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Acoustic Metrology
Chapter 4
Impedance Tube  part 1.
Surface impedance, reflection and absorption coefficients measurements
Diego Turo, Joseph Vignola and Aldo Glean
June 21, 2012
http://mason.gmu.edu/~dturo/collaborations/CUA_Lecturer_ME_661.html
Impedance tube – part 1
Basic theory 3
Acoustic characteristic and surface impedances 3
Superposition of two waves propagating in opposite directions 3
Impedance variation along a direction of propagation 3
Impedance at normal incidence of a layer of fluid backed by an impervious rigid wall 4
Impedance at normal incidence of a multilayered fluid 5
Reflection coefficient and absorption coefficient at normal incidence 5
Reflection coefficient 5
Absorption coefficient 5
Definition and Symbols 5
Sound absorption coefficient at normal incidence 5
Sound pressure reflection coefficient at normal incidence 5
Normal surface impedance 6
Wave number 6
Complex sound pressure 6
Cross spectrum 6
Auto spectrum 6
Transfer function 6
Calibration factor 6
Basic principle of measurements performed with an impedance tube 7
Limitations of the impedance tube measurements. 7
Preliminary tests 8
Determination of the speed of sound , wavelength and characteristic impedance 8
Calibration of the measurement setup 8
Selection of the signal amplitude 8
Correction for microphone mismatch 8
Measurement repeated with the microphones interchanged 8
Calibration factor 10
Determination of the reflection coefficient 12
Determination of the sound absorption coefficient 12
Determination of the acoustic surface impedance ratio 13
References 14
Matlab codes 15
Transfer functions 15
Reflection and Absorption coefficients and surface impedance measurements 15
Basic theory
The acoustic impedance at a particular frequency indicates how much sound pressure is generated by the vibration of molecules of a particular acoustic medium at a given frequency.
The ratio of acoustic pressure in a medium to the associated particle velocity is defined as specific impedance (or surface impedance if referred to an interface between two fluids or fluidsolid):
It is usually a complex quantity. However it is a real quantity for progressive plane waves (because pressure and particle velocity are in phase).
The product of the fluid density by the speed of sound in that fluid, , defines a characteristic property of the medium and therefore is often called characteristic impedance. For standing plane waves and diverging waves specific impedance is a complex quantity.
Superposition of two waves propagating in opposite directions
The pressure and the velocity, for a wave propagating toward the positive abscissa are, respectively,
The pressure and the velocity, for a wave propagating toward the negative abscissa are, respectively,
If the acoustic field is a superposition of the two waves described by the above equations, the total pressure and the total velocity are
A superposition of several waves of the same ω and k propagating in a given direction is equivalent to one resulting wave propagating in the same direction. The ratio is called the impedance at x.
Impedance variation along a direction of propagation
In Figure 1., two waves propagate in opposite directions parallel to the axis. The impedance at is known. The impedance can be written

Layer of fluid.
Whereas at , the impedance can be written
From the above equations one can evaluate the following expression
which gives
where is equal to . The above equation is known as the impedance translation theorem.
Impedance at normal incidence of a layer of fluid backed by an impervious rigid wall
A layer of fluid 2 is backed by a rigid plane of infinite impedance at as shown in in the figure below. The impedance at at the surface of the layer of fluid 2 is obtained from
where is the characteristic impedance and the wave number in fluid 2.

Layer of fluid backed by a rigid wall.
The pressure and the velocity are continuous at the boundary. The impedance at both sides of the boundary are equal, the velocities and pressures being the same on either side of the boundary.
Impedance at normal incidence of a multilayered fluid
The impedance of a multilayered fluid can be easily evaluated applying the previous equations layer by layer. Starting from a known impedance at , is evaluated and used as know impedance for the next layer and so on.
Reflection coefficient and absorption coefficient at normal incidence Reflection coefficient
The reflection coefficient at the surface of a layer is the ratio of the pressures and created by the outgoing and the ingoing waves at the surface of the layer. For instance, at , in Figure 2., the reflection coefficient is equal to
This coefficient does not depend on because the numerator and the denominator have the same dependence on . Using previous equations, the reflection coefficient can be written as
where is the characteristic impedance in fluid 1. The ingoing and outgoing waves at have the same amplitude if . This occurs if is infinite or equal to zero. If is greater than 1, the amplitude of the outgoing wave is larger than the amplitude of the ingoing wave. More generally, the coefficient can be defined everywhere in a fluid where an ingoing and an outgoing wave propagate in opposite directions.
Absorption coefficient
The absorption coefficient is related to the reflection coefficient as follows
The phase of is removed, and the absorption coefficient does not carry as much information as the impedance or the reflection coefficient. The absorption coefficient is often used in architectural acoustics, where this simplification can be advantageous. It can be rewritten as
where and are the average energy flux through the plane of the incident and the reflected waves, respectively.
Definition and Symbols Sound absorption coefficient at normal incidence
It is the ratio of sound power entering the surface of the test object (without return) to the incident sound power for a plane wave at normal incidence.
Sound pressure reflection coefficient at normal incidence
It is the complex ratio of the amplitude of the reflected wave to that of the incident wave in the reference plane for a plane wave at normal incidence.
Normal surface impedance
It is the ratio of the complex sound pressure to the normal component of the complex sound particle velocity at an individual frequency in the reference plane ().
It is the variable defined by
where
is the angular frequency;
is the frequency;
is the speed of sound;
is the wavelength.
NOTE: In general the wave number is complex, so
where
is the real component ();
is the imaginary component which is the attenuation constant, in Nepers per metre.
Complex sound pressure
It is the Fourier transform of the temporal acoustic pressure
It is the product , determined from the complex sound pressures and at two microphone positions.
NOTE: * means the complex conjugate.
Auto spectrum
It is the product , determined from the complex sound pressure at microphone position one.
NOTE: * means the complex conjugate.
Transfer function
It is the transfer function from microphone position one to two, defined by the complex ratio:
or , or
Calibration factor
It is the factor used to correct for amplitude and phase mismatches between the microphones.
An impedance tube is a straight, rigid, smooth cylindrical pipe composed by two main sections or tubes: transmitting and receiving tube. The test sample is mounted at one end of the impedance tube (receiving tube). Plane waves are generated in the transmitted tube by a sound source (random, pseudorandom sequence, or chirp), and the sound pressures are measured at two locations near to the sample (preferably less than 3 times the diameter of the tube). The complex acoustic transfer function of the two microphone signals is determined and used to compute the normalincidence complex reflection coefficient , the normalincidence absorption coefficient , and the surface impedance of the test material .
The quantities are determined as functions of the frequency with a frequency resolution which is determined from the sampling frequency and the record length of the digital frequency analysis system used for the measurements. The usable frequency range depends on the width of the tube and the spacing between the microphone positions.
The measurements may be performed by employing one of two following techniques:

twomicrophone method (using two microphones in fixed locations);

onemicrophone method (using one microphone successively in two locations).
Technique 1 requires a pretest or intest correction procedure to minimize the amplitude and phase difference characteristics between the microphones; however, it combines speed, high accuracy, and ease of implementation. This technique is recommended for general test purposes.
Technique 2 has particular signal generation and processing requirements and may require more time; however, it eliminates phase mismatch between microphones and allows the selection of optimal microphone locations for any frequency. It is recommended for precision.
Limitations of the impedance tube measurements.
As all instruments, the impedance tube presents some limitations about which acoustic properties can be measured and in which range of frequency.

Measurements performed in an impedance tube are at normal incidence. It is important to keep in mind that in real life this condition is often not satisfied. However, characteristic impedance and wavenumber of a porous media can be measured with this instrument and used to predict acoustic behavior of the material at oblique incidence.

Plane wave can be generated in a tube only if the excitation frequency is below the smallest acoustic mode (cutoff frequency, see Figure 3.) of the tube. This condition defines the upper working frequency limit of this instrument.

Cutoff frequencies of a circular duct filled with air. Cutoff frequencies are evaluated using the following equation: where satisfy ,_{ } is speed of sound in air (343 m/s) and is diameter of the duct in meters.

Microphones spacing defines both upper and lower working frequencies of the tube. Microphone spacing is 5% of the longest measurable wavelength and 95% of the shortest one (keep in mind that the length of the tube has to be long enough so that at least half of the longest wavelength can fit in it).
Preliminary tests Determination of the speed of sound , wavelength and characteristic impedance
Before starting a measurement, the velocity of sound, , in the tube has to be determined, after which the wavelengths at the frequencies of the measurements has to be calculated.
The speed of sound can be assessed accurately with knowledge of the tube air temperature from:
where T is the temperature, in Kelvin.
The wavelength then follows from:
The density of the air, , can be calculated from
where
T is the temperature, in Kelvin;
is the atmospheric pressure, in kPa;
K;
kPa;
kg/m^{3}.
The characteristic impedance of the air is the product .
Calibration of the measurement setup Selection of the signal amplitude
The signal amplitude has to be at least 10 dB higher than the background noise at all frequencies of interest, as measured at the chosen microphone locations.
During a test, any frequency having a response value 60 dB lower than the maximum frequency response value has to be rejected.
Correction for microphone mismatch
When using the twomicrophone technique, one of the following procedures for correcting the measured transfer function data for channels mismatch must be used: repeated measurements with channels interchanged, or predetermined calibration factor. A channel consists of a microphone, preamplifier and analyzer channel.
Measurement repeated with the microphones interchanged
Correction for microphone mismatch is done by interchanging channels for every measurement on a test specimen. This procedure is highly preferred when a limited number of specimen are to be tested. Place the test specimen in the tube and measure the two transfer functions and , using the same mathematical expressions for both. Place the microphones in configuration I (standard configuration, see Figure below) and store the transfer function . Interchange the two microphones A and B.
When interchanging the microphones, ensure that microphone A in configuration II (microphones interchanged) occupies the precise location that microphone B occupied in configuration I (standard configuration), and vice versa. Do not switch microphone connections to the preamplifier or signal analyzer.
Measure the transfer function and compute the transfer function using equation:

Impedance tube configurations. Configuration I: microphone A in position 1 and microphone B in position 2. Configuration II: microphone B in position 1 and microphone A in position 2.
If the analyzer is only able to measure transfer functions in one direction (e.g from microphone A to microphone B), can be computed using:
In Figure 5. are shown of transfer functions measured using Configuration I and Configuration II. Notice that is whereas is and is .

Transfer functions measured with different configurations.
Calibration factor
The calibration procedure uses a special calibration specimen and the correction is valid for all successive measurements. This procedure is performed once and after calibration the microphones remain in place.
Place an absorptive specimen in the tube to prevent strong acoustic reflections and measure the two transfer functions and .
Compute the calibration factor using the following expression
or, if the analyzer is only able to measure transfer functions in one direction (e.g from microphone A to microphone B), can be computed using:
For subsequent tests, place the microphones in configuration I (standard configuration). Insert the test specimen and measure the transfer function
where
is the uncorrected transfer function and is the uncorrected phase angle;
Correct for mismatch in the microphone responses using the following equation:
In Figure 6. is plotted a calibration factor evaluated using the data shown in Figure 5.. In Figure 7. is instead shown a corrected transfer function using both microphone interchange and the calibration factor techniques.

Correction Factor.

Transfer functions corrections.
Determination of the reflection coefficient
Calculate the normal incidence reflection coefficient using the following expression:
where
is the distance between the sample and the further microphone location;
is the phase angle of the normal incidence reflection coefficient;
is the transfer function of the incident wave alone;
is the transfer function of the reflected wave alone;
is microphone spacing.
NOTE: Complex pressure at position 1, , can be expressed as summation of the incident and reflected waves at location , . Whereas the pressure at position 2, ,can be expressed as superposition of incident and reflected waves at location , . From expressions of and derivation of and is straightforward.
The reflected wave pressure amplitude , can be written in terms of reflection coefficient as
.
The transfer function between two microphones is given by
from which
Q.E.D. (quod erat demonstrandum [En: “which was to be demonstrated”]).
The normal incidence sound absorption coefficient is given by the following equation:

Reflection and absorption coefficients of a layer of porous foam of thickness d = 2.5 cm.
Determination of the acoustic surface impedance ratio
The acoustic surface impedance ratio is the surface impedance normalized respect to the characteristic impedance of the air:

Surface impedance ratio at normal incidence of a layer of porous foam of thickness d = 2.5 cm.
References 
British Standards, “Acoustics — Determination of sound absorption coefficient and impedance in impedance tubes —Part 1: Method using standing wave ratio”, BS EN ISO 105341, 2001.

British Standards, “Acoustics — Determination of sound absorption coefficient and impedance in impedance tubes —Part 2: Transferfunction method”, BS EN ISO 105342, 2001.

Allard, J.F. and Atalla, N., “Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials”, Second Edition, Wiley, 2009.

Chung et al., “Transfer function method of measuring induct acoustic properties. I. Theory”, J. Acoust. Soc. Am., 68, 907913, 1980.

Chung et al., “Transfer function method of measuring induct acoustic properties. II. Experiment”, J. Acoust. Soc. Am., 68, 914921, 1980.

Utsuno et al., “Transfer function method for measuring characteristic impedance and propagation constant of porous materials”, J. Acoust. Soc. Am., 86, 637643, 1989.
Matlab codes Transfer functions
Here is an example of transfer functions measured between two microphones at positions AB and BA, respectively. Measurement have been performed with a B&K impedance tube and using a sample of microlite 22 mm thick.

Transfer function recorded between microphones AB and BA, respectively. Measurement performed with a B&K impedance tube and using a sample of microlite 22 mm thick.
Reflection and Absorption coefficients and surface impedance measurements
% Reflection and Absorption coefficients and surface impedance measurements
% Define constants:
freq = []; % frequency vector (Hz)
rho = 1.21; % density of air (kg/m^3)
c = 343; % speed of sound in air at 23 Celsius (m/s)
s = 0.1; % microphone spacing (m)
Zair = rho*c; % characteristic impedance of air (kg/m^2/s)
k = (2*pi*freq)/c; % wavenumber in air (m^1)
x1 = ?; % distance between the sample and the farther microphone
% Reflection coefficient
R = ( H12  exp(j.*k.*s) )./(exp(j.*k.*s)  H12).*exp(2.*j.*k.*x1);
% H12 is Transfer function measured between two mics
% Absorption coefficient
alpha = 1  abs(R).^2;
% Surface impedance
Zs = Zair*((1+R)./(1R));
% Normalized Surface Impedance
Zs_n = ((1+R)./(1R));
% Plots
figure(1)
plot(freq,alpha,'b','LineWidth',2)
axis([0 1600 0 1])
title('Absorption Coefficient','FontSize',12)
xlabel('Frequency (Hz)'), ylabel('Absorption Coefficient')
grid on
figure(2)
plot(freq,abs(R),'b','LineWidth',2)
axis([0 1600 0 1])
title('Reflection Coefficient','FontSize',12)
xlabel('Frequency (Hz)'), ylabel('Reflection Coefficient')
grid on
figure(3)
subplot(2,1,1)
plot(freq,real(Zs),'b','LineWidth',2)
xlim([0 1600])
title('Surface Impedance – Real part','FontSize',12)
xlabel('Frequency (Hz)')
grid on
subplot(2,1,2)
plot(freq,imag(Zs),'b','LineWidth',2)
xlim([0 1600])
title('Surface Impedance – Imaginary part','FontSize',12)
xlabel('Frequency (Hz)')
grid on
figure(4)
subplot(2,1,1)
plot(freq,real(Zs_n),'b','LineWidth',2)
xlim([0 1600])
title('Surface Impedance Ratio– Real part','FontSize',12)
xlabel('Frequency (Hz)')
grid on
subplot(2,1,2)
plot(freq,imag(Zs_n),'b','LineWidth',2)
xlim([0 1600])
title('Surface Impedance Ratio – Imaginary part','FontSize',12)
xlabel('Frequency (Hz)')
grid on
Plots

Reflection coefficient. Measurement performed with a B&K impedance tube and using a sample of microlite 22 mm thick.

Absorption coefficient. Measurement performed with a B&K impedance tube and using a sample of microlite 22 mm thick.
