Acoustic levitation is a method used to suspend small objects in the air by utilizing the pressure generated by sound waves. The phenomenon relies on forming standing waves and manipulating acoustic radiation pressure to counteract gravity.

 Sound Waves and Pressure

Sound waves are longitudinal waves that travel through a medium, causing particles in the medium to oscillate. The oscillation of particles creates alternating regions of high and low pressure, known as compressions and rarefactions. Mathematically, a sound wave can be represented by the following equation:

where:

• P(x,t) is the sound pressure as a function of position x and time t,

• P0​ is the maximum pressure amplitude,

• f is the frequency of the sound wave,

• k=2πλ/k​ is the wave number,

• λ is the wavelength of the sound wave.

Standing Waves and Pressure Nodes

When two sound waves of equal frequency and amplitude propagate in opposite directions, they interfere to form a standing wave. A standing wave can be described by the following equation:

In this standing wave, pressure nodes (points of zero pressure fluctuation) and antinodes (points of maximum pressure) are created. The distance between successive nodes is half the wavelength (λ/2\​).

For ultrasonic frequencies, such as 40 kHz, the wavelength is very small, which allows for precise positioning of objects at pressure nodes. Objects placed at these nodes experience minimal acoustic pressure, allowing them to remain suspended.

Acoustic Radiation Pressure

In addition to standing waves, acoustic radiation pressure is a key factor in acoustic levitation. The acoustic radiation pressure is the net force exerted by the sound wave on an object, and it can be expressed as:

where:

• Fac​ is the acoustic radiation force,

• I is the intensity of the sound wave,

• c is the speed of sound in the medium.

The intensity III is related to the sound pressure as follows:

where:

• ρ is the density of the medium (air, in this case).

The object will levitate when the upward acoustic radiation force Fac​ balances the gravitational force Fg=mg, where mmm is the mass of the object and g is the acceleration due to gravity.

By controlling the amplitude of the sound wave and ensuring that the object is positioned at a pressure node, acoustic levitation is achieved.

Levitating Objects at Pressure Nodes

The standing wave produced by two out-of-phase ultrasonic transducers forms nodes where the acoustic pressure is minimal. By adjusting the distance between the transducers to match half the wavelength (λ/2), the object is suspended at a node.

where d is the distance between the two transducers, this ensures the formation of standing waves with well-defined pressure nodes and antinodes, allowing stable levitation at the pressure node.

Oscillation and Stability

Once suspended, the object may oscillate slightly around the node due to the balance of forces between gravity and acoustic radiation pressure. The oscillatory behavior can be described by the following equation for simple harmonic motion:

where:

• x(t) is the displacement of the object over time,

• A is the amplitude of oscillation,

• ω is the angular frequency of oscillation,

• ϕ is the phase angle.

By observing and analyzing the object's oscillation over time, we can study the stability of the levitated system.

 In conclusion, acoustic levitation involves creating standing waves and balancing acoustic radiation pressure with gravitational forces. Positioning the object at a pressure node makes it possible to achieve stable levitation and analyze its oscillatory behavior over time.

When considering the motion of a small object under the influence of acoustic levitation, we need to account for forces acting on the object, including gravity, acoustic radiation pressure, and air resistance. Therefore, the equation of motion will include a term for the drag force caused by air resistance, which acts in the opposite direction of the object's velocity.

 Forces Acting on the Object

1. Gravitational Force (Fg​): The downward force acting on the object due to gravity is given by:

where mmm is the mass of the object, and g is the acceleration due to gravity.

 Acoustic Radiation Force (Fac​): The upward force generated by the acoustic standing wave, responsible for levitating the object. As described earlier:

This force counteracts gravity and works to stabilize the object at a pressure node.

 Air Resistance (Drag Force, Fd​): The air resistance or drag force is proportional to the velocity of the object and opposes its motion. For small objects moving slowly, the drag force can be approximated using Stokes' Law for a spherical object:

where:

1. • η is the dynamic viscosity of air,

   • r is the radius of the object,

   • v is the velocity of the object.

This equation assumes the object is spherical and moving at low Reynolds numbers.

Equation of Motion with Air Resistance

Taking into account all forces, we can now write the net force acting on the object. According to Newton's second law of motion, the sum of the forces is equal to the object's mass times its acceleration:

Substituting the expressions for each force:

Rearranging this equation to isolate the acceleration a=dv/dt

Dividing through by mmm to express the acceleration explicitly:

This is the equation of motion for the object under the influence of gravity, acoustic radiation pressure, and air resistance.

Steady-State Condition (Terminal Velocity)

In the case where the object reaches a steady state, the net acceleration becomes zero (dv/dt=0). The velocity at this point is known as the terminal velocity, vt, and can be obtained by setting the left-hand side of the equation to zero:

Solving for the terminal velocity vt

This equation represents the terminal velocity, where the upward acoustic radiation pressure balances both gravity and air resistance.

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Time-Dependent Motion

For time-dependent motion, the solution to the differential equation can be found using methods for solving first-order linear differential equations. The velocity as a function of time v(t) can be obtained by solving:

This yields a solution of the form:

where vt​ is the terminal velocity. Initially, the object accelerates due to the net force but eventually reaches its terminal velocity as air resistance increases with speed.