Sound Intensity: Theory and Measurement

Acoustic measurements and acoustic theory have not always progressed side by side. The publication of Lord Rayleigh's influential work, "The Theory of Sound", laid the foundations of modern acoustics. The quantity sound intensity was fundamental to this theory. But a full hundred years were to elapse before the emergence of a thoroughly practical method of measuring sound intensity.
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SOUND INTENSITY
BY BRÜEL & KJÆR

What is Sound Intensity

Sound intensity describes the rate of energy flow through a unit area. Sound intensity also gives a measure of direction as there will be energy flow in some directions but not in others. Therefore sound intensity is a vector quantity as it has both magnitude and direction. On the other hand, the pressure is a scalar quantity as it has magnitude only. Any piece of machinery that vibrates radiates acoustical energy.

• Sound power is the rate at which energy is radiated [energy per unit time).
• Sound intensity describes the rate of energy flow through a unit area.

In the SI system of units, the unit area is 1 m2. And hence the units for sound intensity are Watts per square meter. Sound intensity also gives a measure of direction as there will be energy flow in some directions but not in others. Therefore sound intensity is a vector quantity as it has both magnitude and direction. On the other hand, the pressure is a scalar quantity as it has magnitude only.

Usually, we measure the intensity in a direction normal (at 90°) to a specified unit area through which the sound energy is flowing. We also need to state that sound intensity is the time-averaged rate of energy flow per unit area. In some cases, energy is transferred or traveling back and forth. This will not be measured; if there is no net energy flow there will be no net intensity.

In the diagram, the sound source is radiating energy. All this energy must pass through an area enclosing the source. Since intensity is the power per area, we can easily measure the normal spatial-averaged intensity over an area which encloses the source. We then multiply the intensity by the area to find the sound power.

Note that intensity (and pressure) follows the inverse square law for free field propagation. As shown in the diagram, with a distance of 2r from the source, the area around the source is 4 times larger, than the area at a distance r. Still, the power radiated must be the same whatever the distance, and consequently the intensity, the power per area, must decrease.

Why Measure Sound Intensity?

On the factory floor, we can make sound pressure measurements and find out if the workers risk hearing damage. But once we have found this, we may well want to reduce the noise.

To do this, we need to know how much noise is being radiated and by what machine. We, therefore, need to know the sound power of the individual machines and rank them in order of highest sound power. Once we have located the machine making the most noise we may want to reduce the noise by locating the individual components radiating noise.

We can do all this with sound intensity measurements.

Previously we could only measure pressure which is dependent on the sound field. Sound power can be related to sound pressure only under carefully controlled conditions where special assumptions are made about the sound field.

Specially constructed rooms such as anechoic or reverberant chambers fulfill these requirements. Traditionally, to measure sound power, the noise source had to be placed in these rooms. Sound intensity, however, can be measured in any sound field. No assumptions need to be made.

This property allows all the measurements to be done directly in situ. And measurements on individual machines or individual components can be made even when all the others are radiating noise. This is because steady background noise makes no contribution to the sound power determined when measuring intensity.

Because sound intensity gives a measure of direction as well as magnitude it is also very useful when locating sources of sound. Therefore the radiation patterns of complex vibrating machinery can be studied in situ.

How is Sound Intensity Measured?

The Euler Equation

Finding the Particle Velocity Sound intensity is the time-averaged product of the pressure and particle velocity. A single microphone can measure pressure — this is not a problem.

But measuring particle velocity is not as simple. The particle velocity, however, can be related to the pressure gradient (the rate at which the instantaneous pressure changes with distance) with the linearized Euler equation.

With this equation, it is possible to measure this pressure gradient with two closely spaced microphones and relate it to particle velocity. Euler's equation is essentially Newton's second law applied to a fluid. Newton's Second Law relates the acceleration given to add mass to the force acting on it.

If we know the force and the mass we can find the acceleration and then integrate it with respect to time to find the velocity. With Euler's equation, it is the pressure gradient that accelerates a fluid of density p.

With knowledge of the pressure gradient and the density of the fluid, the particle acceleration can be calculated. Integrating the acceleration signal then gives the particle velocity.

Approximation of the Finite Difference

The pressure gradient is a continuous function, that is, a smoothly changing curve. With two closely spaced microphones, it is possible to obtain a straight-line approximation to the pressure gradient.

This is done by taking the difference in pressure and dividing by the distance between them. This is called a finite difference approximation. It can be thought of as an attempt to draw the tangent of a circle by drawing a straight line between two points on the circumference.

Calculation of sound intensity

The pressure gradient signal must now be integrated to give the particle velocity. The estimate of particle velocity is made at a position in the acoustic center of the probe, between the two microphones. The pressure is also approximated at this point by taking the average pressure of the two microphones. The pressure and particle velocity signals are then multiplied together and time averaging gives the intensity.

A sound intensity analyzing system consists of a probe and an analyzer. The probe simply measures the pressure at the two microphones. The analyzer does the integration and calculations necessary to find the sound intensity.

The equations are not new. What is new is the use of modern signal processing techniques to implement the equation and produce instant measurement results.

This can be done in two ways:

1. Directly using integrators and filters (analog or digital) to implement the equation step by step.
2. Using an FFT analyzer.

The latter relates the intensity to the imaginary part of the cross-spectrum (a mathematical term) of two microphone signals. The formulations are equivalent - and both reveal the sound intensity...