Highly directive acoustic beams applied to the characterization of sound absorbing materials
B. Castagnède 1 , M.Saeid 1, A. Moussatov 1 , V. Tournat 1 , V. Gusev 1,2
1 Laboratoire d'Acoustique de l'Université du Maine, LAUM, UMR 6613, Le Mans, France 2 Laboratoire de Physique de l'Etat Condensé, Université du Maine, UMR 6087, Le Mans, France
This presentation reviews some recent work dealing with the application of parametric antennae for the characterization of air-saturated porous materials. Parametric arrays are based on amplitude modulated powerful pump waves which propagate in the nonlinear medium. When the high-frequency (HF) component is damped out by attenuation in the medium one observes the remaining demodulated (or rectified) low frequency (LF) component.
It is only recently that these ideas were effectively applied in the area of audio acoustics. The application of highly directive acoustic beams produced by parametric effect in air for the characterization of poroelastic noise damping materials such as cellular plastic foams and fibrous mats is very promising.
Some preliminary work was performed by our group both in the transmission as well as in the retrodiffusion mode with a 170 kHz standard ultrasonic transducer working in the parametric regime.
Dispersion curves and reflection coefficients were measured for compressed thermal felts over the 5 to 30 kHz bandwidth. Some work has also been done with dedicated 40 kHz parametric arrays in the audio range covering 200 Hz – 3 kHz to measure dispersion and absorption curves.
1. Introduction
During the last few years many efforts have been devoted to the applications of audio techniques using highly directive acoustic beams.
These works are based on the use of the so-called parametric arrays, which was initially discovered and applied in the field of non linear underwater acoustics [1]. One start with a high power ultrasonic beam, which is amplitude modulated by an audio signal. During the propagation of the resulting mixed wave in some non linear medium, one observes the attenuation of the ultrasonic pump wave as well as the demodulation (or rectification) of the low component audio signal.
This demodulation effect is a cumulative process taking place during the propagation of the wave.
As a result, one obtains an acoustic audio beam having a highly directivity similar to the one of the ultrasound for the given configuration (i.e.
frequency and aperture of the parametric antenna).
Any ultrasonic transducer can work in the parametric regime, even if the efficiency, which is proportional to the square of the ratio between the modulation frequency and the ultrasonic frequency, is often quite poor at low frequency. Applications in the field of audio engineering were envisioned quite early [2], but efficient systems were available only recently, during the last five years. These sources of highly directive acoustic beams are presently used for numerous audio applications such as spatially localised messages, remote communications, industrial warnings, multiple conferences, etc.
Additionally, these sources of highly directive audio beams present a great potential for applications in the field of the metrology of sound absorbing materials, as it was clearly demonstrated during the last two years in our research group [3, 4]. The present report describes some of the achieved work in the area of the characterisation of air-saturated porous media.
2. Experimental procedures
In this work we have used some commercial parametric arrays, specifically the HSS “ directed audio sound system ” product from the American Technology Corporation (San Diego, CA), and an European prototype from the Sennheiser company.
The first item works with PVDF technology at 47
kHz for the ultrasonic pump wave. The second
item 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
material, while a detection system in the form of
an audio microphone is mounted either on the
same side of the material when reflection or
absorption 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). A computerized acquisition
and processing system is implemented in order to
obtain and process the various acoustic data of
interest. The very definitive advantage of the use of parametric arrays lies on the fact it allows to work in the free field, without the need of an anechoic chamber. In order to achieve quick measurements, one uses short bursts, most often with one single modulation period.
Figure 1: Basic set-up for free-field characterisation of poroelastic materials. a) reflection configuration
; b) transmission configuration.
One of the main features of the treatment is related to the short duration of the used signals, allowing to obtain the acoustical measurements over the usable bandwidth. Some examples of the temporal waveforms used are provided in the next sections, as well as the amplitude spectra.
3. Results in transmission
The measurements performed in transmission allow to obtain either the coefficient of transmission T( ω ), or the dispersion curve (i.e. the phase wavespeed c( ω ) versus frequency). The American HSS sound projector has been used is this work, with a modulation frequency of 1.4 kHz, covering the bandwidth (at – 6 dB) between 500 Hz and 2.5 kHz as shown on Figure 2b. The tested material is a 5 mm thick thermal felt used in the automotive industry. It is a compressed material having the following physical parameters : porosity, φ = 0.77 ; resistivity, σ = 320000 Nsm -4 ; tortuosity, α∞ = 1.3 ; viscous characteristic length, Λ = 14 µ m ; thermal characteristic length, Λ ’ = 42 µ m. The experimental results are
presented onto Figure 2c. It clearly shows that the wavespeed is very small at low frequency, e.g.
around 15 m/s at 200 Hz. This trend is totally predicted by some numerical simulations [6] based on the standard « equivalent fluid » model originally designed 15 years ago by J.F. Allard [7].
Figure 2: Transmission measurements inside a porous felt material, performed with the HSS sound projector. a) temporal waveforms, incident signal in black, reflected signal in red ; b) amplitude spectra, in black for incident, in red for reflection ; c) phase wavespeed over the 200 Hz – 2.5 kHz bandwidth.
4. Results in reflection
Extensive results in the reflection configuration were presented with parametric arrays working at higher frequencies, i.e. with a probe working at 175 kHz, covering a demodulated wave over the 4 – 30 kHz bandwidth [6]. Here we have used a Sennheiser sound projector, which allows to work at very low frequencies. In order to demonstrate the performances of that probe, Figure 3 shows a pair of temporal waveforms recorded at different modulation frequencies. In fact, the Sennheiser Sound projector
Audio microphone
Porous material
Rigid reflector
a)
Sound projector
b)
Audio microphone Porous
material
k k
2 0 0 0
1 5 0 0 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0 1 5 0 0
2.0E-4 0.0E+0 5.0E-5 1.0E-4 1.5E-4
signals
3.0E-2
0.0E+0 5.0E-3 1.0E-2 1.5E-2 2.0E-2 2.5E-2
2 . 5
0 . 5 1 . 0 1 . 5 2 . 0
amplitude spectrum
1 20
0 2 0 4 0 6 0 8 0 1 00
2 5
0 2 0 5 1 0 1 5 2 0
phase wavespeeds (m/ s)
a)
b)
c)
s
kHz kHz
incident spectrum
incident wave reflected wave
reflected spectrum
sound projector is correctly working down to 200 Hz, even if the efficiency becomes very small (it is 25 times smaller at 200 Hz when compared to 1kHz).
The reflection configuration allows to directly measure the coefficient of absorption when the porous material is mounted over a rigid reflector. As a matter of fact, the energy which is not damped out inside the porous layer is then reflected back towards the audio microphone.
Accordingly, the energy conservation law allows very easily to derive the following equation, where α(ω) represents the coefficient of absorption versus frequency :
α ( ω ) = 1 – R( ω ) 2 . (1)
Figure 3 : Some recorded temporal waveforms in the reflection configuration. The first signal is the incident wave, while the second one having a smaller amplitude is the reflected wave. a) modulation frequency = 1 kHz ; b) modulation frequency = 1.5 kHz.
In fact, because the coefficient of absorption is always very small at low frequency, the reflection coefficient should be close to one. Moreover, it should decrease when enlarging the applied frequency. This is exactly what is seen on Figure 3, where the amplitude of the reflected wave diminishes versus frequency. A better description is given when these signals are processed in the computer. Such image is provided on Figure 4 showing the temporal waveforms as well as the amplitude spectra which are then used to obtain the coefficient of reflection by a simple division versus frequency over the attainable spectral bandwidth. In the present example, the modulation frequency is 1.5 kHz and the plotted bandwidth extends from 0 to 3 kHz. This division of the
amplitude spectra allows to obtain the reflection coefficient versus frequency, and then the coefficient of absorption by using equation (1), as shown from the computer output on Figure 5.
Because the amplitude spectra are superimposed at low frequency, the coefficient of reflection is 1 and absorption is 0. When increasing frequency, the difference between the two amplitude spectra (for the incident and the reflected wave) increases, and accordingly, the coefficient of absorption increases as well, as shown on Figure 5.
Figure 4 : a) Temporal waveforms and b) amplitude spectra used for the numerical treatment.
Figure 5: Absorption coefficient measured with a Sennheiser sound projector. This plot is directly processed from data shown on Figure 4 with the help of equation (1).
These measurements have been compared on one hand with some data obtained with a resonance Kundt tube, and on the other hand with numerical predictions provided by the “equivalent fluid”
0 . 0 4
- 0 . 0 6 - 0 . 0 4 - 0 . 0 2 0 . 0 0 0 . 0 2
2.0E-3 0.0E+0 5.0E-4 1.0E-3 1.5E-3
référence échantillon
signaux
4.0E-4
0.0E+0 1.0E-4 2.0E-4 3.0E-4
3.0E+0
0.0E+0 1.0E+0 2.0E+0
référence échantillon
spectres d'amplitude
a)
b)
s
kHz
1 . 0 0 0
0 . 0 0 0 0 . 1 0 0 0 . 2 0 0 0 . 3 0 0 0 . 4 0 0 0 . 5 0 0 0 . 6 0 0 0 . 7 0 0 0 . 8 0 0 0 . 9 0 0
3 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5
coefficient d'absorption
kHz
0.02
0.03 0.02 0.01 0.00 0.01
5
0 1 2 3 4
ms V
1 kHz
0.04
0.04 0.02 0.00 0.02
5
0 1 2 3 4
ms V
1.5 kHz
a)
b)
incident wave reflected wave
incident wave
reflected wave
model. The significant physical parameters of the tested porous material (a 30 mm thick cellular
Figure 6 : Coefficient of absorption for a 30 mm plastic foam obtained with the Sennheiser sound projector. Red circles are the Kundt tube measurements ; Blue squares are the numerical predictions based on the “equivalent fluid” model ; Triangles (green and orange respectively) are the data obtained at a modulation frequency of 1.5 kHz and 3 kHz respectively.
plastic foam manufactured by the Tramico company) were measured independently, and no correction or numerical fit was implemented, the raw data being directly processed. The parameters are: porosity, φ = 0.96 ; resistivity, σ = 6000Nsm - 4 ; tortuosity, α∞= 1.12 ; viscous characteristic length, Λ = 80 µ m ; thermal characteristic length, Λ ’ = 240 µ m. Some results are collected on Figure 6. They show a good agreement between the various measurements and the numerical predictions.
5. Conclusions and perspectives
The work presented here demonstrates the very strong potentialities of commercial parametric antennae which are available since 2000 for multi- purposes audio applications. The very high directivity allows new opportunities in order to perform free field measurements of the coefficient of absorption of poroelastic materials by using very short pulses covering large bandwidths. This opens new opportunities for “in-situ”
measurements as well as for “on-line” monitoring of porous materials directly during their manufacturing. One should emphasise that some limitations exist when working at very low frequencies. For instance, with the Sennheiser sound projector, one should presently work at a minimum distance around 4 or 5 m, in order to sufficiently decrease the amount of the ultrasonic fields. Evidently, for other applications, for instance at higher frequencies a simple laboratory bench with standard ultrasonic transducers is amply sufficient to determine some physical parameters such as tortuosity, shape factors and dispersion curves.
References
[1] V.A. Zverev, How the idea of a parametric acoustic array was conceived, Acous. Phys., 45, 611-618 (1999) ; see also V.A Zverev, Modulation technique for measuring the dispersion of ultrasound, Dokl. Akad. Nauk.
SSSR, 91, 791 (1953).
[2]. V. Tournat, B. Castagnède, V. Gusev, P . Béquin, Self-demodulation acoustic signatures for nonlinear propagation in glass beads, C.R.
Mécanique, 331, 119-125 (2003).
[3] D.T. Blackstock, Audio application of parametric array, J. Acous. Soc. Am., 102, 3106 (1997) ; F.J. Pompei, The audio spotlight : put sound wherever you want it, J. Audio Eng.
Soc., 47, 726-731 (1999).
[4] B. Castagnède, V. Tournat, A. Moussatov, V. Gusev, "Procédé et dispositif de mesure acoustique de caractéristiques physiques de matériaux poroélastiques", French patent submitted to INPI (n° 0303913) on 28th march 2003. International extension pending.
[5] M. Saeid, B. Castagnède, A. Moussatov, V.
Tournat, V. Gusev, Measurements of the acoustical coefficient of reflection for air saturated materials via a non linear demodulation technique, C. R. Mécanique , 332, 849-858 (2004).
[6] B. Castagnède, M. Saeid, A. Moussatov, V.
Gusev, V. Tournat, Retrodiffusion and transmission at normal incidence onto air- saturated poroelastic materials and direct measurements based on parametric demodulated ultrasonic waves, submitted to Ultrasonics (2005).
[7] J.F. Allard, « Propagation of sound in porous media : modelling sound absorbing materials », Elsevier Applied Science, London (1993).
0 0.2 0.4 0.6 0.8 1
0 500 1000 1500 2000 2500 3000
Kundt+Senn/MD40cm/cor.dat
B D F H
Coefficient d'absorption
Fréquence (Hz)
Mesures tube de Kundt Simulation numérique Manip Senn modul 1.5 kHz Manip Senn modul 3 kHz
