number of half wavelengths.
• Acoustic energy density E = p0² / (4 ρ0 c²).
• Gor'kov trapping energy U = V [ f1 <p²> / (2 ρ0 c²) - (3/4) f2 ρ0 <v²> ], with f1 = 1, f2 = 2(ρ̃ - 1)/(2ρ̃ + 1), contrast factor Φ = f1/3 + f2/2.
• Axial force F = 4π Φ k a³ E sin(2kz). The bead is held when ρp g / (3 Φ k E) < 1 and sits at sin(2kδ) = ρp g / (3 Φ k E) below the node; the bead radius cancels.

Checks run in the page

• Cavity resonance found by summing 400 reflections while scanning the gap.
• Force from a numerical derivative of the energy built from the pressure and velocity fields against the closed form.
• A bead released above a node, integrated with gravity, force and Stokes drag, against the analytic offset.
• Node spacing by scanning the field, and the sound speed against the ideal gas formula.

Honest simplifications

The bead is treated as incompressible and small compared with the wavelength (plane standing wave, no transverse trapping, no wave scattering, no streaming, no nonlinear distortion). The cavity is a single lumped round-trip loss. The 3D kit is a visual illustration with its own arbitrary frequency and gap units, so its sliders do not map to the numbers here. The animation is slowed 100 times.

• Use the tuned gap preset, then drag the gap away from a whole number of half wavelengths and watch the cavity gain and the trap collapse.
• Choose Too dense to hold: the same sound level cannot hold a bead of that density and the beads fall out in red.
• Raise the temperature to 40 C and see the wavelength, and the gap, grow.