Design Acoustic Levitation

Design + validate acoustic levitation. Determine radiation pressure → balance gravity, select transducer+reflector → standing wave, compute node positions + trapping strength, verify stability vs lateral/axial perturbations.

Use When

• Contactless sample holder → chem/bio
• Levitator demo → education
• Eval: can object levitate (size, density, freq)?
• Single-axis (transducer-reflector) vs phased array
• Calc node positions + trapping forces
• Extend single → multi-axis w/ phased arrays

In

• Required: Object props (mass, density, radius, compressibility if known)
• Required: Medium (air, water, inert gas) + density + speed of sound
• Optional: Transducer freq (default 40 kHz)
• Optional: Transducer power/voltage
• Optional: Manipulation cap (static vs dynamic)

Do

Step 1: Object props + acoustic contrast

Characterize → fundamental feasibility:

1. Object params: m, density rho_p, radius a, bulk modulus K_p (kappa_p = 1/K_p). Rigid metal → K_p effectively inf.
2. Medium params: rho_0, c_0, K_0 = rho_0 * c_0^2.
3. Contrast factors (Gor'kov → node vs antinode):
   • Monopole: f_1 = 1 - (K_0 / K_p) = 1 - (rho_0 * c_0^2) / (rho_p * c_p^2)
   • Dipole: f_2 = 2 * (rho_p - rho_0) / (2 * rho_p + rho_0)
   • Most solids in air: f_1 ~ 1, f_2 ~ 1 → trapped at pressure nodes (velocity antinodes).
4. Size constraint: a << lambda = c_0 / f (Gor'kov requires a << lambda, typically a < lambda/4). Else ray acoustics or full num sim.

## Object and Medium Parameters
- **Object**: [material, mass, density, radius, bulk modulus]
- **Medium**: [gas/liquid, rho_0, c_0, K_0]
- **Contrast factors**: f_1 = [value], f_2 = [value]
- **Wavelength**: lambda = [value] at f = [frequency]
- **Size ratio**: a / lambda = [value] (must be << 1)
- **Trapping location**: [pressure node / pressure antinode]

→ Complete character. Object confirmed → pressure nodes. Size constraint OK.

If err: a / lambda > 0.25 → Gor'kov breaks. Use FEM acoustic sim or calibration. If f_1, f_2 opposite signs → intermediate position, map Gor'kov potential.

Step 2: Required radiation pressure

Field intensity → balance gravity:

1. Radiation force (small sphere at node, 1D standing wave):
   • F_ax = -(4 * pi / 3) * a^3 * [f_1 * (1 / (2 * rho_0 * c_0^2)) * d(p^2)/dz - (3 * f_2 * rho_0 / 4) * d(v^2)/dz]
   • Plane standing wave p(z,t) = P_0 * cos(kz) * cos(omega*t) near node:
   • F_ax = (pi * a^3 * P_0^2 * k) / (3 * rho_0 * c_0^2) * Phi * sin(2kz)
   • Phi = f_1 + (3/2) * f_2, k = 2*pi/lambda.
2. Force balance: Max force (sin(2kz) = 1, at lambda/8 from node) = gravity:
   • F_ax_max = (pi * a^3 * P_0^2 * k) / (3 * rho_0 * c_0^2) * Phi = m * g = (4/3) * pi * a^3 * rho_p * g
   • Solve: P_0 = sqrt(4 * rho_p * rho_0 * c_0^2 * g / (k * Phi))
3. Intensity: I = P_0^2 / (2 * rho_0 * c_0). Compare transducer rating.
4. SPL: L = 20 * log10(P_0 / 20e-6). Typical air levitation 150-165 dB SPL.

## Acoustic Requirements
- **Required pressure amplitude**: P_0 = [value] Pa
- **Required intensity**: I = [value] W/m^2
- **Sound pressure level**: L = [value] dB SPL
- **Safety note**: [hearing protection required if > 120 dB at audible frequencies]

→ P_0 min in Pa, W/m^2, dB SPL. Achievable w/ specified or commercial transducer.

If err: P_0 exceeds available → reduce mass/density, lighter material, denser medium (SF6). Multi transducers focused array.

Step 3: Transducer-reflector geometry

HW → stable standing wave:

1. Transducer: Ultrasonic at f (28 kHz, 40 kHz, 60-80 kHz piezo). Higher f → smaller lambda, tighter trap, smaller max obj. Verify P_0 at operating distance.
2. Reflector: Flat or concave opposite transducer. Acoustically hard (high impedance mismatch). Metal/glass in air. Concave → concentrates sound + increases P at axis.
3. Cavity length: L = n * lambda/2 (n int). Creates n nodes spaced lambda/2.
4. Node positions: z_j = (2j - 1) * lambda/4 from reflector, j = 1..n. Node nearest center = most stable.
5. Resonance tuning: Fine-tune L w/ micrometer stage, monitor force or P w/ mic. Optimal → strongest standing wave.

## Geometry Design
- **Transducer**: [model, frequency, rated power or SPL]
- **Reflector**: [material, shape (flat/concave), dimensions]
- **Cavity length**: L = [n] x lambda/2 = [value] mm
- **Number of nodes**: [n]
- **Node positions from reflector**: z_1 = [value], z_2 = [value], ...
- **Selected trapping node**: z_[j] = [value]

→ Complete HW spec + node positions + selected trap node.

If err: No standing wave (L not precise n*lambda/2) → 0.1 mm increments. Temp shifts c_0 + lambda → re-tune. Beam diverges → horn/waveguide or reduce L.

Step 4: Trapping potential + restoring forces

Quantify strength + spatial extent:

1. Gor'kov potential:
   • U(r) = (4/3) * pi * a^3 * [(f_1 / (2 * rho_0 * c_0^2)) * <p^2> - (3 * f_2 * rho_0 / 4) * <v^2>]
   • Object trapped at min of U(r) + mgz.
2. Axial restoring:
   • F_z ~ -k_z * delta_z, k_z = (2 * pi * a^3 * P_0^2 * k^2) / (3 * rho_0 * c_0^2) * Phi
   • omega_z = sqrt(k_z / m).
3. Lateral restoring (Gaussian beam waist w):
   • k_r ~ k_z * (a / w)^2 (lateral weaker than axial)
   • Lateral = limiting factor for stability.
4. Trap depth: Max displacement before escape. Axial well: Delta_U = F_ax_max * lambda / (2 * pi). Express as × k_B*T if relevant.

## Trapping Analysis
- **Axial stiffness**: k_z = [value] N/m
- **Axial natural frequency**: omega_z / (2*pi) = [value] Hz
- **Lateral stiffness**: k_r = [value] N/m
- **Lateral natural frequency**: omega_r / (2*pi) = [value] Hz
- **Axial well depth**: Delta_U = [value] J = [value] x k_B*T
- **Stiffness ratio**: k_z / k_r = [value] (lateral is weaker)

→ Stiffness for both axes + freqs + well depth. Lateral confirmed positive.

If err: Lateral neg/negligible → drifts sideways. Wider transducer (bigger waist), add lateral transducers, phased array, concave reflector for converging wavefront.

Step 5: Stability vs perturbations

Reliable trap + hold:

1. Gravity offset: delta_z = m * g / k_z. Must be << lambda/4. If ~lambda/4 → falls out.
2. Air currents: F_drag = 6 * pi * eta * a * v_air (Stokes). Max tolerable: v_max = k_r / (6 * pi * eta).
3. Acoustic streaming (Rayleigh, steady circulation): v_stream ~ P_0^2 / (4 * rho_0 * c_0^3 * eta) * lambda. Drag on object. Must be < lateral restoring.
4. Thermal: Absorption heats medium → c_0 shifts → nodes drift. High-intensity (>160 dB SPL) → estimate temp rise + drift over time.
5. Phased array ext (dynamic): Replace single pair w/ phased array. Adjusting phases → nodes move continuously, carry object. Phase resolution: delta_z ~ lambda / (2 * pi * N_phase_bits).

## Stability Verification
| Perturbation | Magnitude | Restoring Force | Margin | Stable? |
|-------------|-----------|----------------|--------|---------|
| Gravity offset | delta_z = [val] | k_z * delta_z | delta_z / (lambda/4) = [val] | [Yes/No] |
| Air currents | v_air = [val] m/s | F_lat = [val] N | F_lat / F_drag = [val] | [Yes/No] |
| Acoustic streaming | v_stream = [val] | F_lat = [val] N | F_lat / F_stream_drag = [val] | [Yes/No] |
| Thermal drift | Delta_T = [val] K | Re-tune interval | [time] | [Acceptable/No] |

→ All perturbations quantified + w/in margins. Gravity offset small frac of lambda/4. Air + streaming don't overwhelm lateral trap.

If err: Gravity offset too big → increase P_0 or higher freq. Air currents → draft shield. Streaming destabilizes → reduce amplitude, shallow concave reflector minimizes vortices.

Check

• a << lambda (Gor'kov applicable)
• Contrast factors + node/antinode identified
• P_0 calc + achievable w/ HW
• Cavity L = n * lambda/2 + nodes computed
• Axial + lateral stiffness both positive
• Gravity offset small frac of lambda/4
• Air + streaming w/in margins
• Safety for high-SPL docs
• Phased array: phase res + precision specified

Traps

• Violating small-particle: Gor'kov assumes a << lambda. Approaching lambda/4 → point-particle breaks, force diffs (mag + direction). Full-wave sim for large.
• Ignoring lateral: Most treatments focus axial, neglect lateral. Lateral instability = primary failure mode near size limit.
• Forget streaming: High-intensity → steady streaming drag vs radiation force. Not small — dominant destabilizer at high SPL.
• Temp sensitivity: c_0 in air → 0.6 m/s per °C. 10° swing → lambda shifts ~2% → nodes drift mm in typical cavity. Long runs need active comp or temp ctrl.
• Pressure vs velocity nodes: P nodes = v antinodes + vice versa. Solids w/ positive contrast → P nodes (P min, v max). Reversed → wrong position.
• Nonlinear at high amp: >155-160 dB SPL → harmonic gen, shock formation → reduces trapping vs linear theory.

→

• evaluate-levitation-mechanism — compare acoustic vs magnetic, electrostatic, aerodynamic
• analyze-magnetic-levitation — complement for compare
• derive-theoretical-result — radiation pressure from first principles