40.1 Sound Intensity Measurement

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Every noise control project starts with the identification and ranking of the noise sources. Several

methods have been proposed for the purpose and have proved to be useful and widely utilized. In this

chapter, sound intensity measurement and directional measuring devices such as the mirror–

microphone system and microphone array are introduced and their principles and applications are

described. Other useful measurements, such as acoustic holography method [1,2] and spatial

transformation of sound fields [3], are described in the literature.

40.1.1 Theoretical Background

Sound intensity is a measure of the magnitude and direction of the flow of sound energy. The

instantaneous intensity vector, IðtÞ; is given by the product of the instantaneous sound pressure, pðtÞ; and

the corresponding particle velocity, uðtÞ; that is, IðtÞ ¼ pðtÞuðtÞ:

In practice, the time-averaged intensity, I􀀊; is more important, and is given by the equation:

I􀀊 ¼ lim

T!1

1

T

ðT=2

2T=2

pðtÞuðtÞdt ð40:1Þ

The intensity vector denotes the net rate of flow of energy per unit area (watts/m2). Thus, the acoustic

power, W ; of the source located in a closed surface, S; is given by the integral of the intensity passing

through the surface, S; as

W ¼

ð ð

s

I􀀊·dS ð40:2Þ

40-1

© 2005 by Taylor & Francis Group, LLC

Equation 40.2 indicates that the measurement of sound intensity over a surface enclosing a source enables

the estimation of its sound power, which shows the usefulness of the sound intensity concept.

40.1.2 Measurement Method

The principle of intensity measurement systems in

commercial production employs two closely

spaced pressure microphones [4,5], as shown in

Figure 40.1.

The particle velocity, ur ðtÞ; in a particular

direction, r; can be approximated by integrating

over time the difference of sound pressures at two

points separated by a distance Dr in that direction:

ur ðtÞ ¼ 2

1

r0

ðt

21

p2ðtÞ 2 p1ðtÞ

Dr

dt ð40:3Þ

where p1 and p2 are the sound pressure signals

from the two microphones. The sound pressure at

the center of two microphones is approximated by

pðtÞ ¼

p1ðtÞ þ p2ðtÞ

2 ð40:4Þ

Thus, the intensity in the direction r can be calculated as

Ir ðtÞ ¼ 2

1

2r0Dr ½p1ðtÞ þ p2ðtÞ􀀉

ðt

21 ½p2ðtÞ 2 p1ðtÞ􀀉dt ð40:5Þ

Some commercial intensity analyzers use Equation 40.5 to measure the intensity. Another type of

analyzer uses the equation in the frequency domain:

Ir ðvÞ ¼ 2

Im½G12􀀉

vr0Dr ð40:6Þ

where G12 is the cross spectrum between the two microphone signals. Equation 40.6 makes it possible to

calculate sound intensity with a dual-channel fast fourier transform (FFT) analyzer.

40.1.3 Errors in Measurement of Sound Intensity

The principal systematic error of the two-microphone method is due to the approximation of the

pressure gradient by a finite pressure difference. When the incident sound is a plane wave, the ratio of the

measured intensity, I^r , and the true intensity, Ir , is given by

I^r =Ir ¼

sinðkDr cos uÞ

kDr cos u ð40:7Þ

where the angle u is as defined in Figure 40.1 and k is the wave number. Equation 40.7 indicates that the

upper frequency limit is inversely proportional to the distance between the microphones.

Another serious error is caused by the phase mismatch between the two measurement channels. In the

calculation of intensity from Equation 40.5, the phase difference, w; between the two microphone signals,

p1 and p2; is very important. Hence, the phase mismatch between the two measurement channels, Dw;

must be much smaller than w: Since w increases with frequency, this error is serious in lower frequencies.

Other possible errors, such as in the sensitivity of microphones and random errors associated with a

given finite averaging time, are usually less serious.

θ

p1(t) p2(t)

Δr r

Microphones

Sound

FIGURE 40.1 Microphone arrangement used to

measure sound intensity.

40-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

40.1.4 Applications

One important application of sound intensity measurement is the determination of the sound power

level using Equation 40.2. Furthermore, measurement of the intensity in the very near field of a source

surface makes it possible to identify and rank the noise-sources. Plots of the sound intensity measured on

a surface near a sound source are useful for investigating noise source distributions. Figure 40.2 shows

sound intensity of noise from a wheel of a railway car. An intensity probe is located in the vicinity of the

wheel and the normal component of sound intensity is measured by traversing the probe on a plane

100 mm away from the side surface of the wheel. These figures show a free vibration behavior of the

wheel at each frequency; the wheel vibrates with one nodal diameter at 700 Hz and with three nodal

diameters at 1150 Hz. Visualization by intensity vectors also gives valuable information about a noise

source. Figure 40.3 shows the sound intensity vectors at each octave band measured in the vicinity of a

railway car running at 120 km/h. These results suggest that the main radiator of rolling noise is the rail at

the 500 Hz to 1 kHz band and the wheels at the 2 to 4 kHz band.

FIGURE 40.2 Measurements of the sound intensity radiated by a wheel of a railway car (1 dB contour).

FIGURE 40.3 Sound intensity vectors measured in the vicinity of a railway car running at 120 km/h.

Instrumentation 40-3

© 2005 by Taylor & Francis Group, LLC

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