ME661 Lecture Notes ImpedanceTube Part1 Chapter 4

8/11/2019 ME661 Lecture Notes ImpedanceTube Part1 Chapter 4

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THECATHOLICUNIVERSITYOFAMERICASchool 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, 2012http://mason.gmu.edu/~dturo/collaborations/CUA_Lecturer_ME_661.html

CUA
http://mason.gmu.edu/~dturo/collaborations/CUA_Lecturer_ME_661.htmlhttp://mason.gmu.edu/~dturo/collaborations/CUA_Lecturer_ME_661.htmlhttp://mason.gmu.edu/~dturo/collaborations/CUA_Lecturer_ME_661.html


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Impedance tubepart 1BASIC 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................................................................................. 5Reflection coefficient ............................................................................................................................................................................................... 5

Absorption coefficient ............................................................................................................................................................................................ 5

DEFINITION AND SYMBOLS ............................................................................................................................................................ 5

SOUND ABSORPTION COEFFICIENT AT NORMAL INCIDENCEw( ) .............................................................................................................. 5

SOUND PRESSURE REFLECTION COEFFICIENT AT NORMAL INCIDENCE Rw( )............................................................................................ 5

NORMAL SURFACE IMPEDANCE Zs w( ) ............................................................................................................................................................. 6

WAVE NUMBER k0

................................................................................................................................................................................................. 6

COMPLEX SOUND PRESSURE Pw( ) .................................................................................................................................................................... 6

CROSS SPECTRUM S12 w

( ) ................................................................................................................................................................................... 6AUTO SPECTRUM S

11 w( ) ..................................................................................................................................................................................... 6

TRANSFER FUNCTION H12

w( ) ........................................................................................................................................................................... 6

CALIBRATION FACTOR Hc w( ) ........................................................................................................................................................................... 6

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 Z0

........................................ 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 ......................................................................................................................................................................................... 15REFLECTION AND ABSORPTION COEFFICIENTS AND SURFACE IMPEDANCE MEASUREMENTS ................................................................. 15

PLOTS ...................................................................................................................................................................................................................... 17


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Basic theory

Acoustic characteristic and surface impedancesThe 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):

,

,s

p x tZ

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).

0

0

,

,

j t kx

s cj t kx

p x t j cA eZ c Z

v x t j A e

The product of the fluid density by the speed of sound in that fluid, 0c , 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( )=Aej kx+w t( )

v x, t( ) =A

Zce

j kx+w t( )

The pressure and the velocity, for a wave propagating toward the negative abscissa are, respectively,

p* x, t( )=A*e

j kx+w t( )

v* x, t( )=A*

Zce

j kx+w t( )

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

pT x, t( )=Aej kx+ t( )

+A*e

j kx+ t( )

vT x, t( )=Ae

j kx+w t( ) A

*ej kx+w t( )

Zc

A superposition of several waves of the same 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.

Impedance variation along a direction of propagation

InFigure 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( )=pT x2, t( )

vT x2, t( )=Zc

Aej kx2+w t( )

+A*ej kx2+w t( )

Aej kx2+w t( )

A*ej kx2+w t( )

= ZcAe

j kx2( )+A*e

j kx2( )

Aej kx2( )

A*ej kx2( )


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Figure 1.

Layer of fluid.

Whereas at1

x , the impedance 1Z x can be written

Z x1( )=

pT x1, t( )

vT x1, t( )=Zc

Aej kx1( )

+A*ej kx1( )

Aej kx1( )

A*ej kx1( )

From the above equations one can evaluate the following expression

A

A*=

Z x2( ) Zc

Z x2( )+Z

c

e 2j kx2( )

which gives

Z x1( )=Zc

jZ x2( )cot kd( )+ZcZ x

2( ) jZc cot kd( )

where dis equal to x2 x

1. 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 x1at the surface of the layer of fluid 2 is obtained from

Z x1( )= lim

Z x2( )Z

c

jZ x2( )cot kd( )+ZcZ x2( )

jZc cot kd( )

=Z

c

jZ x2( )cot kd( )

Z x2( )= jZ

ccot kd

( )

where Zcis the characteristic impedance and kthe 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.


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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 nextlayer 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

R x1( )=p* x1, t( )p x1, t( )

This coefficient does not depend on tbecause the numerator and the denominator have the same dependenceon t. Using previous equations, the reflection coefficient R x

1( ) can be written as

R x1( )=

Z x1( ) Zc1

Z x1( )+

Zc1

where Zc1

is the characteristic impedance in fluid 1. The ingoing and outgoing waves at x1 have the same

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, thecoefficient R can be defined everywhere in a fluid where an ingoing and an outgoing wave propagate in

opposite directions.

Absorption coefficient

The absorption coefficient x1( )is related to the reflection coefficient R x1( ) as follows

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

x1( ) =1E

*x1( )

E x1( )

where E x1( ) and E

* x1( )are the average energy flux through the plane x =x1of the incident and the reflected

waves, respectively.

Definition and Symbols

Sound absorption coefficient at normal incidencew( )

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 Rw( )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.


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Normal surface impedance Zs w( )

It is the ratio of the complex sound pressure Pw( )x=0

to the normal component of the complex sound particle

velocity V w( )x=0

at an individual frequency in the reference plane (x = 0 ).

Wave number k0

It is the variable defined by

k0= wc0

= 2p

fc0

= 2p

l

0

where

w

is the angular frequency;

fis the frequency;

c0is the speed of sound;

l

0is the wavelength.

NOTE: In general the wave number is complex, so k0=Re k

0( )+j Im k0( ) where

Re k0( ) is the real component ( Re k0( )= 2p / l 0 );

Im k0( )is the imaginary component which is the attenuation constant, in Nepers per metre.

Complex sound pressure Pw( )

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 P

1 w( ) and P2 w( ) at two

microphone positions.

NOTE: * means the complex conjugate.

Auto spectrum S11

( )

It is the product P1 w( )P1 w( )

*, determined from the complex sound pressure P

1 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( )

P1 w

( )=

S12

w( )

S11 w

( )or

S22 w( )

S21 w

( ), or

S12 w( )

S11 w

( )

S22 w( )

S21 w

( )

Calibration factor Hc w( ) It is the factor used to correct for amplitude and phase mismatches between the microphones.

Basic principle of measurements performed with an impedance tubeAn 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, orchirp), and the sound pressures are measured at two locations near to the sample (preferably less than 3 times


7/17

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 Rw( ), the normal-incidence

absorption coefficientw( ), and the surface impedance of the test material Zs w( ) .

The quantities are determined as functions of the frequency with a frequency resolution which is determinedfrom 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 bemeasured 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 wavenumberof 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, seeFigure 3)of the tube. This condition defines the upper working frequencylimit of this instrument.

Figure 3.Cut-off frequencies of a circular duct filled with air. Cut-off frequencies are evaluated using the following equation: nm nm

cf

d

where

nm satisfy ' 0m nmJ ,

c is speed of sound in air (343 m/s) and dis diameter of the duct in meters.

3. Microphones spacing defines both upper and lower working frequencies of the tube. Microphonespacing 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).


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Preliminary tests

Determination of the speed of sound c0, wavelength

l 0and 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:

c0= 343.2

T

293

where T is the temperature, in Kelvin.

The wavelength then follows from:

l 0=c0

f

The density of the air,r 0

, can be calculated from

r = r0

pa

p0

T0

T

whereT is the temperature, in Kelvin;

pa is the atmospheric pressure, in kPa;

T0= 293K;

p0=101.325kPa;

0 1.186 kg/m

3.

The characteristic impedance Z0of the air is the product 0 0c .

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, asmeasured 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 mismatchWhen 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 testspecimen. 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 H12

Iw( ) and H12

IIw( ) , 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:


9/17

H12

w( ) = H12I

w( )H12II

w( )= H12 ej

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:

H12

w( )=H12

Iw

( )H

21

IIw( )

= H12 ej

InFigure 5 are shown of transfer functions measured using Configuration I and Configuration II. Notice that

12I

H is2 1

/ /B A

P P P P whereas 12II

H is2 1

/ /A B

P P P P and 21II

H is1 2

/ /B A

P P P P .


10/17

Figure 5.

Transfer functions measured with different configurations.

Calibration factorThe 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 H12

Iw( ) and H12

IIw( ) .

Compute the calibration factor using the following expression

Hc w

( )=

H12

Iw( )

H12IIw

( )=

Hc e

j

or, if the analyzer is only able to measure transfer functions in one direction (e.g from microphone A to

microphone B), Hc w( ) can be computed using:

Hc w( ) = H12I

w( )H21II

w( )= Hc ej

For subsequent tests, place the microphones in configuration I (standard configuration). Insert the test specimen

and measure the transfer function

H12

w( ) = H12 ej =Re H12( )+j Im H12( )

where

H12

( ) is the uncorrected transfer function and

is the uncorrected phase angle;Correct for mismatch in the microphone responses using the following equation:

H12 w( )=H12 ej

=

H12 w( )

Hc w( ) In Figure 6 is plotted a calibration factor evaluated using the data shown in Figure 5. InFigure 7 is insteadshown a corrected transfer function using both microphone interchange and the calibration factor techniques.

200 400 600 800 1000 1200 1400 1600 1800 2000

100

Frequency, Hz

|H|

200 400 600 800 1000 1200 1400 1600 1800 2000-4

-2

0

2

4

Frequency, Hz

(H),rad

H12I

H12II

H21II

H12I

H12II

H21II


11/17

Figure 6.

Correction Factor.

Figure 7.Transfer functions corrections.

200 400 600 800 1000 1200 1400 1600 1800 2000

100

100.01

Frequency, Hz

|Hc

|

200 400 600 800 1000 1200 1400 1600 1800 2000-4

-2

0

2

4

Frequency, Hz

(H

c),rad

200 400 600 800 1000 1200 1400 1600 1800 2000

100

Frequency, Hz

|H12

|

200 400 600 800 1000 1200 1400 1600 1800 2000-4

-3

-2

-1

0

Frequency, Hz

(H

12)

H12I

H12II

H12

Mics interchanged

H12

Correction factor

H12I

H12II

H12

Mics interchanged

H12

Correction factor


12/17

Determination of the reflection coefficientCalculate the normal incidence reflection coefficient using the following expression:

Rw( )= R e

j R=

H12 w( ) Hi

Hr H12 w( )e2jk0x1

where

x1is the distance between the sample and the further microphone location;

R

is the phase angle of the normal incidence reflection coefficient;

Hi=Pi2

Pi1= e

jk0 x1 x2( )= e jk0s is the transfer function of the incident wave alone;

Hr=Pr2

Pr1= e

jk0 x1 x2( )= ejk0s is the transfer function of the reflected wave alone;

s =x1 x

2is 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( )=Piejkox1

+Pre jkox1

=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( )=Piejkox2

+Pre jkox2

=Pi2+Pr2

From expressions of P1 w

( ) and P2 w

( ) derivation of Hi w

( ) and Hr w

( )is straightforward.The reflected wave pressure amplitude Pr w( ), can be written in terms of reflection coefficient as

Pr w( )=Rw( )Pi w( ).The transfer function between two microphones is given by

H12=

P2 w( )

P1 w( )

=

Piejk0x2

+Rw( )Pie jk0x2

Piejk0x1

+Rw( )Pie jk0x1

=

ejk0x2 +Rw( ) e jk0x2

ejk0x1 +Rw( ) e jk0x1

from which

H12

ejk0x1 +R w( )e jk0x1

=ejk0x2 +R w( ) e

jk0x2

H12

R w( )

e jk0x1 R w( )

e jk0x2 =ejk0x2 H12

ejk0x1

R w( )=ejk0x2 H

12ejk0x1

H12

e jk0x1 e jk0x2=

ejk0x1 e jk0 x1 x2( )

H12( )

e jk0x1 H12 e

jk0 x1 x2( )( )

R w( )=e jk0s H

12

H12 ejk0s

e2jk0x1

Q.E.D. (quod erat demonstrandum [En: which was to be demonstrated]).

Determination of the sound absorption coefficientThe normal incidence sound absorption coefficient is given by the following equation:

w( )=1 R2


13/17

Figure 8.Reflection and absorption coefficients of a layer of porous foam of thickness d = 2.5 cm.

Determination of the acoustic surface impedance ratioThe acoustic surface impedance ratio is the surface impedance normalized respect to the characteristic

impedance of the air:

Zs w( )r c0

=

Zs w( )Z0

=1+R

1 R

Figure 9.Surface impedance ratio at normal incidence of a layer of porous foam of thickness d = 2.5 cm.

200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Frequency, Hz

ReflectionCoefficient,R

200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Frequency, Hz

AbsorptionCoefficient,

200 400 600 800 1000 1200 1400 1600 1800 2000-1

0

1

2

3

4

Frequency, Hz

Re(Zs/Z

air

)

200 400 600 800 1000 1200 1400 1600 1800 2000-4

-3

-2

-1

0

1

Frequency, Hz

Im(Zs/Z

air

)


14/17

References[1] British Standards, Acoustics Determination of sound absorption coefficient and impedance in

impedance tubesPart 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 tubesPart 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.


15/17

Matlab codes

Transfer functionsHere 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.

Figure 10. 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 = []; % 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 coefficientR = ( 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

0 200 400 600 800 1000 1200 1400 160010

-2

10-1

100

101

102

Transfer Functions

Frequency (Hz)

H12

H12

A-B

H12

B-A


16/17

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 ImpedanceReal part','FontSize',12)

xlabel('Frequency (Hz)')grid on

subplot(2,1,2)

plot(freq,imag(Zs),'b','LineWidth',2)xlim([0 1600])

title('Surface ImpedanceImaginary 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 RatioReal 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 RatioImaginary part','FontSize',12)xlabel('Frequency (Hz)')

grid on


17/17

Plots

Figure 11. Reflection coefficient. Measurement performed with a B&K impedance tube and using a sample of microlite 22 mm thick.

Figure 12. Absorption coefficient. Measurement performed with a B&K impedance tube and using a sample of microlite 22 mm thick.

0 200 400 600 800 1000 1200 1400 16000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Reflection Coefficient

Frequency (Hz)

ReflectionCoefficien

t

0 200 400 600 800 1000 1200 1400 16000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Absorption Coefficient

Frequency (Hz)

AbsorptionCoefficient

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