Acoustic Levitation Node Calculator

Overview: Using Standing Waves for Acoustic Levitation

Acoustic levitation uses intense sound waves to suspend small objects at pressure nodes of a standing wave. Between two opposing emitters (for example, ultrasonic transducers), the superposition of their waves creates a stationary pattern of nodes (minimal pressure fluctuation) and antinodes (maximal pressure fluctuation). Small particles can be trapped and held near these nodes when the acoustic radiation force balances gravity.

This calculator helps you plan a simple one-dimensional levitation setup by estimating:

• Wavelength of the sound in the chosen medium
• Spacing between consecutive pressure nodes
• Number of usable nodes between two opposing emitters
• Approximate acoustic radiation force on a small spherical particle
• Whether that force can support the particle’s weight in this simplified model

The tool is intended for conceptual design, education, and quick back-of-the-envelope checks, not for final engineering of safety-critical equipment.

Core Formulas and Definitions

The underlying model is a one-dimensional standing wave between two opposing emitters separated by a distance L, in a medium with sound speed c. The emitters operate at an acoustic frequency f and create a pressure amplitude P at the levitation region.

The main quantities are:

• Wavelength λ in the medium
• Node spacing between adjacent pressure nodes
• Wavenumber k
• Acoustic energy density in the standing wave
• Radiation force on a small sphere
• Weight of the particle and a simple levitation margin

With frequency entered in kilohertz and sound speed in metres per second, we first convert to SI units. A concise reference is:

• Frequency in Hz: f = fkHz × 1000
• Wavelength: λ = c / f
• Node spacing: dnodes = λ / 2
• Approximate node count between emitters: N ≈ L / dnodes (with L converted from cm to m)
• Wavenumber: k = 2π / λ

MathML Reference Block

The following MathML block summarises several of the key relationships used by the calculator:

The calculator evaluates a specific proportionality for the radiation force using your inputs and compares it with the particle weight.

Interpreting the Calculator Outputs

When you enter the frequency, sound speed, emitter separation, acoustic pressure, particle radius, and particle density, the tool reports (labels may vary slightly):

• Wavelength in the medium – gives a sense of how “fine” the standing-wave structure is. Higher frequency or lower sound speed leads to shorter wavelengths.
• Node spacing – equal to half the wavelength. This is the ideal distance between consecutive levitation planes along the beam axis.
• Estimated number of nodes between emitters – tells you how many distinct levitation planes can exist along the path between the opposing emitters, based on the separation you specify.
• Acoustic radiation force – an approximate axial force acting to trap the particle at a node.
• Particle weight – the gravitational force pulling the particle downward, computed from its radius and density.
• Levitation margin – a dimensionless ratio, often defined as margin = Facoustic / (m g). Values greater than 1 indicate that the simplified model predicts enough acoustic force to counteract gravity along the main axis.

Use the levitation margin as a qualitative indicator only. A value slightly above 1 does not guarantee a stable trap in practice, because lateral stability, misalignment, and non-ideal behaviour can all reduce the effective force.

Worked Example

To illustrate how the calculator can be used, consider a typical air-based ultrasonic levitation experiment:

• Frequency: 40 kHz
• Speed of sound: 343 m/s (air at about 20 °C)
• Emitter separation: 4 cm
• Acoustic pressure amplitude: 1 kPa
• Particle radius: 0.5 mm
• Particle density: 1000 kg/m³ (roughly water)

Step by step:

1. Convert frequency to Hz
   f = 40 × 1000 = 40,000 Hz
2. Compute wavelength
   λ = c / f = 343 / 40,000 ≈ 0.008575 m (about 8.6 mm).
3. Node spacing
   dnodes = λ / 2 ≈ 4.3 mm.
4. Emitter separation
   Convert 4 cm to metres: L = 0.04 m. The approximate number of half-wavelength intervals is L / dnodes ≈ 0.04 / 0.0043 ≈ 9.3, so roughly 9–10 nodes can fit along the axis between emitters.
5. Particle volume and mass
   Radius in metres: r = 0.5 mm = 0.0005 m.
   Volume: V = (4/3) π r³ ≈ (4/3) π (5 × 10−4 m)³.
   Mass: m = V ρp with ρp = 1000 kg/m³.
6. Weight
   W = m g, where g ≈ 9.81 m/s².
7. Acoustic radiation force
   The calculator uses your pressure amplitude and the wave parameters to estimate a radiation force that roughly scales like F &propto r² k E, with E the acoustic energy density.
8. Levitation margin
   The tool then reports margin = F / W. If this is, for example, 2.5, it suggests the axial acoustic force is 2.5 times the weight in the simplified model, leaving some room for non-idealities.

By changing the inputs (for example, increasing pressure amplitude or frequency, or choosing a lower-density particle), you can see how the levitation margin responds and identify more favourable operating points.

Practical Design Guidance

Keep the following points in mind when using the calculator as a design aid:

• Frequency range: Ultrasonic levitation commonly uses frequencies from 20 kHz to 100 kHz. Higher frequencies give shorter wavelengths and tighter node spacing but typically require more specialised hardware.
• Emitter separation: Choose a separation that fits an integer (or near-integer) number of half-wavelengths. This helps form a strong standing wave and stable node positions.
• Acoustic pressure amplitude: The pressure amplitude you enter should be consistent with what your transducers and driver electronics can safely provide. Manufacturer datasheets typically specify maximum sound pressure levels at a given distance.
• Particle properties: Smaller, lower-density particles are easier to levitate. Liquids and biological samples often behave like dense droplets; their shape and surface tension can also matter.
• Environment: Air temperature, humidity, and medium composition all affect sound speed and damping. The calculator assumes a fixed sound speed value that you provide.

Comparison of Key Parameters

The table below qualitatively compares how different parameter changes affect node spacing and levitation feasibility in this simplified model.

Assumptions and Limitations

The calculator is based on a simplified physical model. It is important to understand its assumptions and limitations before applying the outputs to real hardware:

• One-dimensional standing wave: The model assumes two opposing emitters along a single axis, forming a purely one-dimensional standing wave. Real setups often exhibit three-dimensional field structures and off-axis forces.
• Small, rigid, spherical particles: The radiation force estimate assumes a small sphere whose radius is much smaller than the wavelength and that behaves as a rigid body. Non-spherical, deformable, or comparable-to-wavelength objects may behave very differently.
• Homogeneous, lossless medium: The medium is treated as uniform with constant density and sound speed. Absorption, scattering, and flow in the medium (for example, air currents) are ignored.
• Linear acoustics: The model assumes linear wave propagation. At very high intensities, nonlinear effects, shock formation, and cavitation in liquids can occur, none of which are captured here.
• Approximate force model: The radiation force expression used is an approximation valid only in a restricted regime (small particle, far from boundaries, near a pressure node). It does not account for all multipole contributions, viscous effects, or near-field corrections.
• No lateral stability analysis: The tool considers only axial trapping along the standing-wave axis. Lateral stability and rotational dynamics are not evaluated.
• Ideal alignment: Perfect alignment between the two emitters and no phase errors are assumed. In practice, misalignment can significantly reduce the effective field strength and shift node positions.
• No safety or regulatory checks: The calculator does not check whether your chosen parameters comply with exposure limits, equipment ratings, or regulatory standards.

Because of these assumptions, the results should be viewed as indicative, not definitive. Experimental measurements and more detailed simulations are required for precise design work.

Safety and Responsible Use

Acoustic levitation experiments, especially at ultrasonic frequencies, can involve high sound pressures, high voltages on transducers, and potential exposure to inaudible but intense sound fields. Keep in mind:

• High-intensity ultrasound can pose risks to hearing and may damage sensitive components.
• Liquids and biological specimens may heat up or undergo cavitation under strong acoustic fields.
• Always follow laboratory safety protocols, manufacturer datasheets, and applicable regulations when energising transducers.
• Use the calculator as an educational and planning tool, not as a substitute for professional engineering design or safety evaluation.

How to Use This Calculator Effectively

To get the most value from the tool:

• Start with realistic defaults for the medium, such as 343 m/s for air at room temperature.
• Vary one parameter at a time (for example, frequency or pressure amplitude) to see how it affects wavelength, node spacing, and levitation margin.
• Look for combinations where the levitation margin is significantly above 1 while keeping pressure amplitudes within safe and realistic limits for your hardware.
• Use the predicted node spacing to design mechanical supports that allow emitters to be placed at separations corresponding to an integer number of half-wavelengths.
• Treat the results as a starting point; refine them with measurements, more detailed modelling, or published experimental data.

By understanding both the capabilities and the limitations of this simplified standing-wave model, you can use the calculator to explore acoustic levitation concepts, plan lab demonstrations, and guide early-stage design decisions before committing to more complex analysis.