CUA
THE CATHOLIC UNIVERSITY OF AMERICA
School of Engineering
Department of Mechanical Engineering
620 Michigan Ave.
Washington DC 20064
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 a (w ) .............................................................................................................. 5
SOUND PRESSURE REFLECTION COEFFICIENT AT NORMAL INCIDENCE R (w ) ............................................................................................ 5
NORMAL SURFACE IMPEDANCE Z s (w ) ............................................................................................................................................................. 6
k0 ................................................................................................................................................................................................. 6
COMPLEX SOUND PRESSURE P (w ) .................................................................................................................................................................... 6
CROSS SPECTRUM S12 (w ) ................................................................................................................................................................................... 6
AUTO SPECTRUM S11 (w ) ..................................................................................................................................................................................... 6
TRANSFER FUNCTION H12 (w ) ........................................................................................................................................................................... 6
CALIBRATION FACTOR H c (w ) ........................................................................................................................................................................... 6
WAVE NUMBER
BASIC PRINCIPLE OF MEASUREMENTS PERFORMED WITH AN IMPEDANCE TUBE ................................................... 6
LIMITATIONS OF THE IMPEDANCE TUBE MEASUREMENTS............................................................................................................................... 7
PRELIMINARY TESTS ........................................................................................................................................................................ 8
DETERMINATION OF THE SPEED OF SOUND c0 , WAVELENGTH l0 AND CHARACTERISTIC IMPEDANCE Z 0 ........................................ 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
PLOTS ......................................................................................................................................................................................................................17
Basic theory
Acoustic characteristic and surface impedances
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 fluid-solid):
Zs 
p  x, t 
v  x, t 
It is usually a complex quantity. However it is a real quantity for progressive plane waves (because pressure and
particle velocity are in phase).
p  x, t  j0cA   e j t  kx 
Zs 

 0c  Z c
v  x, t 
j A   e j t  kx 
The product of the fluid density by the speed of sound in that fluid, 0 c , 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,
p ( x, t ) = Ae (
j -kx+wt )
A j(-kx+wt )
e
Zc
The pressure and the velocity, for a wave propagating toward the negative abscissa are, respectively,
v ( x, t ) =
p* ( x, t ) = A*e (
j kx+wt )
A* j(kx+wt )
e
Zc
If the acoustic field is a superposition of the two waves described by the above equations, the total pressure
pT ( x, t ) and the total velocity vT ( x, t ) are
v* ( x, t ) = -
pT ( x,t ) = Ae (
j -kx+w t )
j -kx+wt )
j kx+wt )
- A* e ( )
Zc
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 pT ( x,t ) / vT ( x, t ) is called the impedance at x.
vT ( x, t ) =
Ae (
+ A* e (
j kx+w t
Impedance variation along a direction of propagation
In Figure 1, two waves propagate in opposite directions parallel to the x -axis. The impedance Z ( x2 ) at x2 is
known. The impedance Z ( x2 ) can be written
Z ( x2 ) =
j -kx +wt
j kx +wt
j -kx
j kx
pT ( x2, t )
Ae ( 2 ) + A*e ( 2 )
Ae ( 2 ) + A*e ( 2 )
= Zc j(-kx2 +wt)
=
Z
c
j kx +wt
j -kx
j kx
vT ( x2 , t )
Ae
- A* e ( 2 )
Ae ( 2 ) - A*e ( 2 )
Figure 1. Layer of fluid.
Whereas at x1 , the impedance Z  x1  can be written
j -kx
j kx
pT ( x1, t )
Ae ( 1) + A*e ( 1)
Z ( x1 ) =
= Zc j(-kx1)
j kx
vT ( x1, t )
Ae
- A*e ( 1 )
From the above equations one can evaluate the following expression
A Z ( x2 ) - Zc -2 j(kx2 )
=
e
A* Z ( x2 ) + Z c
which gives
- jZ ( x2 ) cot ( kd ) + Zc
Z ( x1 ) = Zc
Z ( x2 ) - jZc cot ( kd )
where d is equal to x2 - x1 . 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 x2 = 0 as shown in in the figure below.
The impedance at x1 at the surface of the layer of fluid 2 is obtained from
é - jZ ( x ) cot ( kd ) + Z ù
- jZ ( x2 ) cot ( kd )
2
c
Z ( x1 ) = lim êZc
= - jZc cot ( kd )
ú = Zc
Z ( x2 )®¥ ê
Z
x
jZ
cot
kd
Z
x
ú
(
)
(
)
(
)
ë
û
2
c
2
where Z c is the characteristic impedance and k the wave number in fluid 2.
Figure 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 xn = 0 , Z ( xn-1 ) 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 R at the surface of a layer is the ratio of the pressures p* and p created by the
outgoing and the ingoing waves at the surface of the layer. For instance, at x1, in Figure 2, the reflection
coefficient R ( x1 ) is equal to
p* ( x1, t )
p ( x1, t )
This coefficient does not depend on t because the numerator and the denominator have the same dependence
on t . Using previous equations, the reflection coefficient R ( x1 ) can be written as
R ( x1 ) =
Z ( x1 ) - Zc1
Z ( x1 ) + Zc1
where Z c1 is the characteristic impedance in fluid 1. The ingoing and outgoing waves at x1 have the same
R ( x1 ) =
amplitude if R ( x1 ) =1. This occurs if Z ( x1 ) is infinite or equal to zero. If Z ( x1 ) is greater than 1, the
amplitude of the outgoing wave is larger than the amplitude of the ingoing wave. More generally, the
coefficient R can be defined everywhere in a fluid where an ingoing and an outgoing wave propagate in
opposite directions.
Absorption coefficient
The absorption coefficient a ( x1 ) is related to the reflection coefficient R ( x1 ) as follows
a ( x1 ) =1- R ( x1 )
2
The phase of R ( x1 ) 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
E * ( x1 )
a ( x1 ) =1E ( x1 )
where E ( x1 ) and E * ( x1 ) are the average energy flux through the plane x = x1 of the incident and the reflected
waves, respectively.
Definition and Symbols
Sound absorption coefficient at normal incidence a (w )
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 R (w )
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 Zs (w )
It is the ratio of the complex sound pressure P (w )
velocity V (w )
x=0
x=0
to the normal component of the complex sound particle
at an individual frequency in the reference plane ( x = 0 ).
Wave number k0
It is the variable defined by
w 2p f 2 p
k0 = =
=
c0
c0
l0
where
w is the angular frequency;
f is the frequency;
c0 is the speed of sound;
l0 is the wavelength.
NOTE: In general the wave number is complex, so k0 = Re ( k0 ) + j Im ( k0 )
where
Re ( k0 ) is the real component ( Re ( k0 ) = 2p / l0 );
Im ( k0 ) is the imaginary component which is the attenuation constant, in Nepers per metre.
Complex sound pressure P (w )
It is the Fourier transform of the temporal acoustic pressure p ( t )
Cross spectrum S12 (w )
It is the product P2 (w ) P1 (w ) , determined from the complex sound pressures P1 (w ) and P2 (w ) at two
microphone positions.
NOTE: * means the complex conjugate.
*
Auto spectrum S11 (w )
It is the product P1 (w ) P1 (w ) , determined from the complex sound pressure P1 (w ) at microphone position one.
NOTE: * means the complex conjugate.
*
Transfer function H12 (w )
It is the transfer function from microphone position one to two, defined by the complex ratio:
P2 (w ) S12 (w )
S (w )
S12 (w ) S22 (w )
or 22
, or
=
P1 (w ) S11 (w )
S21 (w )
S11 (w ) S21 (w )
Calibration factor H c (w )
It is the factor used to correct for amplitude and phase mismatches between the microphones.
Basic principle of measurements performed with an impedance tube
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, pseudo-random 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 normal-incidence complex reflection coefficient R (w ) , the normal-incidence
absorption coefficient a (w ) , and the surface impedance of the test material Z s (w ) .
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:
1. two-microphone method (using two microphones in fixed locations);
2. one-microphone method (using one microphone successively in two locations).
Technique 1 requires a pre-test or in-test 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.
1. 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.
2. Plane wave can be generated in a tube only if the excitation frequency is below the smallest acoustic
mode (cut-off frequency, see Figure 3) of the tube. This condition defines the upper working frequency
limit of this instrument.
Figure 3. Cut-off frequencies of a circular duct filled with air. Cut-off frequencies are evaluated using the following equation: f nm  nm
nm satisfy J m '  nm   0 , c is speed of sound in air (343 m/s) and d is diameter of the duct in meters.
c
where
d
3. 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 c0 , wavelength l0 and characteristic impedance Z 0
Before starting a measurement, the velocity of sound, c0 , 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:
T
c0 = 343.2
293
where T is the temperature, in Kelvin.
The wavelength then follows from:
c
l0 = 0
f
The density of the air, r 0 , can be calculated from
p T
r = r0 a 0
p0 T
where
T is the temperature, in Kelvin;
pa is the atmospheric pressure, in kPa;
T0 = 293 K;
p0 =101.325 kPa;
0  1.186 kg/m3.
The characteristic impedance Z 0 of the air is the product 0 c0 .
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 two-microphone 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 H12I (w ) and H12II (w ) , using the same mathematical
expressions for both. Place the microphones in configuration I (standard configuration, see Figure below) and
store the transfer function H12I (w ) . 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 H12II (w ) and compute the transfer function using equation:
H12 (w ) = H12I (w ) H12II (w ) = H12 e jf
Figure 4. 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), H12 (w ) can be computed using:
H12I (w )
H12 (w ) =
= H12 e jf
II
H 21 (w )
In Figure 5 are shown of transfer functions measured using Configuration I and Configuration II. Notice that
H12I   is P2 / P1  PB / PA whereas H12II   is P2 / P1  PA / PB and H 21II   is P1 / P2  PB / PA .
|H|
HI12
HII12
0
10
HII21
200
400
600
800
1000
1200
1400
1600
1800
2000
Frequency, Hz
 (H), rad
4
HI12
2
HII12
0
HII21
-2
-4
200
400
600
800
1000
1200
1400
1600
1800
2000
Frequency, Hz
Figure 5. 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 H12I (w ) and H12II (w ) .
Compute the calibration factor using the following expression
H12I (w )
= H c e jf
II
H12 (w )
or, if the analyzer is only able to measure transfer functions in one direction (e.g from microphone A to
microphone B), H c (w ) can be computed using:
H c (w ) =
H c (w ) = H12I (w ) H 21II (w ) = H c e jf
For subsequent tests, place the microphones in configuration I (standard configuration). Insert the test specimen
and measure the transfer function
Ĥ12 (w ) = Ĥ12 e jf̂ = Re Ĥ12 + j Im Ĥ12
( )
( )
where
Ĥ12 (w ) is the uncorrected transfer function and fˆ is the uncorrected phase angle;
Correct for mismatch in the microphone responses using the following equation:
H12 (w ) = H12 e jf =
Ĥ12 (w )
H c (w )
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.
0.01
|Hc|
10
0
10
200
400
600
800
1000
1200
1400
1600
1800
2000
1400
1600
1800
2000
Frequency, Hz
 (Hc), rad
4
2
0
-2
-4
200
400
600
800
1000
1200
Frequency, Hz
Figure 6. Correction Factor.
|H12|
0
10
HI12
HII12
H12 Mics interchanged
H12 Correction factor
200
400
600
800
1000
1200
1400
1600
1800
2000
1400
1600
1800
2000
Frequency, Hz
0
 (H12)
-1
HI12
-2
HII12
-3
H12 Mics interchanged
H12 Correction factor
-4
200
400
600
800
1000
1200
Frequency, Hz
Figure 7. Transfer functions corrections.
Determination of the reflection coefficient
Calculate the normal incidence reflection coefficient using the following expression:
H (w ) - H i 2 jk0 x1
R (w ) = R e jfR = 12
e
H r - H12 (w )
where
x1 is the distance between the sample and the further microphone location;
f R is the phase angle of the normal incidence reflection coefficient;
P
- jk x -x
H i = i2 = e 0 ( 1 2 ) = e- jk0s is the transfer function of the incident wave alone;
Pi1
P
jk x -x
H r = r 2 = e 0 ( 1 2 ) = e jk0s is the transfer function of the reflected wave alone;
Pr1
s = x1 - x2 is microphone spacing.
NOTE: Complex pressure at position 1, P1 (w ) , can be expressed as summation of the incident and reflected
waves at location x1, P1 (w ) = Pi e jko x1 + Pr e- jko x1 = Pi1 + Pr1 . Whereas the pressure at position 2, P2 (w ) ,can be
expressed as superposition of incident and reflected waves at location x2 , P2 (w ) = Pi e jko x2 + Pr e- jko x2 = Pi2 + Pr2 .
From expressions of P1 (w ) and P2 (w ) derivation of H i (w ) and H r (w ) is straightforward.
The reflected wave pressure amplitude Pr (w ) , can be written in terms of reflection coefficient as
Pr (w ) = R (w ) Pi (w ) .
The transfer function between two microphones is given by
P2 (w ) Pi e jk0 x2 + R (w ) Pi e- jk0 x2 e jk0 x2 + R (w ) e- jk0 x2
H12 =
=
=
P1 (w ) Pi e jk0 x1 + R (w ) Pi e- jk0 x1 e jk0 x1 + R (w ) e- jk0 x1
from which
H12 éëe jk0 x1 + R (w ) e- jk0 x1 ùû = e jk0 x2 + R (w ) e- jk0 x2
H12 R (w ) e- jk0 x1 - R (w ) e- jk0 x2 = e jk0 x2 - H12 e jk0 x1
(
e jk0 x1 e 0 ( 1 2 ) - H12
e jk0 x2 - H12 e jk0 x1
R (w ) =
=
H12 e- jk0 x1 - e- jk0 x2 e- jk0 x1 H12 - e jk0 ( x1-x2 )
- jk x -x
(
)
)
e- jk0s - H12 2 jk0 x1
e
H12 - e jk0s
Q.E.D. (quod erat demonstrandum [En: “which was to be demonstrated”]).
R (w ) =
Determination of the sound absorption coefficient
The normal incidence sound absorption coefficient is given by the following equation:
a (w ) =1- R
2
Reflection Coefficient, R
Absorption Coefficient, 
1
0.8
0.6
0.4
0.2
0
200
400
600
800
1000
1200
Frequency, Hz
1400
1600
1800
2000
400
600
800
1000
1200
Frequency, Hz
1400
1600
1800
2000
1
0.8
0.6
0.4
0.2
0
200
Figure 8. 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:
Z s (w ) Zs (w ) 1+ R
=
=
rc0
Z0
1- R
4
Re(Zs/Zair)
3
2
1
0
-1
200
400
600
800
1000
1200
Frequency, Hz
1400
1600
1800
2000
400
600
800
1000
1200
Frequency, Hz
1400
1600
1800
2000
1
Im(Zs/Zair)
0
-1
-2
-3
-4
200
Figure 9. Surface impedance ratio at normal incidence of a layer of porous foam of thickness d = 2.5 cm.
References
[1] British Standards, “Acoustics — Determination of sound absorption coefficient and impedance in
impedance tubes —Part 1: Method using standing wave ratio”, BS EN ISO 10534-1, 2001.
[2] British Standards, “Acoustics — Determination of sound absorption coefficient and impedance in
impedance tubes —Part 2: Transfer-function method”, BS EN ISO 10534-2, 2001.
[3] Allard, J.F. and Atalla, N., “Propagation of Sound in Porous Media: Modelling Sound Absorbing
Materials”, Second Edition, Wiley, 2009.
[4] Chung et al., “Transfer function method of measuring in-duct acoustic properties. I. Theory”, J. Acoust.
Soc. Am., 68, 907-913, 1980.
[5] Chung et al., “Transfer function method of measuring in-duct acoustic properties. II. Experiment”, J.
Acoust. Soc. Am., 68, 914-921, 1980.
[6] Utsuno et al., “Transfer function method for measuring characteristic impedance and propagation constant
of porous materials”, J. Acoust. Soc. Am., 86, 637-643, 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 Functions
2
10
H12 A-B
H12 B-A
1
H
12
10
0
10
-1
10
-2
10
Figure 10.
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
Transfer function recorded between microphones A-B and B-A, 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 = [];
rho = 1.21;
c = 343;
s = 0.1;
Zair = rho*c;
k = (2*pi*freq)/c;
x1 = ?;
% frequency vector (Hz)
% density of air (kg/m^3)
% speed of sound in air at 23 Celsius (m/s)
% microphone spacing (m)
% characteristic impedance of air (kg/m^2/s)
% wavenumber in air (m^-1)
% 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)./(1-R));
% Normalized Surface Impedance
Zs_n = ((1+R)./(1-R));
% 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
1
0.9
0.8
Reflection Coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 11.
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
Reflection coefficient. Measurement performed with a B&K impedance tube and using a sample of microlite 22 mm thick.
Absorption Coefficient
1
0.9
Absorption Coefficient
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 12.
0
200
400
600
800
1000
Frequency (Hz)
1200
1400
1600
Absorption coefficient. Measurement performed with a B&K impedance tube and using a sample of microlite 22 mm thick.