Acoustical Measurement and Instrumentation: Paper ICA2016-421

Development of a measurement method foroblique-incidence sound absorption coefficient using a

thin chamber

Naohisa Inoue(a), Tetsuya Sakuma(b)

(a)University of Tokyo, Japan, inoue@env-acoust.k.u-tokyo.ac.jp(b)University of Tokyo, Japan, sakuma@k.u-tokyo.ac.jp

Abstract

There have been a number of methods proposed for the measurement of oblique-incidencesound absorption coefficients. The objective of this paper is to present a novel method thatutilize the propagation mode expansion of two-dimensional acoustic field in a thin rectangularchamber. This paper is organized as follows. In the first, measurement principle is presented.One of the greatest problem in practical is to guarantee accuracy and efficiency of multipoint mea-surement of the complex pressures. Thus, secondly, an elaborate prototype of the measurementsystem is introduced. Thirdly, numerical simulation of the measurement demonstrates the validityof the proposed procedures to extract mode amplitude from measured complex pressures. Fi-nally, some examples of measured results are shown. In general, good agreement was observedbetween measured and theoretical values.

Keywords: Oblique incidence absorption coefficient, Measurement, Numerical simulation, Prop-agation mode expansion

Development of a measurement method foroblique-incidence sound absorption coefficient using

a thin chamber

1 IntroductionThere have been a number of methods proposed for measuring oblique-incidence sound ab-sorption coefficients. Most of them are performed in the free or semi-free field, and the meth-ods proposed in references [1,2] seem to be widespread with the development of digital mea-surement technique. On the other hand, Terao et. al. numerically demonstrated an alterna-tive method to measure oblique-incidence absorption coefficients in a thin duct [3]. There aremainly two advantages in this method: one is that required test chamber is considerably smallin comparison with the open space methods, the other is that this method does not involveeffects of diffracted wave from edges of a specimen in principle. Despite the theoretical ef-fectiveness of this method, practicality and viability as an actual measurement are not fullydiscussed.

The objective of this paper is to present an implementation of a measurement method based onthe propagation mode expansion of two-dimensional acoustic field in a thin rectangular cham-ber. Furthermore, its validity is examined both numerically and experimentally in comparisonwith theoretical results for infinite area material.

2 Measurement principle2.1 Analytical expression of 2D rectangular acoustic field

Consider a two-dimensional rectangular acoustic field depicted in Fig. 1. Complex sound pres-sure in the field is expressed as the superposition of propagation modes, which is derived fromthe plane wave solution of Helmholtz equation with rigid boundary conditions on the x = 0 andx =W . It follows that

p(x,y) =∞

∑n=0

[an exp(− jknyy)+bn exp( jkn

yy)]cos(knxx) (1)

where k is wave number in propagation direction of plane wave and knx and kn

y are n th or-der wave number in x and y direction respectively. They are defined as kn

x = nπ/W andkn

y =√

k2− kn2x . The n th propagation mode represents the plane wave that propagates in the

direction where the phase change along x axis becomes just nπ as depicted in Fig. 1. an andbn, respectively, are the complex amplitude of incidence and reflection plane wave of nth mode.Furthermore, incidence angle θn and oblique incidence absorption coefficient α in the angleare obtained for each propagation mode as follows.

θn = sin−1(

knx

k

)(2a) α( f ,θn) = 1−

∣∣∣∣bn

an

∣∣∣∣2 (2b)

2

Note that this method assumes that incidence and reflection angles correspond one-to-one, i.e.it cannot be applied to materials that induce scattering.

Rig

id B

oundar

y

θ1 θ 2

x

y

0th order mode 1st order mode 2nd order mode

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

0

+

-

0

00 W

k = 2π/(2W) =π/W 1x k = 2π/W2

x

Figure 1: Geometrical interpretation of propagation modes.

2.2 Extraction of mode amplitudes from complex sound pressure distributions

Consider the following equation obtained by multiplying Eq.(1) by mth mode distribution in x-direction and integrating over x-axis.∫ W

0p(x,y)cos(km

x x)dx =∞

∑n=0

{[an exp(− jkn

yy)+bn exp( jknyy)]

∫ W

0cos(kn

xx)cos(kmx x)dx

}(3)

With orthogonality of eigenfunction, the integral term in right hand side of Eq.(2) falls to zeroexcept for the case m = n, hence

pm(y) = am exp(− jkmy y)+bm exp( jkm

y y) (4)

where weighted-average complex sound pressure pm(y) is defined as

pm(y) = εm/W∫ W

0p(x,y)cos(km

x x)dx, (5)

and the mode normalization coefficient εm is 1 for m = 0 and 2 for m > 0. Accordingly, am

and bm can be deduced from complex sound pressures measured on two lines along x axis inrectangular chamber as follow.{

am

bm

}=

12 j sinkm

y ∆y

[exp( jkm

y y2) −exp( jkmy y1)

−exp(− jkmy y2) exp(− jkm

y y1)

]{pm(y1)pm(y2)

}(6)

3

3 A proto-type measurement systemFigure 2 shows a specification of a proto-type measurement system.

3.1 Specification of a chamber

As mentioned in the above, this measurement requires multi-point measurement of complexsound pressures in the chamber. Receivers are located at intervals of 2 centimeters in aline, so that there are 128 receivers in total. However, measurement is divided and performedin plural times due to limited numbers of microphones. Then, microphone placement shouldbe conducted as efficiently and accurately as possible in order to reduce influences of timevariance and uncertainty of receiving position. Therefore, measurement is conducted as ex-changing alternately the microphone holder and neighbor dummy lids.

In addition, material placement should be conducted carefully in order not to involve circum-ferential constraint or air-gap when setting a specimen in the chamber. Then, movable wall isinstalled behind the specimen to avoid pushing material unnecessarily. It is possible to measureup to about a total thickness of 100 mm.

Furthermore, source position can affect degree of excitation of each propagation mode. Thus,10 positions are available for the speaker, and an influence of the source position is partiallydiscussed in the section 4.

Figure 2: A geometrical specification of a proto-type chamber.

3.2 Measurable frequency range and incidence angle

The inner height of chamber defines upper limit frequency, fulim = c0/2H, below which theacoustic field in the thin chamber can be regarded as two-dimensional. In this proto-typesystem, the upper limit frequency is about 3,400 Hz.

4

Furthermore, the distance of two measurement lines causes another limitation. When the ydirection phase change along one line to the other line becomes π , i.e. km

y ∆y = π, incidenceand reflection waves cannot be decomposed. This frequency is expressed for each mth modeas

f mis =

c0

2

√(mW

)2+

(1

∆y

)2

. (7)

Although it seems efficient in terms of wider range of measurable frequency that f 0is is set as

equal to fulim, the distance between two lines is set as 8 centimeter in this proto-type chamber,i.e. f 0

is < fulim. This is because wider ∆y makes phase difference between two lines detectedvalidly to as lower frequency as possible in the actual measurement.

Figure 3 shows the relation between incidence angle and frequency in the proto-type chamber.It is important to note that incidence angle obtained in this method is ruled by Eq.(2a), not inarbitrary incidence angle.

Figure 3: Relation between frequency and incidence angle in the proto-type chamber.

3.3 Measurement of complex sound pressures

Impulse responses(IRs) are measured at each receiving point using 1.37-second log sweptsine signal with 8 times synchronous averaging. Subsequently, complex sound pressures arecalculated by discrete Fourier transform of the IRs. Note that spatial phase differences amongthe 128 receivers are taken into account by absolute time difference of IRs.

4 Numerical investigation on the proposed methodThe proposed method is examined by the finite element analysis [4] simulating the measure-ment in this section.

5

4.1 Extrapolation of boundary sound pressures

Sound pressures on the lateral rigid boundaries have relatively large impact on the accuracyof pm(y). However, in the proto-type chamber, the receiver cannot be placed on the x-directionwalls. Accordingly, this paper presents an extrapolation procedure to approximate the complexsound pressure on the boundary.

As depicted in Fig. 4, it is supposed that complex sound pressure p(x) varies quadraticallyin the region x0 ≤ x ≤ x2. Complex sound pressure on the boundary p(x0) is extrapolated bythe two values, p(x1) and p(x2), neighboring along x axis and the rigid boundary condition,∂ p(x)/∂x|x=x0 = 0, as follow.

p(x0) =43

p(x1)−13

p(x2) (8)

p(x) = c1x2 + c2x + c3

p(x1) : Known

p(x2) : Known

p(x0) : Unknown∂p

∂x x=x0

=0

: Known

x

p

x0 x1 x2

Δx Δx

Figure 4: Assumption for extrapolation of sound pressure on the rigid boundary.

4.2 Results and Discussion

Error analysis is performed by using glass wools with three different combinations of bulk den-sity and thickness: (ρb kg/m3, t mm) are (32, 25), (32, 50) and (96, 25). The complex bulkmodulus and the effective density within materials are evaluated by Kato model [5] with ma-terial density 2,400 kg/m3 and fiber diameter 7 µm. Configuration for the numerical analysisis depicted in the Fig. 5(a). In the following, pFEM denotes sound pressure calculated withthe finite element method, whereas pEx denotes sound pressure extrapolated by the aboveprocedure.

Figure 5(a) shows average relative errors ep of sound pressures pFEM and pEx at both edgesof two lines. Furthermore, figure 5(b) shows average relative errors eαθ

of oblique incidenceabsorption coefficients between theoretical and numerical values, where there are two patternsof numerical values regarding the treatment of the boundary values: one is calculated by usingpFEM and the other by pEx. The integral expression in Eq.(5) is numerically calculated by thetrapezoidal integration formula. Average relative errors, ep, tend to increase as the frequencygrows, whereas average relative errors, eαθ

, are almost same between two patterns of numeri-cal values except for those at 1.6 and 2.0 kHz. The amount of error would be permissible forthe absorption coefficient below 1.6 kHz. Furthermore, dependency of the errors on material

6

properties is generally small.

Figure 6(a) shows amplitudes of each mode at 1.25, 1.6 and 2.0 kHz. Obviously, 10 th modeis not excited in the results for 1.6 and 2.0 kHz. Oblique incidence absorption coefficients arenot identified correctly only at the unexcited modes (See Fig. 6(b)). Hence, increase of theaverage relative errors, eαθ

at high frequency range is mainly caused by the unexcited modes.It is found that in the 10th mode, the node of the mode coincides with the center axis ofthe speaker. Furthermore, even if the sound source has a certain size, phase mismatch stillremains between the piston-like vibration and the 10th mode shape. Schematic of this phe-nomenon is depicted in the Fig. 7. Through additional numerical analyses and measurements,it is confirmed that the 2nd, 6th, 14th modes are also not excited when placing the vibratingsurface at position 3. As a conclusion of this section, the proposed extrapolation is valid atleast until 2.0 kHz.

10-2

10-1

100

Av

erag

e R

elat

ive

Err

or

[%]

500 1000 2000Frequency [Hz]

(a)

GW32K 25mm GW32K 50mm GW96K 25mm

Evaluation Points

500 1000 200010-2

10-1

100

101

Frequency [Hz]

Av

erag

e R

elat

ive

Err

or

[%]

(b)

Boundary sound pressure is calculated with

(N : the num. of propagation modes)

FEM (directly)

Extrapolation

1.3src.

Figure 5: Average relative errors of (a) sound pressure on the boundary and (b) oblique-incidence absorption coefficients.

5 Experimental resultsMeasurement was performed by using glass wools with three different combinations of bulkdensity and thickness: (ρb kg/m3, t mm) are (32, 25), (32, 50) and (96, 25).

5.1 Normal incidence absorption coefficient

Figure 8 shows normal incidence absorption coefficients measured for the three different mate-rials, together with theoretical ones. Theoretical values are calculated with Kato model, whereparameters are estimated by visual curve fitting within typical ranges. Estimated parametersare also indicated in each figure.

Measured values fluctuate remarkably in the low frequency range. One likely reason is thatthe distance between two measurement lines is still smaller than the y direction wavelength,so that phase difference is not detected correctly. In addition, vibration of the chamber bodyis another error factor especially at its resonance frequencies. Fluctuation of measured values

7

0 10 20 30 40 50 60 70 80 90Propagation Angle [degree]

0 10 20 30 40 50 60 70 80 90Propagation Angle [degree]

0 10 20 30 40 50 60 70 80 900

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Propagation Angle [degree]

Obli

que−

Inc.

Abso

rp. C

oef

. []

10

15

20

25

30

35

40

45

50

Sound P

ress

ure

Lev

el [

dB

]

(a) 1250 Hz (a) 1600 Hz (a) 2000 Hz

(b) 1250 Hz (b) 1600 Hz (b) 2000 Hz

am

bm

Numerical

Theoretical

10 th mode 10 th mode

Figure 6: (a) Amplitudes for each mode and (b) oblique incidence absorption coefficientscalculated for glass wool with bulk density 96 kg/m3 and thickness 25 mm at 1.25, 1.6 and2.0 kHz. All values are calculated by FEM.

2nd

6th

10th

14th

Speaker Position

2 3 4 5 6 7 8 9 101

Material

Air Space in Chamber

0 1.3Spatial Distribution of Phase and Amplitude

mo

de

ord

er

node

Figure 7: Schematic of a chamber and eigen-mode shapes of 2nd, 6th, 10th and 14th mode.

becomes considerably smaller above 600 Hz except for around f 0is (See Eq.(7)).

Regarding the GW96K specimen, height of specimen was slightly larger than inner height ofthe chamber, which induced constraint of upper and bottom faces. For this reason, the theo-retical values based on the Biot’s poroelastic theory agree well with the measured values. Thisdemonstrates that the frame vibration greatly affects the results as a well-known problem in theimpedance tube measurement. For an analogy to the impedance tube measurement, there canbe great sensitivity to material cutting in order to attain the slip support condition where one

8

can measure acoustic characteristics that is equivalent to those for the infinite area material.Furthermore, this measurement should be carefully performed especially for materials with highflow resistivity.

5.2 Oblique incidence absorption coefficient

Figure 9 shows oblique incidence absorption coefficients measured for the same three speci-mens in the above. Theoretical values are calculated by using Kato model and the parametersestimated in the above.

In general, measured values for GW32K capture correctly the theoretical ones both quantita-tively and quantitatively. However, the remarkable discrepancies from the theoretical values arealso observed at the same frequency and incidence angle as in numerical results shown inthe previous section. Regarding the results for GW96K, differences between theoretical andmeasured values are relatively large, which is considered due to the frame vibration and edgeconstraint of the specimen. These results demonstrate the validity of the proposed measure-ment theory and implementations, although there remain some problems.

125 250 500 1000 2000 4000

Frequency [Hz]

125 250 500 1000 2000 4000

Frequency [Hz]

125 250 500 1000 2000 40000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Norm

al−

Inc.

Abso

rption C

oef

fici

ent [

]

Measured Theoretical (Rigid Frame) Theoretical (Elastic Frame)

Fiber Diameter : 7 μm Fiber Diameter : 9 μm

Fiber Diameter : 8 μm

Young s Modulus : 6.0×106 N/m2

Loss Factor : 0.4

’Estimated Parameter Estimated Parameter

Estimated Parameters

GW32K 25mm GW32K 50mm GW96K 25mm

Figure 8: Normal incidence absorption coefficient measured for three different material.

0 10 20 30 40 50 60 70 80 90Incidence angle [degree]

0 10 20 30 40 50 60 70 80 90

1

Incidence angle [degree]0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Incidence angle [degree]

Obli

que−

Inc.

Abso

rpti

on C

oef

fici

ent

[]

630 Hz500 Hz 800 Hz 1000 Hz 1250 Hz 1600 Hz 2000 Hz

Lines : Theoretical (Rigid Frame). Markers : Measured

GW32K 25mm GW32K 50mm GW96K 25mm

Figure 9: Oblique incidence absorption coefficient measured for three different material.

9

6 ConclusionsThis paper has described an implementation of a method for measuring oblique incidenceabsorption coefficient based on the propagation mode expansion of a rectangular acoustic fieldin 2-D. Numerical simulation by the finite element method confirmed the following remarks.

• Boundary sound pressures can be extrapolated from neighboring two values and the rigidboundary condition with enough accuracy for determining absorption coefficient.• Degree of excitation of some modes depends on the size of an excitation source as well

as its location.

Actual measurement is also performed and the following results are obtained.

• In general, measured values capture correctly the theoretical ones both quantitatively andquantitatively in the high frequency range.• In particular, normal incidence values fluctuate remarkably in the low frequency range,

and they may be unreliable at this stage.• There may be an analogy to the impedance tube measurement regarding the support

conditions of the specimen, and typical effects of the frame vibration are observed in theresults for GW96K.

Future work should be dedicated to investigation and interpretation of measured surface impedancesthat may include rich information on the errors clarified here.

Acknowledgements This project has been funded by the Grant-in-Aid for Challenging Ex-ploratory Research from Japan Spciety for the Promotion of Science(No. 15K14072).

References

[1] Kimura, K.; Yamamoto, K. A method for measuring oblique incidence absorption coefficientof absorptive panels by stretched pulse technique., Applied Acoustics, Volume 62, 2001,617-632.

[2] Allard, J. F.; Bourdier, R.; Bruneau, A. M.; The measurement of acoustic impedance atoblique incidence with two microphone, Journal of Sound and Vibration, Volume 101(1),1985, 130-132.

[3] Terao, M.; Sekine, H; Ohkawa, S. A basic study on an induct measurement method ofoblique incidence reflection factors of a surface., Proceedings of Spring Meeting of Acous-tical Society of Japan, 2001, pp 765-766. (in Japanese)

[4] Sakuma, T.; Sakamoto, S; Otsuru. T; Editors. Computational Simulation in Architectural andEnvironmental Acoustics, Springer Japan, 2014.

[5] Kato, D.; (2008) Predictive model of sound propagation in porous material: Extension ofapplicability in Kato model. Journal of Acoustical Society of Japan, Volume 64, 2008, 339-347. (in Japanese)

10