A measurement technique of sound absorption coefficient and impedance
using an impedance tube and two cardioid microphones
Kazuma HOSHI(1) , Toshiki HANYU(2)(1) Nihon University, Japan, hoshi.kazuma@nihon-u.ac.jp(2) Nihon University, Japan, hanyu.toshiki@nihon-u.ac.jp

Abstract
Two omnidirectional microphones and an acoustic tube are usually used to measure the surface normal impedance
and absorption coefficient of materials. This measurement technique is specified in ISO10354-2, JIS1405-2,
and so on. In this paper, we tried to obtain the surface normal impedance of material using two cardioid microphones in the tube. As a result, it was clarified that the ratio of sound pressure and velocity effect on the valueof impedance measurement. Therefore when the ratio is equalized, the values of impedance and absorption
coefficient can be obtained as with the values using two omnidirectional microphones.
Keywords: Cardioid microphone, Absorption coefficient, Surface normal impedance

1INTRODUCTION

The measurement method of normal incident impedance in a tube is specified ISO10354-2[1], and this method
has some limitations caused by examined in a laboratory. Recently, our needs are increasing yearly for obtaining
absorption coefficients of mounted materials in in-situ. Accordingly, several methods for measuring in a freefield[2, 3] or a random incident noise field[4, 5] have been proposed, and these methods were reviewed[6].
On the other hand, we are focusing on the sensor for acoustic impedance measurement. The most major
technique is known as using two omnidirectional microphones (called P-P sensor) and the transfer function ofobtained from the receiving signals. However, the distance of these microphones affects measurement precision.
A specific acoustic impedance at the measurement position can be directly obtained by a P-U sensor which has
an omnidirectional microphone and a sound velocity sensor[7]. Humidity significantly affects, however, on thevalue of sound velocity measured by the P-U sensor[?]. Therefore the P-U sensor is too delicate and hard totreat in in situ conditions because the calibration procedure must be done before the measurement daily.
Hence, we have been trying to use two cardioid microphones for in-situ impedance measurement[9, 10, 11], thistechnique, what is called C-C method[12, 13], is based to pick out sound pressure and velocity. In this study,
we propose how to measure an acoustic impedance and absorption coefficient using the cardioid microphones
and acoustic impedance tube, and report the measurement results with the proposed method.

2ESTIMATION METHOD OF THE SURFACE IMPEDANCE OF THE MATERIAL BY
SOUND PRESSURE AND VELOCITY IN A TUBE

Jacobsen et al. have proposed the estimation technique of the surface impedance, which is calculated from thevalues of sound pressure and velocity using the P-U sensor[14].
Let us consider a acoutic tube like shown in Figure 1. When the sound source which has complex amplitude A
is exposed from a loudspeaker located x = 0, sound pressure p(x) and x direction velocity ux (x) are describedas follows:

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x0L
xlSpecimen

Sound Source

Figure 1. An acoustic impedance tube for pressure and sound velocity measurement at x.
p(x) = A e− jkx + r e jk(x−2L) ,A  − jkxux (x) =e− r e jk(x−2L) ,ρc(1)(2)

where r is the reflection coefficient, ρc is a specific impedance of the air at the position x. Thus, the specificimpedance Z(x) at x is given by
Z(x) =p(x)1 + r e2 jk(x−L)= ρc.
ux (x)
1 − r e2 jk(x−L)(3)

Therefore, when l is defined as l = L − x, the reflection coefficient r is expressed asr=Z(x) − ρc 2 jkle .
Z(x) + ρc(4)Finally, The surface impedance Z(L) and absorption coefficint α can be obtained as follows:1+r,
1−r
α = 1 − |r|2 .

Z(L) = ρc(5)(6)

These equations indicate that the surface impedance can be obtained from both values of specific impedanceZ(x) and the distance from the sensor to the material. This method was verified by Jacobsen et al. using P-U
regular sensor (Microflown Co. Ltd.) which was calibrated in an anechoic room[14].

3ESTIMATION METHOD OF THE SURFACE IMPEDANCE OF THE MATERIAL USING CARDIOIDMICROPHONE(S)In this section, we propose how to measure acoustic impedance and absorption coefficient of material using one
or two cardioid microphone(s) and acoustic impedance tube.
3.1 Measuring twice with a cardioid microphone
The measurement condition is shown in Figure 2. The signal M + (t) is received at x with a cardioid microphonewhich heads the directivity heads for the specimen, and the signal M − (t) is done with it which heads thedirectivity for the loudspeaker. The receiving signals consist of both sound pressure and sound velocity, so thatreceiving signals M + and M − are described with complex constant α, β :M + (x) = α p(x) − β ρc ux (x),−M (x) = α p(x) + β ρc ux (x).

1374(7)(8)

Thus, the specific impedance at x equals
Z(x) =p(x)M+ + M− β= −· · ρc.
ux (x) M − M + α(9)Hence, we can get the surface impedance and absorption coefficient from Eqs. (4), (5), (6), and (9).

x0Lxl

M–M+Specimen

Sound Source

Figure 2. Acoustic impedance measurement schematic diagram using one cardioid microphone and recording
twice at x.

4EFFECT OF THE BARANCE OF BOTH SOUND PRESSURE AND VELOCITY ON
THE SPECIFIC IMPEDANCE

The previous section, we can obtain a sound pressure and velocity from one or two cardioid micrphone, and canalso obtain the specific impedance at measureing position. However, the real microphone has several detecitive
barance of the sound pressure and velocity. In this section, when we use one cardioid microhne, the effect ofthe ratio β /α in Equaiton (9) is examined numerically.
4.1 Effect of the barance of both sound pressure and velocity included in a cardioid microhoneThe even barance of both pressure and velocity mixing siganl can be obtaine dby an ideal cardioid microphone.
The ideal microphone has the ratio β /α = 1/2+0 j. Where j is imaginary unit. When we use the ideal cardioid
microphone, the measured specific impedance Zt (x) is given by
Zt (x) =M + (x) + M − (x)ρc.
M − (x) − M + (x)(10)

Therefore, when we define the specific impedance Zm (x) measured by a real cardioid microhone, the relation ofZm (x) and Zt (x) is expressed as:Zm (x) =βZt (x).
α(11)

Equation (16) express that the ratio of both sound pressure and vecity included in a cardioid micphone gives
the error to the specific impedance.
When, We express ε = β /α, rt and rm are respectively given by
Zt (x) − ρc 2 jkl
e ,Zt (x) + ρc
εZt (x) − ρc 2 jkl
rm =e .
εZt (x) + ρc
rt =(12)
(13)

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Z(x)/ρc
50REF.
2 deg.
5 deg.

510 2
10Frequency [Hz]103

101.0Z(L)/ρc5α0.6

00.4

REF.
2 deg.
5 deg.

−5−10102

0.8

Absorption coefficientSpecific impedance ratio [-]10Surface impedance (ratio) [-]4.2 Numerical examination for revealing the effect of the retio of sound pressure and velocity
The numerical examination of specific impedance measurement using Delany-Bazlay model[15]. The simulated
porus material has 32 kg/m3 , 80,000 Ns/m4 flow registance and 50 mm thick. In this exmamination, the error
factors (1) the ratio of amplitute between pressure and velocity, (2) the difference of phase (degree.) is given
to receiving signals. Additionally, (3) the distance l from the specimen to receiving postion aloso assumes tobecome the error factor, l is changed stepwisely.
The examnination ressults of phace difference are shown in Figure 3. A little bit of error in Z(x) can be seenin the left figure, however the errors become bigger in surface impdedance Z(L) and absorption coefficient α.
When the periodical peaks shown in Z(x) are made to flat to calculate Z(L). In this procedeure, the error
could make bigger. So that, These errors has periodical flactuation not to depend on the phase difference. The
exmamination results of amplitude difference are shown in Figure 4. These errors also become bigger in surface

Frequency [Hz]103REF.
2 deg.
5 deg.

0.2

0.0 2
10Frequency [Hz]103

Z(x)/ρc50REF.
0.5 dB
1.0 dB

510 2
10Frequency [Hz]

103

10

1.0Z(L)/ρc5

α

0.6

00.4

REF.
0.5 dB
1.0 dB−5−101020.8

Absorption coefficientSpecific impedance ratio [-]10Surface impedance (ratio) [-]Figure 3. Effect of phase difference of sound pressure p and velocity u inherent in a cardioid macrophone on
specific impedance at x, surface impedane and absorption coefficient (l = 36 cm).

Frequency [Hz]

103

REF.
0.5 dB
1.0 dB

0.2

0.0 2
10

Frequency [Hz]103

5Z(x)/ρc0REF.
l = 1 cml = 24 cm
l = 36 cm−5−10102

Frequency [Hz]103

1.010Z(L)/ρc
50.8

Absorption coefficientSpecific impedance ratio [-]10Surface impedance (ra io) [-]Figure 4. Effect of amplitude difference of pressure p and sound velocity u inherent in a cardioid microphoneon specific impedance at x, surface impedane and absorption coefficient (l = 36 cm).

α0.6

0REF.
l = 1 cml = 24 cm
l = 36 cm−5−10 210Frequency [Hz]103

0.4REF.
l = 1 cml = 24 cm
l = 36 cm

0.2

0.0 2
10Frequency [Hz]103

Figure 5. Effect of distance between measurement position and material on specific impedance at x, surface
impedane and absorption coefficient when phase difference of pressure p and sound velocity u inherent in a
cardioid microphone is 2 degrees.

1376impdedance Z(L) and absorption coefficient α. The exmamination results also has periodical flactuation and thisperoid is same as the previous examination of phase difference. The examination results of the difference ofdistance from receiving position to the specimen are shown in Figure ??. The longer distance, the periodof flactuation becomes longer. Hence, the flacuation depends on the distance, which is only 1 cm from thespecimen, the distance affects on the measument result. This error occur not only using cardioid microphone,
but also using P-U sensor, because the difference between pressure and velocity causes this error. This problem
has been also indicated by Jacobsen[14].

5ESTIMATION METHOD OF THE COMPLEX RATIO OF PRESSURE TO VELOCITY INHERENT IN A CARDIOID MICROPHONEIn previous section, the numerical experiments clarified that the ratio of pressure and velocity inherent in acardioid microhone affects on the specific impedance measurement in a tube. Therefore, The ratio called ε haveto be measued before a impedance measurement. In this section, the mesurement technique of ε of a cardioidmicrohone using an impedance tube will be discussed.
When a rigid boundary is installed at x = L, the Eqs. (1) and (2) becomep(x) = A e− jkx + e jk(x−2L) ,(14)A
e− jkx − e jk(x−2L) ,(15)
ux (x) =ρcso that, the ideal specific impedance Zt (x) also becomes
Zt (x) =p(x)cos(kl)= − j ρc.
ux (x)
sin(kl)(16)

On the other hand, when the measured specific impedance Zm (x) using one cardioid microphone in a tube,
theese impedances Zt (x) and Zm (x) are corresponded ideally, hence, the ε is given from Eqs. (9) and (16),ε=(M − − M + ) cos(kl)β= −j.
α(M + + M − ) sin(kl)(17)

Equaiton (22) tells us that we can measure ε of a cardioid microphne using a impedance tube, however, thisequaiton consist of the ratio cos to sin, so that the upper limit frequency is decided by l < λ /4.

6EXAMINATION FOR VERIFICATION

The possivility of mesurement about the surface impedance using both a cardioid microphone recorded twice
and an impedance tube is verified.
6.1 conditions for surface impedance measurement
A impedance tube made from 15 mm thick MDF wooden boards is ready for measurement. The section diagram is drawn in Figure 6. A grasswool that has 50 mm thick, 32 kg/m3 density is ready for specimen.
The measurement technique which uses one cardioid microphone and receiving signals twice is adopted. Two
different manifucture’s cardioid microhpnes are ready for the measurement. One is ISOMAX II-C (CountrymanCo. Ltd.), the other is one 4188 (DPA). Two ISOMAX II-C and one 4188 are ready for comparison. The samepinknoise is exposed twice from the loudspeaker for one minute respectively. The directivity of cardioid microhone heads for the specimen and head for the loudspeaker in each recording with 48 kHz sampling rate. The
distance l from receiving position to the specimen is 24 cm and 36 cm for the possibility of measurement at
any position. The measuremnt using two omnidirecitional microphone is also examined for obtaining reference
data with transfer function method.

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SpecimenSound Source
M–M+x [mm]0450 570 810
240360Figure 6. Section diagram of an acoustic impedance tube and measurement positions for verifying tests.
6.2 Measuring the ratio of puressure to velocity inherent in cardioid microhone
Before impedance measurement, the ratios β /α = ε of pressure to velocity inherent in the three cardiod microphones are measured using the same impedance tube and rigid surface end. In these measurements, the distancel of microphone to rigid wall is ajusted 1.5 cm and 24 cm respectively. The same pinknoise are exposed fromthe loudspeaker for 1 minitute and we recorded twice when the directivity of the microphone heads to the loudspeaker and to the rigid wall at each position. The ε is calculated step by 16384 data (8192 data overlapping)
and averaged with Eq. (17). We chose the ε measured with l = 24 cm under 240 Hz, and that measure withl = 1.5 more than 240 Hz. The phase values of ε for 240 to 800 Hz are not enable to be obtained suitablevalue, these values are generated through liner interpolation.
6.3 Analysis procedure
After recording for impedance mesurement, we conducted following calclulations. Due to check the effect of ε,the specific impedance Z(x) at x is calculated two patterns. One is no correcting with ε, Z(x)is calculated asZ(x) = ρc(M + + M − )/(M − − M + ). The other is corrected with ε, Z(x) is calculated as Z(x) = (ρc/ε) · (M + +M − )/(M − − M + ). The Z(x) is calculated step by 16384 data (8192 data overlapping) and averaged in the both
calculations procedure. Reflection coefficent r and absorpotion coefficient α are calculated with Eqs. (8) and(9).
6.4 ResultsThe result of l = 24 cm shows in Figure 7 and that of l = 36 cm shows in Figure 8. The upper graph showssurface impedance and the lower graph shows absorption coeffient in each figure. Focusing on (a) originalof absorption coefficient in Figure 7 and 8, we can see the periodical flactuation in measured any cardioid
microphone. The period of flactuation is decided by the distance l and the phase of fractuation differ frommicrophone type. These flactuations occur from the ratio of pressure to velocity inherent in cardioid microphoneown. The errors result in microphone type, and do that the indivitual difference is very small.
Hence, let us focus on the results of absorption coefficient (b) corrected by ε. The all of values measured bycardioid microphone correspond to measured by P-P method (transfer function method) at all frequencies. These
results clearly show that the amplitude and phase both of corrected cardioid microphone can use for impedance
measurement. It was founded that the ε is unique value in each microphone because the ε can use in anydistance l.

7CONCLUSION

We trid to measure surface impedance of a material using one or two cardioid microhone and a impedance tube.
As a result, we found that the cardioid microphone has the error of balance between pressure and velocity, and1378

(a) original(b) corrected

Figure 7. Measurement results of surface impedance and absorption coefficient in an impedance tube
(l = 24 cm). The results (a) directedly calculated from receiving signals. The results (b) derived
from corrected data by ε(a) original(b) corrected

Figure 8. Measurement results of surface impedance and absorption coefficient in an impedance
tube (l = 36 cm). The results (a) directedly calculated from receiving signals. The results (b) arecalculated from corrected data by ε1379

if the error is corrected we can measure the impedance with the cardioid microphone in the tube. In thistrial we chose the technique using one cardioid microphone and recoding twice. Next, we will try to use two
cardioid microhone and recording simaltaneously.

ACKNOWLEDGEMENTS
This work supported by JSPS KAKENHI Grant Number 17K06689, Grant-in-Aid for Scientific Research(C).

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