Instrumentation for Noise Measurement and Analysis
3.13 INTENSITY METERS
3.13.2 Sound Intensity by the p–p Method
In the p–p method, the determinations of acoustic pressure and acoustic particle velocity are both made using a pair of high quality condenser microphones. The microphones are generally mounted side by side or facing one another and separated by a fixed distance (6 mm to 50 mm) depending upon the frequency range to be investigated. A signal proportional to the particle velocity at a point midway between the two microphones and along the line joining their acoustic centres is obtained using the finite difference in measured pressures to approximate the pressure gradient while the mean is taken as the pressure at the midpoint.
The useful frequency range of p–p intensity meters is largely determined by the selected spacing between the microphones used for the measurement of pressure gradient, from which the particle velocity may be determined by integration. The spacing must be sufficiently small to be much less than a wavelength at the upper frequency bound so that the measured pressure difference approximates the pressure gradient. On the other hand, the spacing must be sufficiently large for the phase difference in the measured pressures to be determined at the lower frequency bound with sufficient precision to determine the pressure gradient with sufficient accuracy.
Clearly, the microphone spacing must be a compromise and a range of spacings is usually offered to the user.
The assumed positive sense of the determined intensity is in the direction of the centre line from microphone 1 to microphone 2. For convenience where appropriate in the following discussion the positive direction of intensity will be indicated by unit vector n and this is in the direction from microphone 1 to microphone 2.
Taking the gradient of Equation (1.10) in the direction of unit vector n , and using Equation (1.11) gives the equation of motion relating the pressure gradient to the particle acceleration. That is:
where un is the component in direction n of particle velocity u , p and u are both functions of the vector location r and time t, and ρ is the density of the acoustic medium. The normal component of particle velocity, un, is obtained by integration of Equation (3.25) where the assumption is implicit that the particle velocity is zero at time t = – 4:
The integrand of Equation (3.26) is approximated using the finite difference between the pressure signals p1 and p2 from microphones 1 and 2 respectively and ∆ is the separation distance between them:
un(t) ' & n ρ∆m
t
&4
p2(τ)&p1(τ) dτ (3.27)
p(t) ' 1
2 p1(t)%p2(t) (3.28)
Ii(t) ' & n
2ρ∆ p1(t)%p2(t) m
t
&4
p1(τ)&p2(τ) dτ (3.29)
Ii(t) ' n
ρ∆p2(t)m
t
&4
p1(τ) dτ (3.30)
I ' n ρ∆
lim T64 1
Tm
T
0
p2(t)m
t
&4
p1(τ) dτ dt (3.31)
pi(t) ' p0ie j(ωt%θi) i ' 1, 2 (3.32)
Ii(t)' n
4ρ∆ ω p012sin(2ωt%2θ1)&p022sin(2ωt%2θ2)%2p01p02sin(θ1&θ2) (3.33)
I ' n p01p02
2ρ ω ∆ sin(θ1&θ2) (3.34) The pressure midway between the two microphones is approximated as the mean:
Thus the instantaneous intensity in direction n at time t is approximated as:
For stationary sound fields the instantaneous intensity can be obtained from the product of the signal from one microphone and the integrated signal from a second microphone in close proximity to the first (Fahy, 1995):
The time average of Equation (3.30) gives the following expression for the time average intensity in direction n (where n is the unit vector).
Commercial instruments with digital filtering (one third octave or octave) are available to implement Equation (3.31). As an example, consider two harmonic pressure signals from two closely spaced microphones:
Substitution of the real components of these quantities in Equation (3.30) gives for the instantaneous intensity the following result:
Taking the time average of Equation (3.33) gives the following expression for the (active) intensity:
In ' n P1P2
2ρ ω ∆(θ1&θ2) , θ1&θ2« 1 (3.35)
Ir ' 1
4ω ρ ∆ p012 &p022 (3.36) If the argument of the sine is a small quantity then Equation (3.34) becomes approximately:
This equation also follows directly from Equation (1.69), where the finite difference approximation is used to replace Mθp/Mr with (θ1&θ2) /∆ and p02 is approximated by p01p02. The first two terms of the right-hand side of Equation (3.33) describe the reactive part of the intensity. If the phase angles θ1 and θ2 are not greatly different, for example, the sound pressures p01 and p02 are measured at points that are closely spaced compared to a wavelength, the magnitude of the reactive component of the intensity is approximately:
Equation (3.36) also follows directly from the second (reactive) term of Equation (1.68) where p0 is replaced by (p01 + p02)/2 and Mp/Mr is replaced with the finite difference approximation (p01 - p02)/∆. Measurement of the intensity in a harmonic stationary sound field can be made with only one microphone, a phase meter and a stable reference signal if the microphone can be located sequentially at two suitably spaced points. Indeed, the 3-D sound intensity vector field can be measured by automatically traversing, stepwise, a single microphone over an area of interest. Use of a single microphone for intensity measurements eliminates problems associated with microphone, amplifier and integrator phase mismatch as well as enormously reducing diffraction problems encountered during the measurements.
In general the determination of the total instantaneous intensity vector, Ii(t) requires the simultaneous determination of three orthogonal components of particle velocity. Current instrumentation is available to do this using a single p-u probe or single p-p probe. Note that with a p-p probe, the finite difference approximation used to obtain the acoustic particle velocity has problems at both low and high frequencies which means that at least three different microphone spacings are needed to cover the audio frequency range. At low frequencies, the instantaneous pressure signals at the two microphones are very close in amplitude and a point is reached where the precision in the microphone phase matching is insufficient to accurately resolve the difference. At high frequencies the assumption that the pressure varies linearly between the two microphones is no longer valid. The p-u probe suffers from neither of these problems as it does not use an approximation to the sound pressure gradient to determine the acoustic particle velocity.
3.13.2.1 Accuracy of the pBp Method
The accuracy of the p-p method is affected by both systematic and random errors. The systematic error stems from the amplitude sensitivity difference and phase mismatch
eβ(I) ' βs/βf (3.37)
δpIO&δpI ' 10log10*1%(1/eβ(I))* ' 10log10*1%(βf/βs)* (3.38a,b)
βs ' kd10&δpIO/ 10 (3.39)
between the microphones and is a result of the approximations inherent in the finite difference estimation of particle velocity from pressure measurements at two closely spaced microphones.
The error due to phase mismatch can be expressed in terms of the difference, δpI, between sound pressure and intensity levels measured in the sound field being evaluated and the difference, δpIO, between sound pressure and intensity levels measured by the instrumentation in a specially controlled uniform pressure field in which the phase at each of the microphone locations is the same and for which the intensity is zero. This latter quantity is a measure of the accuracy of the phase matching between the two microphones making up the sound intensity probe and the higher its value, the higher is the quality of the instrumentation (that is, the better is the microphone phase matching). The quantity, δpI, is known as the “Pressure-Intensity Index” for a particular sound field and δpIO, is known as the “Residual Pressure- Intensity Index” which is a property of the instrumentation and should be as large as possible. If noise is coming from sources other than the one being measured or there are reflecting surfaces in the vicinity of the noise source, the Pressure Intensity Index will increase and so will the error in the intensity measurement. Other sources or reflected sound do not affect the accuracy of intensity measurements taken with a p-u probe, but the p-u probe is more sensitive than the p-p probe in reactive sound fields (such as the near field of a source - Jacobsen, 2008).
The normalised systematic error in intensity due to microphone phase mismatch is a function of the actual phase difference, βf between the two microphone locations in the sound field and the phase mismatch error, βs. The normalised error is given by Fahy (1995) as:
The Pressure-Intensity Index may be written in terms of this error as:
A normalised error of Equation (3.37) equal to 0.25 corresponds to a sound intensity error of approximately 1 dB (= 10log10(1 +0.25)), and a difference, δpIO - δpI = 7 dB.
A normalised error equal to 0.12 corresponds to a sound intensity error of approximately 0.5 dB and a difference, δpIO - δpI = 10 dB.
The pressure intensity index will be large (leading to relatively large errors in intensity estimates) in near fields and reverberant fields and this can extend over the entire audio frequency range.
The phase mismatch between the microphones in the p-p probe is related to the Residual Pressure-Intensity Index (which is often supplied by the p-p probe suppliers) by:
Phase mismatch also distorts the directional sensitivity of the p-p probe so that the null in response of the probe (often used to locate noise sources) is changed from the 90E direction to that given by (Fahy, 1995):
βm ' cos&1(βs/kd) (3.40)
er(I) ' 10log101%(BT)&1/ 2 (3.41) where k is the wavenumber at the frequency of interest and d is the spacing between the two microphones in the p-p probe.
In state of the art instrumentation, microphones are available that have a phase mismatch of less than 0.05E. In cases where the phase mismatch is larger than this, the instrumentation sometimes employs phase mismatch compensation in the signal processing path.
The error due to amplitude mismatch is zero for perfectly phase matched microphones. However, for imperfectly phase matched microphones, the error is quite complicated to quantify and depends on the characteristics of the sound field being measured.
Fahy (1995) shows that random errors in intensity measurements add to the uncertainty due to systematic errors and in most sound fields where the signals received by the two microphones are random and have a coherence close to unity, the normalised random error is given by (BT) -1/ 2, corresponding to an intensity error of:
where B is the bandwidth of the measurement in Hz and T is the effective averaging time, which may be less than the measurement time unless real-time processing is performed. The coherence of the two microphone signals will be less than unity in high frequency diffuse fields or where the microphone signals are contaminated by electrical noise, unsteady flow or cable vibration. In this case the random error will be greater than indicated by Equation (3.41).
Finally, there are errors due to instrument calibration that add to the random errors. This is approximately ± 0.2 dB for one particular manufacturer but the reader is advised to consult calibration charts that are supplied with the instrumentation.