Determination of Diffuse Field Sound Absorption from a ...

Determination of Diffuse Field Sound Absorption from a Normal Incidence Impedance Measurement

University of Kentucky

June 24, 2021

David HerrinUniversity of Kentucky

The Vibro-Acoustics Consortium

June 24, 2021

Overview

2

• Normal and Diffuse Absorption Coefficient Normal Incidence Absorption – ASTM E1050 Diffuse Field Absorption – ASTM C423

• Sample Size for ASTM C423• Diffusiveness in the Chamber• Sensitivity Studies


June 24, 2021

Normal Incident Absorption ASTM E1050

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𝐴𝐵

Reflection coefficient

𝑅𝐵𝐴

𝑍 1𝑍 1

Normal-incidence absorption

𝛼 1 𝑅

𝑍 𝑅 𝑖𝑋

Characteristic impedance

Sound source


June 24, 2021

Normal Incidence Absorption Variation

Cox and D’Antonio, 2017 adapted from Horoshenkov et al., 2007

Reconstituted Porous Rubber

Glass Fiber

Reticulated Foam

Solid line indicates the mean sound absorption. Error bars indicate the 95% confidence limit in any one laboratory based on round robin tests.

• ASTM E1050 is very repeatable.• 𝛼 is always less than 1.0.

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Diffuse Field Absorption ASTM C423

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𝛼 0.9210𝑉𝑑𝑐𝑆

Sound source

Material of area 𝑆

𝑉: Volume of reverberation room𝑐: speed of sound𝑑: decay rate, dB/s


June 24, 2021

Diffuse Field Absorption

Broadband sourceAbsorbing material of area 𝑆

𝑇55.3𝑉

𝑐 𝛼 𝑆 𝑆 𝛼 𝑆 4𝑉𝑚

𝑆 Sample area𝑆 Room surface area𝛼 Average absorption coefficient of empty room𝑚 Air attenuation𝑉 Volume

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June 24, 2021

Diffuse Field Absorption Variation

100 mm mineral wool

25 mm foam

Solid line indicates the mean sound absorption. Error bars indicate the 95% confidence limit in any one laboratory measurement on round robin tests (13 laboratories).

• ASTM C423 is not as repeatable from to room to room.

• May exceed 1.0.

Cox and D’Antonio, 2017 adapted from Horoshenkov et al., 2007

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June 24, 2021

Absorption Greater than 1.0

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For a grazing wave, sound power is absorbed but there is technically no “incident power”. The absorbed power increases when the sample has a finite size.


June 24, 2021

ASTM C423

Section 5.3 Diffraction effects usually cause the apparent area of a specimen to be greater than its geometrical area, thereby increasing the coefficients measured according to this test method. When the test specimen is highly absorptive, these values may exceed unity.

Section 5.4 Regardless of the differences and the necessity for judgment, coefficients measured by the test method have been used successfully by architects and consultants in the acoustical design of architectural spaces.

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Discrepancy Edge Effect

𝛼 𝛼 𝛽𝐸

𝐸 ratio of edges to area of sample

𝛼 random incidence absorption coefficient of infinitely large sample (E=0)

Sauro, 2010

𝛽 constant that depends on frequency, the absorption, and the room.


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Reverberation Room

Heller, 2013

Wallace Sabine at Harvard University


June 24, 2021

Discrepancy Between

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𝜃

Average for diffusive incidence:

5 cm thick mineral wool(2.4 m x 2.4 m sample)

0

0.2

0.4

0.6

0.8

1

1.2

100 1000

Abso

rptio

n

Frequency (Hz)

Normal Incident Absorption

Averaged for Infinite Panel

ASTM C423

𝛼 𝜃4𝑅 cos𝜃

1 𝑅 cos𝜃 𝑋 cos𝜃

𝛼 2 𝛼 𝜃 sin𝜃 cos𝜃 𝑑𝜃


June 24, 2021

Overview

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• Normal and Diffuse Absorption Coefficient ASTM E1050 ASTM C423



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Radiation Impedance

𝑥

𝑦

𝑧𝑝

𝑥,𝑦

𝑥 ,𝑦

Integral on area :𝑆

𝑝 𝑥,𝑦, 0𝑍

𝑍 𝑍 2𝑝 𝑥,𝑦, 0

~𝑍

𝑍2𝑝 𝑝

𝑅 𝑍 : Radiation Impedance𝑆

Thomasson, 1980

𝑍 is a function of the geometry of the absorber, frequency, and angle of incidence.


June 24, 2021

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Radiation Impedance

𝑥

𝑦

𝑧

𝜑

𝜃

Incident wave Assuming a diffusive field

𝛼 8 Re 𝑍sin 𝜃𝑍 𝑍

/

𝑑𝜃

ZR …0.27+0.55i 0.27+0.55i … 0.24+0.52i

0.79+0.65i 0.77+0.66i … 0.51+0.68i

… … … … …

1.00+0.01i 1.04+0.02i … 3.37+3.41i

𝜃 0 𝜃 𝜋/6 𝜃 𝜋/2

𝑓 63 Hz

𝑓 125 Hz

𝑓 4000 Hz

ZR of rectangular sample (2.44 m x 2.74 m)


June 24, 2021

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Size Correction

𝑒

𝑒

5 cm mineral wool sample

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

100 1000

Diff

use

Fiel

d Ab

sorp

tion

Frequency (Hz)

e = 3.6 m Measuremente = 3.6 m Predictione = 2.4 m Measuremente = 2.4 m Predictione = 1.2 m Measuremente = 1.2 m PredictionNormal Absorption


June 24, 2021

Overview

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Diffuse Field

Diffuse Field means• Sound pressure is uniform inside reverberation chamber• Sound intensity distribution is uniform over all possible

directions

Redstone Arsenal – U.S. Army Technical University of Denmark


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Sound Intensity Distribution

𝑥

𝑦

𝑧

𝜑

𝜃

Incident wave

Jeong, 2010Kang, 2000

Simulation method by Jeong (2010) used beam tracing method.

𝜃: Incident angle𝜑: Azimuthal angle

: Point source: Observation point


June 24, 2021

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Sound Intensity Distribution

Incident angle 𝜃

Nor

mal

ized

inte

nsity

Irregular room

Incident angle 𝜃N

orm

aliz

ed in

tens

ity

Rectangular room

The high frequency intensity is nearly independent of the room geometry, the location of the absorber and the absorption coefficient of the specimen. – Jeong, 2010


June 24, 2021

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Intensity Distribution Correction

𝑒

𝑒

𝑒 2.4 m

5 cm mineral wool sample

0

0.2

0.4

0.6

0.8

1

1.2

100 1000

Abso

rptio

n

Frequency (Hz)

Normal Incident Absorption

ASTM C423

Size Corrected

Size and Angle Corrected for Irregular Room

Size and Angle Corrected for Rectangular Room

Jeong, 2010


June 24, 2021

Overview

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June 24, 2021

ProcedureNormal Incidence Impedance

Sound source

Material of area S

Diffuse Field Absorption

1. Input layered sound absorber properties (thickness, flow resistivity, etc.).

2. Use empirical models and transfer matrix theory to determine normal incidence impedance (𝑍 ) and normal incidence sound absorption (𝛼 ) in 1/6 octave bands.

3. Use normal incidence impedance and sample size to determine diffuse field sound absorption (𝛼 ).

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June 24, 2021

Normal Incidence vs. Diffuse Field

0

0.2

0.4

0.6

0.8

1

1.2

1.4

100 1000

Abso

rptio

n C

oeffi

cien

t

Frequency (Hz)

Normal Incidence

Diffuse Field

10 cm (4 in) plastic foam𝜎 5000 Rayls/m

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June 24, 2021

ASTM C423 Effect of Sample Size

ASTM C423• 6.69 m2 (72 ft2) is recommended. • Sample size 2.44 m

2.74 m (8 ft 9 ft). • Area may not be less than 5.57 m2

(60 ft2).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

100 1000

Abso

prtio

n C

oeffi

cien

t

Frequency (Hz)

1.2 m X 1.1 m 2.3 m X 2.3 m2.4 m X 2.4 m2.5 m X 2.5 m2.6 m X 2.6 m3.5 m X 3.5 m

10 cm (4 in) plastic foam𝜎 20,000 Rayls/m

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June 24, 2021

ASTM C423 Effect of Flow Resistivity

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

100 1000

Abso

rptio

n C

oeffi

cien

t

Frequency (Hz)

5000 rayls/m

10000 rayls/m

20000 rayls/m

30000 rayls/m

10 cm (4 in) plastic foam

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June 24, 2021

ASTM C423 Effect of Mass Layer

0

0.2

0.4

0.6

0.8

1

100 1000

Abso

rptio

n C

oeffi

cien

t

Frequency (Hz)

Foam (Normal Incidence)Foam (Diffuse Field)Foil+Foam (Normal Incidence)Foil+Foam (Diffuse Field)

Foil + FoamFoam Thickness: 2.54 cm (1 inch)Foam Flow Resistivity: 30000 rayls/mFoil Surface Density: 0.05 kg/m2

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June 24, 2021

References

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• ASTM C423-09a, 2009, Standard Test Method for Sound Absorption and Sound Absorption Coefficients by the Reverberation Room Method.

• ASTM E1050-12, 2012, Standard Test Method for Impedance and Absorption of Acoustical Material Using a Tube, Two Microphones and a Digital Frequency Analysis System.

• Thomasson, S. I. (1980). On the Absorption Coefficient. Acta Acustica, 44(4), 265-273.

• Thomasson, S. I. (1982). Theory and Experiments on the Sound Absorption as Function of the Area. Report No. TRITA-TAK 8201, Department of Technical Acoustics, KTH, Stockholm, Sweden.

• Jeong, C. H. (2010). Non-Uniform Sound Intensity Distributions when Measuring Absorption Coefficients in Reverberation Chambers using a Phased Beam Tracing. The Journal of the Acoustical Society of America, 127(6), 3560-3568.

• Kang, H. J., Ih, J. G., Kim, J. S., and Kim, H. S. (2000). Prediction of Sound Transmission Loss through Multilayered Panels by using Gaussian Distribution of Directional Incident Energy. The Journal of the Acoustical Society of America,107(3), 1413-1420.

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