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Chapter 26

Metrology of Acoustical Properties of Absorbing Materials: Surface Impedance

26.1. Introduction

26.1.1. Method of the Kundt tube

Let us suppose that the material has a plane contact surface with air and that a sounding harmonic wave in homogenous air strikes the material with a normal incidence. The material’s reaction is characterized by its surfacic impedance Zs. This acoustical quantity, found by measurement, depends in particular on the complex wavenumber q(ω) and on the characteristic impedance Zc(ω) of the porous medium, these functions being related to the dynamic tortuosity α(ω) and to the normalized compressibility β(ω), respectively, by the relationships:

βαω .)( oqq = [26.1]

βα

oc ZZ = [26.2]

Chapter written by Michel HENRY.

Materials and Acoustics Handbook Edited by Michel Bmneau & Catherine Potel

Copyright 0 2009, ISTE Ltd.

720 Materials and Acoustics Handbook

with a

o

oo Kc

qρωω

== and aoo KZ .ρ= corresponding to the same quantities

for the free fluid characterized by its density oρ and its adiabatic bulk modulus

aK , respectively.

Figure 26.1. Surfacic impedance in normal incidence

The acoustical characterization of a material is thus made through the measurement of the surfaces impedance of this material. Let us consider a layer of porous material of thickness d located between two air layers. Figure 26.1 represents a plane wave propagating in a direction Ox perpendicular to the surface of a material immersed in air. A part of this wave is reflected and yields a combination of two waves, back and forth, in the exterior air. If pe1 is the acoustic pressure and 1ev the particular velocity normal to the material surface, in the air, at x1= 0, the surface impedance Zs1 is defined by:

1

11

e

es v

pZ = . [26.3]

Using the conditions of pressure continuity and flow conservation, we can link this impedance to the acoustical quantities on the surface, immediately within the material, by:

)1(1

1

11 ss Z

vp

Zφφ

== [26.4]

Metrology of Acoustical Properties of Absorbing Materials 721

where φ is the material porosity and Zs(1) the surface impedance at x1 = 0, defined from the acoustic pressure p1 and the particular velocity v1 in air inside the material. The same relationship exists at the second surface at x2 = d and is written in the following form:

)2(1

2

2

2

22 s

e

es Z

vp

vp

Zφφ

=== . [26.5]

If the impedance Zs2 is known, as well as the wavenumber q(ω) and the characteristic impedance Zc(ω), it is possible to obtain the expression of Zs1 as a function of these quantities. This calculation is classic in acoustics and we can refer for example to the work of J. F. Allard [ALL 93]. We obtain the following relationship between Zs(1) and Zs(2):

).cot(..)2().cot().2(.

)1(dqZiZdqZiZ

ZZcs

sccs +

+= . [26.6]

The surface impedance Zs1 is then written:

).cot(..).cot(...

2

21 dqZiZ

dqZiZZZ

cs

sccs +

+=

φφ

φ. [26.7]

It is convenient to introduce a dimensionless surface impedance (or reduced surface impedance), which will be used in the following and is defined generally by:

o

sZZ

Z = , [26.8]

where Zo = ρoco is the characteristic impedance of the unlimited fluid. Let us assume that, if the second air layer is semi-infinite, Zs2 = Zo (reduced impedance Z2 = 1). We can give the expression of the reduced impedance Z1 as a function of the reduced impedance Z2, replacing q(ω) and Zc(ω) by their expression as functions of the dynamic tortuosity α(ω) and of the compressibility β(ω). We finally obtain:


)cot(..

)cot(..1

2

2

1

αββαφ

αβφβα

βα

φdqiZ

dqZiZ

o

o

+

+= . [26.9]

The surface impedance contains information on the pair α(ω) and β(ω). Measuring the impedances with the same material placed in two different configurations, it is possible to obtain the dynamic tortuosity α and the normalized compressibility β.

Finally, we can quote an interesting specific case, when the material is set on a rigid backing. The surface impedance Z2 is then infinite and the reduced impedance for the material becomes:

⎟⎟⎠

⎞⎜⎜⎝

⎛= αβ

ωβα

φd

ci

Zo

cot1 . [26.10]

To have a situation which is really different from the former, the sample can be located in front of a cavity whose length l corresponds, for a given frequency, to λ/4 (up to a multiple of λ/2). The surface impedance of such a cavity being zero, the reduced surface impedance Z1’ of the material is obtained with Z2 = 0, which gives:

)tan('1 αββα

φdq

iZ o

−= . [26.11]

The surface impedance appears as the acoustical quantity characterizing the acoustical behavior of the material, from which it is possible, if the measurements are sufficiently precise, to find the functions α and β.

If R represents the complex reflection coefficient, i.e. the ratio of the complex amplitudes of the reflected wave and incident wave at the surface of the sample, its reduced impedance can be written:

RR

Z−+

=11

. [26.12]

The absorption coefficient in energy Σ, defined as the flux of absorbed energy divided by the flux of incident energy, is thus given by:


21 R−=Σ . [26.13]

Practically, this coefficient is of great interest for industry. But because it depends only on the modulus of R, it brings less information than the surfacic impedance.

26.1.2. T.M.T.C. method of the Kundt tube

The measurement of a surface impedance can be made in free field or using a pipe of stationary waves, also known as Kundt tube. The study of sound propagation in a tube can be found in most works in acoustics (see M. Bruneau [BRU 83]). A sound source placed at an extremity of a tube yields a plane incident wave which is partially reflected by the studied material which closes the other extremity of the tube. The incident and reflected waves interfere, and yield a system of stationary waves. For sufficiently low frequencies, the wave that propagates along the tube’s axis has the characteristics of a plane wave. The use of a tube is thus interesting only for low frequencies. In our case, the tube diameter is 44.36 mm and the cutting frequency is thus around 4,500 Hz, which enables us to cover a frequency range from 50 to 4,000 Hz. The low-frequency limitation is only of technical order as we will see in the following.

There are different methods of performing the measurement of the surface impedance of a material using one, two or several microphones. Each has its own advantages and disadvantages. To obtain some precise measurements, we have developed in our laboratory (Maine University, Le Mans, France) a method using two microphones fixed on the tube. It is the T.M.T.C. method (Two-Microphone-Three-Calibration).

Initially, this method was proposed by V. Gibiat and F. Laloë [GIB 90] for the measurement of the input impedance of musical wind instruments. To calibrate the two fixed microphones, we use three closed empty cavities, and this is the origin of the name chosen for this method: two microphones and three calibrations (T.M.T.C.). We use a closed tube in which we generate a plane harmonic wave, of fixed frequency f, with the help of a loudspeaker located at one of the extremities, the other extremity being occupied by the sample, for which we want to measure the surface impedance. Two microphones (n°1 and n°2) are fixed on the wall of the tube and are separated by a distance e (microphonic spacing). Microphone n°1 is at a distance s from the plane x = 0 that corresponds to the sample’s face near the loudspeaker; this plane is called the measurement plane. The experimental setup is illustrated in Figure 26.3.


Figure 26.2. Surface impedance in normal incidence

If po and vo are the acoustic pressure and velocities in the measurement plane (x = 0), the surface impedance is given by Zs = po/vo. The pressure signals S1 and S2 delivered by the two microphones n°1 and n°2, respectively, are proportional to the corresponding acoustic pressures p1 and p2. The acoustic pressures and velocities p1, v1 and p2, v2 can be expressed as functions of po and vo. The signals S1 and S2 are then a linear combination of po and vo:

oo

oovapaS

vapaS

22212

12111+=+=

, [26.14]

where the coefficients aii are dependent on the geometry of the system (s, e, etc.), on the acoustical properties of air in the pipe (wavenumber q’, characteristic impedance Zo, etc.), on the gain and finally on the two microphones. The signal ratio y = S2 /S1 can be written:

1211

2221

1

2

aZaaZa

SS

ys

s

++

== . [26.15]

Inverting this relationship, we obtain the surface impedance Zs:

2111

1222

..

ayayaa

Z s −−

= . [26.16]

Dividing this relationship by the characteristic impedance of air Zo = ρoco, the reduced impedance of the measurement surface x = 0 is given by the relationship:


oyyByA

Z−+

=.

, [26.17]

with 11

21

11

22

11

12 ;;aa

yaZ

aB

aZa

A ooo

==−= .

The calibration will consist of determining these three coefficients A, B and yo, coefficients depending on the geometrical configuration and on the microphones, in order to then have the reduced impedance of the studied surface by the measurement of the signal ratio S2/S1. Measuring this ratio “y” for three surfaces whose impedances are known, we obtain a system of equations with three unknowns, which has to be solved. A fourth measurement with the studied material enables us to determine its reduced impedance, based on the condition that the system did not endorse any modification between these different measurements (same temperature, frequency, lengths s and e, etc.). The best-known surfacic impedances are the input impedances of cylindrical cavities with a rigid bottom. If dj is the length of the cavity, the reduced input impedance can be written:

)'.cot(. jj dqiZ = , [26.18]

where q’ is the wavenumber of a plane harmonic wave propagating in the tube. This wavenumber is a complex value that takes into account the viscous and thermal effects in the tube and, consequently, differs from the wavenumber qo of the unlimited fluid. Sound tubes have been the subject of many studies and we can refer to the works of J. Kergomard ([KER 91] and [CAU 84]) or M. Bruneau ([BRU 82] and [BRU 83]). We can start with a first cavity of length zero, i.e. put a rigid wall in the measurement plane (see Figure 26.3). Called cap (or cavity n°0) in the following, the surface impedance of this cavity can be considered as infinite and the measurement of the signal ratio corresponds to the coefficient yo. Denoting yj the signal ratio S2j/S1j obtained for the cavity n°j (with j = 0, 1, 2) of reduced input impedance Zj , we can write:

⎪⎪⎩

⎪⎪⎨

⎧

−+

=

−+

=

o

o

yyBAy

Z

yyBAy

Z

2

22

1

11

[26.19]


Figure 26.3. Measurement principle for the T.M.T.C. method

From this linear system, we can find the expression of the coefficients A and B as functions of the measurements yo, y1, y2 and of the reduced impedances of two cavities Z1 and Z2. A fourth measurement “y”, made this time by putting the surface


of a material at the level of the measurement plane, gives us the following reduced surface impedance Z:

))(())(())((

21

122211

yyyyyyyyZyyyyZ

Zo

oo−−

−−+−−= . [26.20]

A detailed study has enabled us to obtain very precise measurements if some conditions are fulfilled. It is necessary to make the measurements at some frequencies for which the space between the two microphones corresponds to an odd number of quarter-waves. Likewise, for the calibration, cavities 2 and 3 must correspond to cavities in λ/8 and 3λ/8, respectively. We thus work with a frequency comb, the fundamental frequency being imposed by the spacing between the two microphones. Finally, it is important to specify that the calibration being made with the three cavities, it can be used for more than two years as long as nothing is modified on the test bench.

– Applications

From the two impedances Z1 (rigid bottom) and Z1’ (quarter-wave cavity) whose expressions have been given formerly, we can extract the two functions α(ω) and β(ω) without any approximation. Consequently, we obtain:

21

11

1

12

'

'tan

.1

)(

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−±

=ZZ

ZZ

Arc

dqoφωβ [26.21]

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−±==

1

11111

2 'tan

'').()(

ZZ

Arcdq

ZZZZ

oφφωβωα , [26.22]

where the sign ± is chosen so that the real part of the impedance Z1’ is positive. The determination of the arctangent function is made by unwrapping the phase.

The relevant parameters φ, σ, α∞ being measurable by various methods detailed above, it is thus possible to have a good estimation of the parameters Λ and Λ’, the viscous and thermal characteristic lengths which are involved in the functions α(ω)


and β(ω), respectively. A comparison between these measured functions and the models enables the determination of these parameters.

Experimental measurements; Models ; Lafarge

Figure 26.4. Reduced surfacic impedance Z1’ for foam n°1

The graphs presented in Figures 26.4 and 26.5 show what we can obtain on two polyurethane foams. For foam n°1 (φ = 0.98, σ = 4,750 s.i. and α∞ = 1.2), the value of Λ, for which the surface impedance curve Z1’ (a quarter-wave cavity being put behind the sample) best coincides with the models proposed in chapter 2, is 2ä10–

4 m. Likewise, the value of Λ’ for foam n°2 (φ = 0.97, σ = 10,000 s.i. and α∞ = 1.04) given by the best coincidence with the measurements of surface impedance Z1 (the sample being put against a rigid wall) is 3ä10–4m. These results have been confirmed by ultrasonic measurements.


Experimental measurements; Models ; Lafarge

Figure 26.5. Reduced surface impedance Z1 for foam n°2 (set on a rigid impervious backing)

26.1.3. Other methods: Chung and Blaser

Some ultrasonic methods for the measurement of parameters such as α∞,, Λ and Λ’ have enabled us to validate these different techniques.

There are also other principles for the measurement of surface impedance. The most classical technique used to determine the surface impedance of a material is the search for a stationary wave by moving a microphone probe inside the tube, in order to reconstitute the wave pattern. We can determine the modulus and the phase of the reflection coefficient R by measuring the maximum and minimum amplitudes in the tube, as well as the position x1 of the nearest minimum of pressure, the distance being measured from the sample [ALL 93]. We can then deduce the surface impedance. This method, simple in appearance, presents several disadvantages. A good determination of maximum and minimum pressures requires a great number of measurements made by moving the microphone probe, and this for a given frequency. To determine the impedance as a function of frequency, it is necessary to repeat these measurements for each different frequency. The time spent making each


measurement can become very important. Moreover, the signal-to-noise ratio becomes very weak near the minimal pressures, which causes a decrease of measurement precision. Another technical problem appears for low frequencies. At 100 Hz for example, we must be able to move the probe along a distance of around 3.5 m. Such a long probe, apart from the load, will attenuate the measured signal, and thus the signal-to-noise ratio.

To avoid the necessity of the probe displacement in the tube, methods using two microphones have been developed [AST 86]. One of them is the method of Chung and Blaser [CHU 80]. In this case, we use a doublet of microphones fixed on the inner surface of the tube. By measuring the transfer function between the two pressure signals delivered by the microphones, it is possible to obtain the reflection coefficient. The obtained result depends on the distance between the two microphones and on that between the first microphone and the surface of the studied material. The responses in amplitude and phase of the two microphones are never perfectly identical, so it is necessary to perform a calibration of the system. We thus make two consecutive measurements with any absorbing material, inverting the two microphones. Using this method, the measurement time is reduced, but the precision of the results obtained is closely related to that of the position of the microphones. In particular, inverting them inevitably introduces some errors.

Finally, we can mention the microphone doublet technique [ALL 92], a measurement technique in free field developed in our laboratory (Le Mans, France). This method, performed in an anechoic room, consists of measuring the acoustic pressure at two positions above the material, and then returning to the surface impedance. This method gives good results for high frequencies, but is not usable below 500 Hz for thicknesses of several centimeters of common materials. More recently, a new method, based on the works of Tamura [TAM 90] and usable for lower frequencies, has been developed by B. Brouard [BRO 94]: the holographic method. However, for precise measurements of surface impedances at low frequencies, the technique in free field is not ideal.

26.2. Bibliography

[ALL 92] J. F. Allard, D. Lafarge, “Free field measurement at high frequencies of the impedance of porous layer”, Second International Congress on Recent Developments in Air-and Structure-Borne Sound and Vibration, p. 4–6, Auburn University, U.S.A., March, 1992

[ALL 93] J.-F. Allard, Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, Chapman & Hall, London, 1993


[AST 86] ASTM, Standard Test Method for Impedance and Absorption of Acoustical Materials Using a Tube, Two Microphones and a Digital Frequency Analysis System, ASTM standard E 1050–86, 1986

[BRO 94] B. Brouard, D. Lafarge J. F. Allard, “Measurement and prediction of the surface impedance of a resonant sound absorbing structure”, Acta Acustica, n°2, 301–6, 1994

[BRU 82] M. Bruneau, J. Kergomard,– “Constante de propagation dans un tuyau cylindrique”, Fortschritte der Akustik, FASE/DAGA’82, 719–22, Göttingen, 1982

[BRU 83] M. Bruneau, Introduction aux Théories de l’Acoustique, Publications de l’Université du Maine, Le Mans, 1983

[CHU 80] J. Y. Chung, D. A. Blaser, “Transfer function method of measuring induct acoustic properties: I-Theory, II-Experiment”, J. Acoust. Soc. Am, n°68, 907–21, 1980

[CAU 84] R. Caussé, J. Kergomard, X. Lurton, “Input impedance of brass musical instruments–comparison between experiment and numerical models”, J. Acoust. Am., Janvier, n°75 (1), 241–54,1984

[GIB 90] V. Gibiat, F. Laloe, “Acoustical impedance measurements by the Two-Microphone-Three-Calibration (T.M.T.C.) method”, J. Acoust. Soc. Am., n°88, 2533–545, 1990

[KER 91] J. Kergomard, J. D. Polack, J. Gilbert, “Vitesse de propagation d’une onde plane impulsionnelle dans un tuyau sonore”, J. Acoustique, n°4, 467–83, 1991

[KO 65] W.L. Ko, “Deformation of foamed elastomers”, J. Cellular Plastics, 1, 45–50, 1965

[TAM 90] M. Tamura, “Spatial Fourier transform method of measuring reflection coefficients at oblique incidence. I: Theory and numerical examples”, J. Acoust. Soc. Amer., vol. 88, 2259–64, 1990

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