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Absorption 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    Old unpublished paper, written circa 1998 as an attmpt to rationalize the phenomenon of absorption lab results for 3D absorber matereials presenting absorption coefficient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 reading....

 
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    CORRECTIONS FOR RANDOM INCIDENCE SOUND ABSORPTION
    COEFFICIENTS REPORTED AS GREATER THAN UNITY.

 Angelo J. Campanella
 Campanella Associates
 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 ).                      (1)

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[1].  The SC values were obtained with an absorber panel 9'x8' in size.
    Northwood, et al.[7] 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 includes frequencies appropriate to the ASTM C 423[1] 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 curves around g = 0.1 for a reactivity of b=+0.2.  More complete reactivity plots are provided in [8].
    Bartel[11] found 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[15] 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. where ìw<<ìs:

   aSs = Sr*ln[1-(Swìw + Stìs)/Sr],


   Guernsey[14] re-interpreted ASTM C 423 reverberation room data with the Eyring relation,

        t = -(24V/c)ln(10)/[S ln(1-ì)].                          (2)

A slightly lower absorption coefficient, called ìs here, resulted.

  Recent Results

   Insight into scaling parameters requires that the diffraction and edge effects in relation to normal incidence measurements be better understood.   ASTM C423 SC results[13] for 125 Hz to 4,000 Hz material on a standard plastic foam specimen from a previous interlaboratory test series (round robin)[12] were used to predict SPAC values, ìc, using the size parameters of the Northwood formulation[8].  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[8].  These results are shown as, ìsp, column H.  The ìt excess over a' is greater than predicted by diffraction[8] 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 [8] 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 specimen.

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  Table 1A: Predicted a"c (via Normal Incidence Data) vs measured a".               ³

  Freq.  C 384 Round Robin Average Results    Pred.    3-D 
   Hz      a'   R/rc   X/rc      g       b     Fig.1   Pred
                                               ìsp    Kosten 
   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    0.45
  1000  0.643   1.1    -1.53   0.31    0.431   0.84    0.80
  2000  0.952   1.26   -0.40   0.722   0.227   0.97    0.95
  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".

  Freq   C423_a"    Abs.Power   Net
   Hz  Rev.Rm[13]     ìs        Edge
          (SC)      (SPAC)     Effect

   125  0.05+.03    0.05+.03   -0.06
   250  0.23+.02    0.22+.02   +0.01
   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


           ³            Table 2: Predicted ìc vs panel size , Ss(ft^2, m^2)     ³
           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          ³

Practical Application
   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 recording studios:

   -S(ln[1-(ìs+ìs')]-ln[1-ìs]) = .161V(1/t'-1/t)

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:

        Sa = S(1-ìs)(1-e-[0.161V(1/t'-1/t)/S])/ìc  (MKS)          (4)
        Sa = 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 material[16].


                                    Summary:

    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 surface area.
    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 accurate.

                                  Conclusion:

    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.


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References

[1]   ASTM C 423, "Standard Test Method for Sound Absorption and Sound Absorption Coefficients by the Reverberation Room Method".

[2]   W.C. Sabine, "Collected Papers on Acoustics", p 23-25, 224-226

[3]   W. C. Sabine, op cit, pp. 22-23

[y]   D.A. Bies, "Some Notes on Sabine Rooms", Acoustics Australia, (23) 3,  (1995), pp 97-103.

[4]   ASTM C384, "Standard Method for Impedance and Absorption of Acoustical Materials by the Impedance Tube Method".

[5]   P. M. Morse, "Vibrations & Sound, McGraw-Hill, N.Y., 1948, p 388

[6]   C. F. Eyring, JASA Jan., 1930, pp 217-241

[7]   T. D. Northwood, M. T. Grisau and M. A. Medcof, JASA (31), 1959,       pp 595-599

[8]   T. D. Northwood, JASA (35) 8, 1963, p 1174

[9]   T. F. W. Embleton, JASA 50, (1970), pp 801-811

[10]  W. B. Joyce, JASA (58) 3, 1975, pp 643-655

[11]  T. W. Bartel, JASA 69(4), April, 1981, pp 1065-1074

[12]  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.

[13]  J. C. Haines, private communication, May, 1995

[14]  R. M. Guernsey, private communication, May, 1995

[15]  Young, JASA (somewhere).

[16] Morse's last graph

===========================================

    If you have acoustical or noise control design problems you would like to discuss, feel free to call or FAX us any time.

Campanella Associates
3201 Ridgewood Drive
Columbus, Ohio 43026

PHONE  6 1 4 - 8 7 6 - 5 1 0 8 & FAX   //  Cell 6 1 4 - 5 6 0 - 0 5 1 9
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For more information on classroom acoustics...
Angelo J. Campanella, P.E., Ph.D. (Principal)
a.campanella@att.net


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Last updated 24-Dec-2014

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