Technical Column

Methods for Measuring Acoustic Intensity

2025年5月26日

Acoustic Power Measurement

Euler’s Equation

Acoustic intensity is defined as the time-averaged product of sound pressure and particle velocity.
While sound pressure can be relatively easily measured with a single microphone, direct measurement of particle velocity can be challenging.

However, particle velocity can be calculated using the pressure gradient (the spatial rate of change of pressure at a given moment) and the linearized Euler’s equation.
By measuring the pressure gradient with two closely spaced microphones, the particle velocity can be derived.

Euler’s equation is essentially an application of Newton’s Second Law in fluid mechanics.
Newton’s Second Law relates the force acting on a mass to its resulting acceleration.
If the force and mass are known, the acceleration can be determined, and by integrating acceleration over time, the velocity can be obtained.

In Euler’s equation, the acceleration of a fluid with density ρ is driven by the pressure gradient.
From the pressure gradient and the fluid density, the particle acceleration can be calculated.
By integrating this acceleration over time, the particle velocity can be obtained.

Finite Difference Approximation

The pressure gradient is a smoothly varying continuous function.
By taking the difference in sound pressure between two closely spaced microphones and dividing it by the distance between them, a linear approximation of the pressure gradient can be obtained.
This method is known as the finite difference approximation.
It is similar to drawing a straight line between two points on a circle to approximate the tangent at that point.

Calculation of Acoustic Intensity

To estimate particle velocity, the components of the pressure gradient must be integrated.
The particle velocity is evaluated at the midpoint between the two microphones, which represents the acoustic center of the probe.
The sound pressure at this point is approximated by averaging the pressures measured by the two microphones.
Finally, the acoustic intensity is calculated by multiplying the sound pressure and particle velocity signals and taking their time average.

An acoustic intensity analysis system consists of a probe and an analyzer.
The probe simply measures the sound pressure at two points using two microphones.
The analyzer performs the necessary integrations and calculations to determine the acoustic intensity.

Learn more about Acoustic Power Measurement

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