Emerging Technologies in Non-Destructive Testing – Busse et al. (eds)
© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46476-5
Use of state of the art parametric arrays for low frequency measurements in sound absorbing porous materials
Bernard Castagnede, Alexei Moussatov, Denis Lafarge & Vincent Tournat Laboratoire d’Acoustique de l’Université du Maine, Le Mans, France
Vitali Gusev
Laboratoire de Physique de l’Etat Condensé, Université du Maine, Le Mans, France
ABSTRACT: The so-called ‘‘parametric arrays’’ have been used in air for audio engineering applications since 2000, with the emergence of commercially available devices such as ‘‘audio spot’’ sources from several US and EC companies. The principle is to use powerful ultrasonic waves, with 40 kHz central frequency, which are electronically amplitude modulated. During propagation, a non linear demodulation (or rectification) process occurs producing a low frequency ultra-directive acoustical beam. Some metrology applications for the characterization of sound absorbing porous materials were considered quite early, and preliminary work was performed with various laboratory benches and devices having a central frequency varying from 40 kHz to 200 kHz. Reflection and transmission coefficients are available versus frequency, as well as dispersion curves. Experimental data are compared to numerical predictions in the frame of the standard poroelastic models for different porous materials. Some dedicated instrumentation has also been proposed for industrial applications, e.g. for ‘‘on-line’’ and ‘‘in-situ’’ measurements of the low frequency coefficient of absorption over the 100 Hz–6 kHz bandwidth.
1 INTRODUCTION
The measurement of the acoustical properties of porous air-saturated materials is a very basic and important engineering problem in the field of applied acoustics. Because the wavelength of the acoustical waves in air are very large (e.g. 3.4 m at 100 Hz), free field measurements are difficult to achieve. For a very long time, going back to the seminal work of Kundt, the acoustic absorption coefficient measurements of porous absorbing materials are generally performed in impedance or resonance tubes, because in such con- figuration the low frequency modes inside the tube propagate as plane waves. This procedure, which is heavily used in the automotive industry, works well but it is time consuming because one needs to cut the tested samples. Additionally, when dealing with inhomoge- neous or compressed materials and panels, it is tricky to describe the spatial variations of the acoustical prop- erties, as measurements are simply done on the tested samples. Accordingly, several researchers have tried to implement in-situ free-field methods. A basic and simple method, designed 20 years ago, is the so-called microphone doublet which works with two single microphones [e.g. 1–3]. In such configuration a loud- speaker is mounted 2 m above the microphone doublet
in order to approximately produce plane waves. Con- sequently, this set-up needs to be mounted in an anechoic chamber, because loudspeakers radiate low frequency (LF) waves in all directions. Various other approaches were also proposed during the 1990’s, as extensions of the two microphones method, with scan- ning devices and with proper inversion procedures [4, 5]. More recently, absorption measurements have been performed with microflown sensors [6].
A very different approach is extensively presented in the present work [7]. It is based on an amplitude auto-demodulation (or rectification) process, taking place in air, the so-called parametric array technique.
One starts by using a powerful ultrasonic source (the
‘‘pump’’ transducer working at some high frequency ω ), which is electronically amplitude modulated, at a low frequency (with ω ). The demodulation process deals with a nonlinear behavior of air which produces an energy transfer towards low frequency . The parametric rectification effect was discovered 50 years ago in the field of underwater acoustics [8, 9].
It is only recently, due to advances in the parametric
array technology that large scale audio range applica-
tions were envisioned [10, 11]. Metrology applications
in the field of characterization of poroelastic mate-
rials were also proposed for routine measurements
of acoustical properties of air saturated porous and poroelastic materials. Any efficient ultrasonic trans- ducer can be used in the parametric regime. Some preliminary results, obtained with ordinary piezo- electric transducers, were described in 2004 for the characterization of porous materials in the reflec- tion configuration as well as in transmission [12].
The advent of powerful LF systems allows to work at lower modulation frequencies, and various mea- surements and numerical simulations were described with standard commercially available devices manu- factured in the USA (e.g. from Ultrasound Technology, San Diego) which uses thin piezoelectric films [13].
Unfortunately, such devices when driven at very low frequencies (below 200 Hz) were not sufficiently effi- cient, as seen in the results of that publication. The measurement of acoustical absorption for industrial applications necessitates to cover the 100 Hz–6 kHz bandwidth, which was not available with PVDF thin film technology. In order to attain the very low frequencies, alternative designs, for instance in the format of numerous PZT individual transducers are needed. In the present work, an original apparatus manufactured by the European company Sennheiser has been used, because it is very powerful at low frequencies down to 100 Hz, or even better in some cases. The present work reviews some of the unique features of such device for the operation and diagno- sis of porous materials at very low frequencies [14], with a special emphasis on the fine metrology of the acoustical absorption coefficient (cf. Figure 1a for a basic sketch) and dispersion curves (i.e. the frequency dependence of phase or group sound wavespeeds), as seen on Figure 1b.
Sound projector Audio microphone
Porous material
Rigid reflector
a)
Sound projector
b)
Audio microphone Porous
material k k
Figure 1. Basic configurations of the method a) Reflec- tion (or absorption configuration); b) Transmission (or dispersion) configuration.
2 EXPERIMENTAL PROCEDURES 2.1 Description of the experimental set-up In this work we have used some commercial para- metric arrays, specifically the HSS ‘‘directed audio sound system’’ product from the American Technol- ogy Corporation (San Diego, CA), and an Euro- pean prototype from the Sennheiser company. The first device works with PVDF technology at 47 kHz for the ultrasonic pump wave. The second device is designed with an array of PZT ceramics tuned at 40 kHz. The size of each sound projector is around 30 or 40 cm. The basic set-up uses the parametric array in front of the tested porous mate- rial, while a detection system in the form of an audio microphone is mounted either on the same side of the material when reflection or absorp- tion coefficient are sought, or on the other side of the porous plate when transmission coefficient or dispersion curves are searched (see details on Figure 1).
Figure 2 provides a schematic view of the exper- imental set-up for absorption coefficient measure- ments. The audio projector is mounted vertically approximately 1 m above the tested porous plate, in order to produce powerful plane wave orthogonal to the surface of the material. An audio microphone is mounted a few cm above the sample. During a pre- liminary calibration procedure, which should be done from time to time, the sample holder (in the shape of a ‘‘wheel-table’’) is firstly removed in order to cap- ture the incident wave field. Next, the table is mounted back in its prior location with a rigid reflector mounted on it. In that second step one records simultaneously the incident and the reflected field, both having the same amplitude. After the calibration two steps pro- cedure is done, one can perform real absorption measurements just by mounting the porous plate above the table. The detected signals are recorded onto a portable computer connected to a low-cost audio card
1 2
4 3
5
6
Figure 2. Schematic of the used set-up. 1: Audio sound projector; 2 Mounting frame; 3: Audio microphone;
4: Poroelastic plate; 5: Removable wheel-table; 6: Connec-
tion towards audio acquisition card and portable computer.
working at a 92 kHz sampling frequency. This is amply sufficient to cover the audio range, let say between a few 100 Hz and 10 kHz.
2.2 Calibration and measurements procedures Before performing absorption curves measurements, a preliminary two steps calibration procedure is nec- essary. Firstly, the wheel-table is removed, in order to make acquisition of the incident wave only. In such case, the wave reflected back onto the ground is evi- dently present as well, but it is approximately delayed by a 6 ms time duration (for a 1 m high table top).
Consequently, the incident and reflected waves are temporally separated as long as the frequency is not too low. As a rule of thumb, for an ultra-short pulse, having one single period, the low frequency limit is around 200 Hz in order to discriminate between the two waves. In that case, the sample size should be large enough, let say in the range of 1 square meter minimum. If a lower frequency range is sought, then the dimensions of the all set-up (high of the table, size of the square sample) should be increased accordingly.
The second step of the calibration procedure is to bring back the wheel-table and to put on its top a flat and per- fectly reflecting panel. In such configuration, because the microphone is mounted just a few cm above that reference plane, and due to the large wavelength (e.g.
34 cm at 1 kHz), then the incident and reflected waves do superimpose. In fact, the numerical treatment will further use the incident wave measured during the first step of the calibration. This incident wave is simply subtracted, in order to get the reflected wave, which is equal in amplitude to the incident one when using the perfect reflector. At that point the calibration proce- dure is completed, and real-time measurements could be processed. This step is done by mounting the tested absorbing panel. When doing the measurements, one obtain this time the incident wave which is unchanged mixed up with the new reflected wave which should be smaller than when compared with the case of the perfect reflector. The numerical procedure is the same as in step #2 of the calibration procedure, with the subtraction of the incident wave in order to deduce the new reflected wave. Then, the reflection coeffi- cient is easily obtained by computing the ratio between the two reflected waves, with and without the insulat- ing panel. As a matter of fact, this ratio is smaller than one, because part of the incident wave has been absorbed inside the porous material. This is exactly the same argument which is done within a resonance (or Kundt) tube, based on the energy conservation law.
In turn, this allows to determine the absorption coef- ficient A(ω) from the coefficient of reflection R(ω), just by writing the energy conservation balance law in the form: A (ω) = 1 − | R (ω)|
2, where | R (ω)|
2denotes
the modulus of the coefficient of reflection in energy versus frequency.
3 DESIGN AND IMPLEMENTATION OF A PORTABLE INSTRUMENT
For ‘‘on-line’’ and ‘‘in-situ’’ measurements a portable version of the instrumentation which has been described in the previous section is needed. Because of the simplicity of the basic configuration, which uses only one parametric array and one single microphone, the design of such portable system is easy to imple- ment. It uses a standard audio card allowing to provide to the parametric array proper modulation signals.
The same audio card enables to perform acquisition of the microphone signal at a 92 kHz sampling fre- quency, which is amply sufficient with audio signals extending from 100 Hz towards 6 kHz. In fact, the audio signals are systematically oversampled. This last feature is fundamental in order to achieve precise time- of-flight measurements of travelling acoustic bursts, a characteristic which is really useful to obtain dis- persion curves, i.e. the phase or group wavespeeds inside the porous material as a function of frequency.
A portable computer system working with LabVIEW (version 7.0 or 8.0) permits to drive the audio card and to synchronize its input and output channels. Sev- eral low cost such audio card have been used. When using LabVIEW version 7.0, one needs to develop spe- cific routines written in C language which bridge the main LabVIEW program to the audio card. Another approach working with LabVIEW 8.0 is to drive such cards directly through an adequate driver, or to use some external LabVIEW cards. Various approaches are working and provide similar results. In any cases, special care should be devoted to the implementation of the equipment. Only microphones, which are insen- sitive to the very powerful ultrasonic fields, should be used. For instance, condenser or capacitive micro- phones are not appropriate, because the microphone membrane is strongly affected by the radiation pres- sure of the ultrasonic wave. In some cases, Larsen effects might exist, and the precise positionning of the microphone becomes tricky as it is the ultrasonic wavelength (in the order of 10 mm or less) which is responsible for constructive (as well as destructive) acoustical interferences over the propagation path.
When the instrumentation is correctly mounted and
when the settings of the electronics is adequate, the
calibration procedure could be done, and the charac-
terization of porous panels could proceed. Figure 3
shows examples of characteristic signals obtained with
such system. The first burst is the incident wave, while
the second wavepacket is its reflection at the surface
of the porous layer.
0 . 1 0
- 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5
10
0 1 2 3 4 5 6 7 8 9
ms V
400 Hz
0 . 4 0
- 0 . 4 0 - 0 . 2 0 0 . 0 0 0 . 2 0
10
0 1 2 3 4 5 6 7 8 9
ms V
1200 Hz 0.3 0
-0 .30 -0 .20 -0 .10 -0 .00 0.1 0 0.2 0
10
0 1 2 3 4 5 6 7 8 9
ms V
800 Hz
Figure 3. Example of temporal signals propagated with the Sennheiser parametric array.
4 MEASUREMENTS OF THE DISPERSION CURVES
The phase wavespeed c(ω) versus frequency (i.e. the dispersion curve) propagating inside a porous material provides some information onto the acoustic char- acteristics of the medium. Such quantity is related to the effective density ρ(ω) and the effective com- pressibility K(ω) through the generic and universal relationship:
c(ω) = K (ω)
ρ(ω) (1)
These effective functions ρ(ω) and K (ω ) are related to various parameters of the fluid saturating the porous structure (e.g. pressure and density at rest of the fluid, ratio of specific heat, viscosity, Prandtl number), as well as to some physical parameters pertaining to the porous microstructure (i.e. porosity φ, resistivity σ, tortuosity α
∞, viscous and thermal characteristic lengths and
). A complete review of the so- called ‘‘equivalent fluid’’ model which describes the acoustical propagation in porous materials when the solid frame is motionless can be found in the refer- ence textbook from J.F. Allard [15] or in summarized version in [16]. The 5 physical parameters related to the microstructure of the porous materials should in turn strongly affect the LF dispersion curves. It is
0 50 100 150 200 250
0 500
wavespeed (m/s)
Frequency (Hz) 5 mm thickness 10 mm thickness 15 mm thickness
a 20 mm thickness
1000 1500 2000 2500 3000 3500 4000
Figure 4. Dispersion curves in various felt materials hav- ing different thicknesses but with the same basic (or surface weight).
Table 1. Physical parameters of the various porous plates, tested on Figure 4.
h (mm)
φ σ(N m
−4s)
α∞(µm)
(µm)
20 0.92 22 000 1.04 100 200
15 0.90 32 000 1.06 75 150
10 0.88 56 000 1.18 50 100
5 0.80 140 000 1.30 25 50
for instance very easy to demonstrate that the phase wavespeed tends to zero when approaching the null frequency (i.e. 0 Hz). On the other hand, the HF asymptotic limit provides the tortuosity, as given by the following relationship:
α
∞= lim
ω→∞
c
0c (ω)
2(2) where c
0is the speed of sound in free air (i.e. 342 m/s at 20
◦C). The shape of the dispersion curve will defi- nitely depends on the various parameters. An example is provided on Figure 4, for various felt materials which are commonly used in the automotive indus- try to damp out the interior noise. The measurements were done by using the phase unwrapping method [17]
originally developed by Sachse and Pao 30 years ago for composite materials.
In such example, the values of the physical param- eters are strongly modified by the compression ratio going from 1 (plate having 20 mm thickness) to 4 (5 mm thickness), as outlined in Table 1. For instance, the characteristic lengths and
are decreased linearly with the compression ratio.
It should be mentioned that the parameters collected
onto Table 1 have also been used to predict the disper-
sion curves. These results are going to be published in
Applied Acoustics [14], showing an excellent agree- ment. The knowledge of the dispersion curves allow to gain access to the effective density and compressibility functions ρ(ω) and K (ω) . These two basic functions are also providing the surface impedance of the porous materials as given by:
Z(ω) =
K(ω)ρ(ω) (3)
Such expression somehow links absorption to disper- sion, because the coefficient of absorption A (ω) is related to the coefficient of reflection R (ω) through the following equation:
A(ω) = 1 − |R(ω)|
2(4)
while the coefficient of reflection R (ω) is linked to the surface impedance Z (ω) by the usual equation:
R (ω) = z(ω) cos k(ω)h − jφ sin k(ω)h
z(ω) cos k(ω)h + jφ sin k(ω)h
(5) where z (ω) represents the normalized acoustic impedance of the porous material to those of air, and where k (ω) is the wavenumber. Consequently, one could understand that there exist some indirect links between absorption coefficient and dispersion curve.
For instance, the low frequency limit of both quantity is zero. This formal link, known as the Kramers-Kronig relationships has been described for porous media by Fellah et al [18], and dispersion curves in the ultra- sonic domain were measured independently by various authors (e.g. see [19, 20]). The extension of such work to the LF audio frequency range was extremely difficult to achieve before the advent of efficient para- metric arrays, as was described in the recent literature [12–14].
5 MEASUREMENTS OF
THE ABSORPTION COEFFICIENT
The measurements of the absorption coefficient can be done with the experimental procedures described in sections 2 and 3. As explained in the introduction such free-field measurements are clearly difficult to achieve with good accuracy at low frequencies, due to the wavelength of the acoustical waves which becomes very large (e.g. 3.40 m at 100 Hz). This causes diffrac- tion effects on the edges of the tested sample or on the supporting table or sample holder. Additionally, one should also consider the LF cut-off frequency of the room which is somehow related to its dimensions. As a matter of fact, in order to obtain absorption mea- surements down to 100 Hz, the space of the testing room should be of sufficient size (let say in the order
of 50 m
3). In this section, we provide some signifi- cant results obtained on glass-wool porous materials.
Again, the various samples have different thicknesses, but they keep constant their basic (or surface weight).
Figure 5a shows some experimental results, while Figure 5b deals with numerical simulations taken from the ‘‘equivalent fluid’’ model for the very same mate- rials [15, 16]. On Figure 5a, one can observe some oscillations in the measurements. They are mainly due to diffraction effects on the edges of the sample. Data are obtained down to very low frequencies although it is well known that the Sennheiser device cannot gener- ate waves below 100 Hz. The comparison of these data with numerical simulations collected onto Figure 5b is correct, but it is never excellent. The physical parame- ters which have been used during such comparisons are collected onto Table 2. As it was already visi- ble on Table 1, during compression, some physical parameters increase (resistivity, tortuosity) while oth- ers decrease (porosity, characteristic lengths). As long
0 0.2 0.4 0.6 0.8 1
Absorption
a
0 1000 2000 3000 4000 5000 6000
25 mm
20 mm 15 mm 10 mm 5 mm
Frequency (Hz)
Figure 5a. Experimental data for the coefficient of absorp- tion for various compressed glass-wools.
0 0.2 0.4 0.6 0.8 1
25 mm 20 mm 15 mm 10 mm 5 mm b
Frequency (Hz)
Absorption
0 1000 2000 3000 4000 5000 6000
Figure 5b. Numerical simulations for the coefficient of
absorption for various compressed glass-wool.
Table 2. Physical parameters of the various porous plates, on Figure 5a and with numerical computations (Figure 5b).
h (mm)
φ σ(N m
−4s)
α∞(
µm)
(
µm)
25 0.98 40 000 1.05 100 200
20 0.975 50 000 1.06 80 160
15 0.97 67 000 1.08 60 120
10 0.95 100 000 1.12 40 80
5 0.90 200 000 1.25 20 40
0 0.2 0.4 0.6 0.8 1
0 1000 2000 3000 4000 5000 6000
Absorption
Frequency (Hz)
Figure 6. Fine measurements of the coefficient of absorp- tion performed on a 15 mm felt material manufactured by Rieter. The measurements were done with a portable sys- tem and by using the Sennheiser audio projector. The crosses are experiment data while the triangles are numerical predic- tions. Big squares are measurements obtained with a Brüel &
Kjaer standard impedance tube. The physical parameters for the felt material, as estimated through the very best fit, are the following:
φ=0.90;
σ=35 000 N m
−4s;
α∞=1.06;
=
