Greater than Unity
Old unpublished paper, written circa 1998 as an attmpt to rationalize
the phenomenon of absorption lab results for 3D absorber matereials presenting
absorption coefficient values greater that unity. Might make interesting
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CORRECTIONS FOR RANDOM INCIDENCE
COEFFICIENTS REPORTED AS GREATER THAN UNITY.
Angelo J. Campanella
3201 Ridgewood Drive
Hilliard, OH 43026 - 2453
In treating rooms for excess reverberation, smaller patches of sound absorbing material result in faster sound decay for a given
total area of material.
Prediction of decay ratedependsonlaboratory data on that material acquired in laboratory rooms which generally differ
from the room to be treated. There
are two different forms of laboratory absorption coefficient, one the result of Sabine coefficient (SC) found
by decay measurements in laboratory
rooms, the other the sound power absorption coefficient (SPAC) observed in a steady sound field in a
room. This paper attempts to consolidate
these two for engineering application of sound absorbing materials.times, t, then determine the requisite acoustical absorption
A' to be added from the
Sabine relation that t = 0.161*V/A, as
A' = 0.161*V*( 1/t'- 1/t ).
The reverberation times t and t' of concern are for sound ranging in frequency from 125 Hz to 4,000 Hz. The requisite value
to be added, A', is determined
as S*a, where S is the treatment material area and a is its SC
obtained at each frequency
by the reverberation room test method. The SC values were obtained with an absorber panel 9'x8' in size.
al. extended the diffraction analysis of Levitas and Lax, giving values of the random incidence absorption coefficient
as a function of frequency
and and conductance, g. His results are shown here on an extended log-log scale using his Eq. (1). The abscissa
appropriate to the ASTM C 423 standard specimen size of 2.7mx2.4m (9'x8') with extension to smaller panels of 1/2,
1/4 and 1/8 times these
original dimensions. Reactivity affects a as exemplified by the dotted
g = 0.1 for a reactivity of b=+0.2. More complete reactivity
plots are provided
that Northwood's diffraction model did not account for all of the excess observed on smaller pathches.
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There is reason to believe
that diffraction theory predicts only the SPAC as discussed by Eyring and Young among others. Young
assigns the name "sound power
absorption coefficient" (SPAC Here), ì, obtained from the SC, a, in a large test room having mostly reflective walls, i.e.
aSs = Sr*ln[1-(Swìw + Stìs)/Sr],
Guernsey re-interpreted ASTM C 423 reverberation room data
with the Eyring relation,
t = -(24V/c)ln(10)/[S ln(1-ì)].
A slightly lower
absorption coefficient, called ìs here, resulted.
Insight into scaling parameters requires that the diffraction and edge effects in relation to normal incidence measurements be better
understood. ASTM C423 SC results for
125 Hz to 4,000 Hz material on a standard plastic foam
specimen from a previous interlaboratory test series (round robin) were used to predict SPAC values, ìc, using the
size parameters of the Northwood formulation.
These are listed in Table 1B, column F (SC). These
are corrected by Eq.(1A) to provide the measured SPAC, ìs and listed
in column G.
Normal incidence impedance tube
data on the same material from the round robin results
are shown in Table 1, columns A through E. These data were used to predict SPAC values, ìc, using the g and b formulation
provided by Northwood. These results are shown
as, ìsp, column H. The ìt excess over a' is greater than predicted by diffraction alone, ìc.
For the round robin plastic foam specimen, at 2,000
Hz the at=1.07 value resulted in ìs = 1.01.
Kosten has shown that the normal incidence data must be corrected for off-normal absorption, which is often greater than
normal absorption. Applying this correction,
with a better estimate based on the reactive part of
the conductance brings full agreement.
The predicted SPAC, ìc, is
less than the measured SPAC, ìsm. Fig. 2 and  indicate that a high value of ìc can occur either by
the applicable value of g (found with the one-dimensional
impedance tube) being underestimated or by the size
of the specimen tested in the reverberation room being
overestimated. Another possibility isthat
higher absorption can occur at frequencies around 500-1,000
Hz since the specimen was segmented comprising a series
of 12" square and disconnected panels each of which can act independently. Higher absorption can also occur in this freqency
range since these squares were not cemented to the
test room floor. The 3-dmensional nature
of the specimen, where added conductance volume is availble
around the perimeter of the specimen, has been the more popular
rationale for this rationale, often called an "edge
effect" which is proportional to the perimeter of the
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Table 1A: Predicted a"c (via Normal Incidence
Data) vs measured a".
Freq. C 384 Round Robin Average
Results Pred. 3-D
a' R/rc X/rc g
b Fig.1 Pred
125 0.047 2.57
-14.0 0.013 0.069 0.11
250 0.091 1.32
-7.17 0.025 0.135 0.21
500 0.259 1.1
-3.52 0.081 0.259 0.53
1000 0.643 1.1
-1.53 0.31 0.431 0.84
2000 0.952 1.26
-0.40 0.722 0.227 0.97
4000 0.871 1.96
0.142 0.508 -0.04 0.97 0.93
Table 1A: Predicted a"c (via diffraction)
vs measured a".
(SC) (SPAC) Effect
500 0.65 "
0.62 " 0.09
1000 1.04 "
0.93 " 0.09
2000 1.07 "
1.01 " 0.04
4000 1.04 "
0.98 " 0.01
2: Predicted ìc vs panel size , Ss(ft^2, m^2)
³ Frequency º ¬ x(9,0.4)³ « x(18,1.7)³1x(72,6.7)³2x(288,27)³
³ 125 º 0.11*
³ 0.11* ³ 0.11*
³ 0.11 ³
³ 250 º 0.22*
³ 0.22* ³ 0.21
³ 0.20 ³
³ 500 º 0.60*
³ 0.60 ³ 0.53
³ 0.47 ³
³ 1000 º 1.29
³ 1.03 ³ 0.84
³ 0.74 ³
³ 2000 º 1.23
³ 1.06 ³ 0.97
³ 0.93 ³
³ 4000 º 1.07
³ 1.00 ³ 0.97
³ 0.95 ³
³ * panel is less than one half wavelength wide
Panel ìc values can be determined
with Fig. 1 according to panel size, admittance, g
and conductance, b. See Table 2. The installed performance of panels can be predicted with the Eyring relation
(2) for relatively large treatment areas, e.g. sound
where ìs' represents the added material
and material panel size. This is solved for the
area, Sa, of added material panels of power absorption coefficient ìc:
= S(1-ìs)(1-e-[0.161V(1/t'-1/t)/S])/ìc (MKS)
= S(1-ìs)(1-e-[0.049V(1/t'-1/t)/S])/ìc (US)
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Suppose for instance that
the reverberation time of a 50'x50'x16' room is found
to be t=2 sec at 2,000 Hz, that t' should be reduced to 1.5 sec for the room use intended and that 4'x4' panels of sound absorber material
of a" = 1.07 measured at 2,000 Hz by the ASTM C 423
method are available. Eq. (1) would require that
A'=305 square feet, or that 19.1 panels 4'x4' be installed.
With the proposed method,
Eq. (3) determines that as=.113 for the existing room.
Eq. (4) corrects a" to a"c = 1.01. By Fig. 2 or Table 2, the applicable value of a"c for a 4'x4' panel is found to be 1.08
for half-sized panels. Eq. (4) predicts that
only 263 square feet, or 16.5 panels are required,
indicating a reduction of 14% or 2.6 fewer panels.
Diffraction enhancement diminishes when the panels are less than one wavelength wide. To be fully effective, separate panels should
be installed with separation between them.
When there is no space between panels (completely
covered walls), the limiting value of a" becomes 0.96 for any
Sabine observed that absorbers
distributed as patches about a room have more effect
than when concentrated in one contiguous area.
Northwood and others modeled the excess absorption as a diffraction phenomenon. His relation, Fig. 2, indicates that for a good
absorber, e.g. moderately dense fiberglass which is
2" or more thick, and at a frequency of 500 Hz, the
C 423 standard specimen size can achieve a reported a" of 1.2.
??? Rewrite this section.....
Eyring accounted for an enhancement
effect that occurs when the room surfaces are covered
with absorption over a significant percentage of available
Bartel showed that the ratio
of the absorber perimeter to absorber area can be related
to the excess absorption, but that relation is frequency dependent. He also concluded that Northwood's method is reasonably
Engineering calculations for
room absorption treatment areas with small absorber
panels are more precise when the Eyring and Northwood corrections
are applied to provide a value of a" appropriate
to the absorber panel size and spacing.
- Page 5 -
 ASTM C 423,
"Standard Test Method for Sound Absorption and Sound Absorption Coefficients by the Reverberation Room Method".
 W.C. Sabine, "Collected Papers
on Acoustics", p 23-25, 224-226
 W. C. Sabine, op cit, pp. 22-23
[y] D.A. Bies, "Some Notes on Sabine
Rooms", Acoustics Australia, (23) 3, (1995),
 ASTM C384, "Standard Method for
Impedance and Absorption of Acoustical Materials by
the Impedance Tube Method".
 P. M. Morse, "Vibrations &
Sound, McGraw-Hill, N.Y., 1948, p 388
 C. F. Eyring, JASA Jan., 1930,
 T. D. Northwood, M. T. Grisau
and M. A. Medcof, JASA (31), 1959,
 T. D. Northwood, JASA (35) 8,
1963, p 1174
 T. F. W. Embleton, JASA 50, (1970),
 W. B. Joyce, JASA (58) 3, 1975, pp
 T. W. Bartel, JASA 69(4), April, 1981,
 ASTM Committee E33 on Environmental
Acoustics Research Report Number RR:E33-1006 on Standing
Wave and Two-Microphone Impedance Tube Round Robin
Test Program. James Haines, Manville Sales Corp. R & E Center, Denver,
CO, February, 1989.
 J. C. Haines, private communication,
 R. M. Guernsey, private communication,
 Young, JASA (somewhere).
 Morse's last graph
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