Acoustic Levitation

The intuition — sound as a cage of nodes

Point a powerful ultrasonic speaker at a hard reflector a few millimetres away. The outgoing wave and its reflection overlap, and where they meet they lock into a standing wave: a fixed pattern of layers where the air pressure swings violently (pressure antinodes) alternating with layers where the pressure barely changes at all (pressure nodes). The pattern doesn't travel — it just breathes in place.

Drop a tiny bead into that field and it doesn't fall. It snaps to one of the quiet layers — a pressure node — and hangs there. Nudge it and it springs back. The standing wave has built an invisible cage of stacked trapping points, each one half a wavelength below the next.

That is the whole trick: a sound field intense enough that its time-averaged push on an object is comparable to the object's weight. The push is called the acoustic radiation force, and it always herds a small dense particle toward the nearest pressure node.

How a standing wave forms the trap

A one-dimensional standing wave between a source and a reflector has a pressure field of the form:

p(z, t) = 2·p₀·cos(k·z)·cos(ω·t)

where k = 2π/λ is the wavenumber, ω = 2πf is the angular frequency, and p₀ is the amplitude of each travelling wave. The factor cos(k·z) is fixed in space — it sets where the nodes and antinodes are, independent of time.

• Pressure nodes sit where cos(k·z) = 0, i.e. z = λ/4, 3λ/4, 5λ/4, … Here the pressure is always near zero, but the air's velocity oscillation is maximal — these are velocity antinodes.
• Pressure antinodes sit where |cos(k·z)| = 1, i.e. z = 0, λ/2, λ, … Here the pressure slams up and down hardest, but the air barely moves (velocity nodes).

Nodes and antinodes alternate every quarter wavelength, so successive trapping points (the nodes) are λ/2 apart. At 40 kHz in air (λ ≈ 8.6 mm) the trapped particles stack about 4.3 mm apart — which is exactly the spacing you see in a multi-droplet acoustic levitator.

The governing physics — Gor'kov's potential

For a particle much smaller than the wavelength (the Rayleigh regime, particle radius a with k·a ≪ 1), Lev Gor'kov showed in 1962 that the time-averaged radiation force derives from a scalar potential:

F_rad = -∇U

U = 2π·a³·ρ₀·[ (1/3)·f₁·⟨p²⟩ / (ρ₀²·c₀²)  −  (1/2)·f₂·⟨v²⟩ ]

Here ⟨p²⟩ and ⟨v²⟩ are the time-averaged squared pressure and squared fluid velocity, ρ₀ and c₀ are the density and sound speed of the surrounding fluid, and the two contrast factors compare the particle to the fluid:

f₁ = 1 − (ρ₀·c₀²) / (ρ_p·c_p²)        (compressibility/monopole term)

f₂ = 2·(ρ_p − ρ₀) / (2·ρ_p + ρ₀)       (density/dipole term)

For a dense, stiff particle in air (ρ_p ≫ ρ₀), both f₁ → 1 and f₂ → 1. The velocity term then wins, and U is minimised where ⟨v²⟩ is maximal and ⟨p²⟩ is minimal — i.e. exactly at the pressure nodes. Because F = -∇U, the force points downhill toward those minima from both sides: a restoring force. That is why the particle is trapped at a node rather than blown away.

The acoustic contrast factor Φ = f₁/3 + f₂/2 summarises the sign: Φ > 0 (the usual case — dense particle in lighter fluid) means trapping at pressure nodes; Φ < 0 (a bubble in water, which is light and very compressible) flips the sign so the object is pushed to pressure antinodes instead.

The levitation condition — beating gravity

Levitation works only when the upward radiation force can match the particle's weight. Near a node the radiation force is approximately linear in displacement, behaving like a spring with an axial stiffness set by the sound field. Equating the peak available radiation force to gravity gives the rough requirement:

F_rad,max  ≈  k·V·E_ac·Φ   ≳   m·g  =  ρ_p·V·g

where V is the particle volume, E_ac = ⟨p²⟩/(2·ρ₀·c₀²) is the acoustic energy density, and g = 9.81 m/s². Two things fall out of this:

1. Volume cancels in scaling. Both the force and the weight scale with V, so making the particle bigger doesn't help on its own — the real lever is the energy density E_ac and the wavenumber k (higher frequency = stronger gradient). You win by cranking the sound intensity or raising the frequency, not by changing particle size.
2. The equilibrium sits below the node. Gravity shifts the trapped particle a little below the geometric node until the spring-like restoring force exactly cancels mg. The stronger the field, the smaller that sag.

Because k = ω/c₀, a higher frequency tightens the trap (steeper potential well) but shrinks the node spacing λ/2 — so there is a practical sweet spot, which is why most rigs land around 20–40 kHz.

By the numbers — a 40 kHz air levitator

Quantity	Symbol	Typical value (air, 40 kHz)
Sound speed in air	c₀	343 m/s
Air density	ρ₀	1.20 kg/m³
Wavelength	λ = c₀/f	8.6 mm
Node spacing	λ/2	4.3 mm
Sound pressure level	SPL	~155–170 dB
Pressure amplitude	p₀	~1–6 kPa (vs 101 kPa ambient)
Max trapped particle	2a	≲ λ/2 ≈ 2–4 mm
Trapped mass (water drop)	m	~1–30 mg

A sound pressure level of 165 dB sounds absurd — it is far above the ~120 dB pain threshold — but at 40 kHz it is well above human hearing, so the chamber is silent to us. The pressure amplitude of a few kilopascals is only a few percent of atmospheric pressure, yet integrated over each cycle it produces a steady radiation force of order tens of micronewtons, comfortably more than the ~0.1–300 µN weight of a millimetre droplet.

Single-axis, phased arrays, and acoustic holograms

Type	Geometry	What it can do	Limitation
Single-axis (Langevin + reflector)	One transducer facing a curved reflector	Stacks particles at fixed nodes; the classic demo	No lateral control; one column of nodes
Phased array (TinyLev / Ultraino)	Dozens of small 40 kHz transducers in a bowl	Moves the trap in 3D by retiming each emitter	Still milligram-scale; needs phase calibration
Twin-trap / vortex / bottle beam	Phased array with engineered phase pattern	Holds larger objects; tweezers individual beads	Trap shape sensitive to object size
Acoustic hologram	3D-printed phase plate over a transducer	Sculpts complex pressure fields, "draws" with particles	Field is fixed once the plate is printed
In-liquid (acoustofluidics)	Standing wave in a microchannel	Sorts cells and microbeads by size/density	Not air levitation; impedance is matched to water

The leap from a single-axis rig to a phased array is what made acoustic levitation a manipulation tool rather than a parlour trick: by adjusting the phase delay sent to each transducer, you steer the node to any point in the working volume and even rotate or merge droplets in mid-air.

Where it shows up

• Containerless processing. Pharmaceutical and materials labs levitate a droplet so it crystallises or reacts without ever touching a wall — no container contamination, no nucleation on glass. The European Space Agency and NASA have flown acoustic levitators to study melts and supercooled liquids.
• Analytical chemistry. A single levitated droplet is a clean, wall-free micro-reactor; spectrometers probe it directly. Evaporation, mixing and protein crystallisation have all been run this way.
• Display and fabrication. Phased arrays form mid-air "acoustic tractor beams" and even volumetric displays that move a single bead fast enough to trace a glowing 3D image (the University of Sussex "Multimodal Acoustic Trap Display").
• Biology, contact-free. Live ants, ladybirds, and fish eggs have been levitated unharmed to demonstrate that the technique works on any impedance mismatch — useful for handling delicate cells without a pipette wall.
• Microassembly. Picking and placing tiny surface-mount electronic components or optical elements without tweezers that could scratch or stick.

Common misconceptions and edge cases

• "The object sits at the loudest point." No — a dense particle sits at a pressure node, the quietest pressure layer (but the busiest velocity layer). Only light, compressible objects like bubbles go to antinodes.
• "You hear a loud hum." The field is typically ultrasonic (≥ 20 kHz), inaudible to humans, even though its SPL would be deafening if it were in the audible band.
• "Bigger sound lifts heavier things linearly." The force scales with acoustic energy density (∝ pressure squared), not pressure itself, and at high intensity nonlinear effects — acoustic streaming, heating, droplet atomisation — set the ceiling, not the linear theory.
• "It only works in air." It works in any fluid; in water the impedance contrast and energy handling let you trap far larger objects, which is the basis of acoustofluidic cell sorting.
• "The particle is perfectly still." Real traps show slow orbiting and shape oscillation; large droplets flatten and can even split (a Rayleigh-like instability) if the field is too strong.
• "It defies gravity." It doesn't — gravity still acts. The radiation force simply supplies an equal, opposite, time-averaged push, exactly like a hand holding the object up, and the equilibrium sits a touch below the node where the two balance.