Flow resistance information for acoustical design

AppliedAcoustics 13 (1980)357-391

FLOW

RESISTANCE

INFORMATION DESIGN

FOR

ACOUSTICAL

D. A. BIES and C. H. HANSEN

Department of Mechanical Engineering, The University of Adelaide, Adelaide, South Australia (Australia) (Received: 10 July, 1979)

SUMMARY

The measurement and use oJ flow resistance information for the calculation of the acoustical properties of porous materials has been examined. It is demonstrated that flow resistance information provides a description of a porous material wh&h is sufficient to characterise its acoustical performance for all common applications. Design charts are presented which facilitate the use of flow resistance information for the three most common applications: (1) the control of the reverberant soundfieM in an enclosure, (2) the improvement of the transmission loss through pipe wrappings and enclosure walls and (3) the attenuation of sound propagating in ducts. Measured values of flow resistance for fibrous and foamed flroducts available in Australia are presented. Information is also presented for estimating the flow resistance of fibrous products of generally uniform fibre diameter. Alternatively, a means for the measurement of flow resistance is described.

INTRODUCTION

The use of porous materials for the absorption of sound is well known. However, what is not well known is that flow resistanCe information about a wide class of porous materials is sufficient to completely describe their acoustical performance. The class of materials which may be so described includes all generally homogeneous fibrous and open cell foamed products. Thus, while the concept of flow resistance is very old and well recognised on the one hand, and frequent reference is made to its use in the literature on the other, no systematic review and assembly of its uses has previously been attempted. Perhaps this has been the case because of the existence of various gaps in information which have prevented such an approach. The purpose of this paper is to present a systematic review of the use of 357 Applied Acoustics 0003-682X/80/0013-0357/$02"25 Printed in Great Britain

© Applied Science Publishers Ltd, England, 1980

358

D . A . BIES, C. H. HANSEN

flow resistance information and to demonstrate the utility of its use for the solution of a wide range of acoustical problems. As an aside, all information is presented in dimensionless form and the parametric dependence of flow resistance on various fundamental measurable parameters is given. Thus, in all applications described in this paper, air at standard room conditions could, in principle, be replaced with any gas at any temperature and pressure. The basis for modelling and for the design of unusual applications is thus established. It is suggested that the work presented here might ultimately prove useful in resolving the very old problem of the difference between the calculated statistical absorption coefficient and the value measured by standard procedures in a reverberant room. Finally, the direction to be taken to extend the work to very high sound pressure levels into what the acoustician calls the non-linear range of behaviour is hinted at in the formalism. However, these and other aspects of the problem are left for future work.

STEADY FLOW THROUGH POROUS MEDIA

Flow resistance and resistivity If a constant differential pressure is imposed across a layer of bulk porous material of open cell structure, a steady flow of gas will be induced through the material. Experimental investigation has shown that, for a wide range of materials, the differential pressure, p, and the induced normal velocity, v, of the gas at the surface of the material are linearly related provided that the normal velocity is small. If the normal velocity is not small, then, for fibrous relatively rigid structured materials, the dependence of the differential pressure on induced normal velocity is quadratic. Experimentation also shows that, for a small velocity, the ratio of the differential pressure to the normal velocity is always linearly related to the thickness, l, of the layer of porous material in the direction of induced flow. These facts and dimensional considerations suggest the following general equation between the pressure differential, p, and the normal velocity, v: 1

P = f~ (Pm/Pf)(lah'/d2 ) + f2 (P,n/Pf)(pIv2/d)

(1)

Further consideration of eqn. (1) is instructive. The observation that the differential pressure, p, is linearly related to the thickness, l, of the porous layer for fixed normal velocity suggests that the relationship between p and v is determined by the average properties of a unit thick layer of the material and this, in turn, suggests that the unit layer be one fibre diameter, d, thick. The empirical functions,f~ and f2, then describe the average properties of a layer of porous material one unit thick. This, in turn, suggests that these functions must be dependent upon the dimensionless ratio of the bulk density divided by the fibre density, p,,./pf. This quantity is approximately the fraction of blocked area in a unit area one fibre diameter thick. The dependence shown in eqn. (1) on the quantities viscosity, p, and

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

359

gas density, p, follows from dimensional considerations. The dependence of the first term on viscosity and fibre diameter has been verified experimentally. The dependence of the second term upon density and fibre diameter is not yet well established; it is at present speculative. The non-linear flow resistance of a porous material i thick is defined as the ratio of the differential pressure, p, divided by the normal velocity, v. The non-linear flow resistance, R,, may be written in dimensionless form by introducing the real quantities, gas density, p, and speed of sound, c, and using eqn. (1) as:

R,,/pc =fl(pm/py)(#l/d2pc) + f2(p,,,/p:)(lv/dc)

(2)

The linear flow resistance, R, is defined using eqn. (2) as the limiting value as the normal velocity, v, tends to zero. 2 Thus, the flow resistance expressed in dimensionless form is:

R/pc = fl (P,,/P/)(#I/d2p c)

(3)

Finally, the flow resistivity, R 1, is defined as the flow resistance per unit thickness, or, in dimensionless form, as:

Rid/pc = f~ (p,/py)(#/dpc)

(4)

We see, as an aside, that flow resistance scales with the reciprocal of the Reynold's number (dpc/l~). The function, f l , may be determined empirically. For example, the following equation holds for fibre glass products: 2

Rld2p~ 1.53 = K

(5)

The constant K has the value 3.18 x 10 - 9 when the fibre diameter, d, is in metres, the bulk density, Pro, is in kg/m 3 and the flow resistivity, R~, is in MKS rayls/m. We proceed to rewrite eqn. (5) in dimensionless form as:

(Rid/pc) = K2(p,,,/py)153(l~/dpc)

(6)

where: K 2 = K(plf'53/Ig )

=

27"3

In determining the value of K 2 we have taken the fibre density pf = 2-5 and the viscosity of air /a = 1-84 x 10 -5 poiseuille at standard conditions. Comparison of eqns (4) and (6) shows that, for fibre glass products: x 103 k g / m 3

fl(P,./Pf) = K2(p,n/Pf) 1"53

(7)

In this paper we will be concerned with the linear flow resistance, R, and the related linear quantity, flow resistivity, R 1. We may safely neglect the second velocity dependent term of eqn. (2) in all of our further considerations, as this term will be quite negligible at ordinary sound pressure levels. Although the discussion

360

D . A . BIES, C. H. HANSEN

has been concerned with steady pressures and induced flow velocities, the formalism adequately describes the ordinary acoustic case of fluctuating pressures and velocities provided that the corresponding sound pressures and particle velocities are not very large. Thus far the discussion has been concerned with fibrous materials characterised by a fibre diameter, d. If the material is fairly rigid then eqn. (1) provides a reasonable description of the relationship between differential pressure and normal velocity. If the material is fairly soft then the flow resistance is probably still reasonably well described by eqn. (3) but the form off~ is different for the hard and soft cases. If the material is not fibrous but is, for example, an open cell foam, then an equivalent fibre diameter can probably be found and eqn. (3) will still hold for the case of semi-rigid and soft foams as well.

Apparatus for the measurement of flow resistance An apparatus suitable for measuring the flow resistance of porous materials which meets the requirements of American Standard ASTM C522-73 is illustrated in Fig. 1. The specimen holder shown in the figure has been designed so that it also serves as the specimen cutter and, when care is taken in cutting a sample of semi-rigid material for test, this arrangement works quite well. If, however, the material is of very low density it is sometimes necessary to prevent leakage of air around the edge of the sample by coating the cutter with silicon grease. If the material is soft it is sometimes necessary to contain the sample between perforated plates of large percentage open area. Alternatively, if the material is rigid and perhaps brittle, a suitable sharp, toothless cutter driven with a drill press may be used to cut the sample. In use, the specimen holder (1) with its porous material specimen (2) is placed in the apparatus (4) as illustrated in Fig. 1. An O-ring seal (3) prevents leakage around the holder. The funnel shape of the apparatus with a gradual taper of only four degrees ensures uniform flow of air through the sample. The pressure on the upstream side of the sample is measured using a barocell (11) and a digital manometer (12) attached to the test apparatus by a tube (5). The barocell and digital manometer are capable of measuring differential pressure with a resolution of 0.014N/m 2. As the smallest pressure which we measured was 1-3N/m 2, the resolution of the measuring apparatus is quite adequate. Air under pressure is introduced at the throat of the conical horn of the apparatus (4). Flow into the apparatus is measured using an air flow meter (7). Flow rates ranging between 5 x 10 -4 and 5 x 10-2m/s have been measured with this apparatus with a resolution of __+2 per cent. In use, control valve (6), which is used to control the flow rate through the sample, is choked. In this way the pressure through the flow rate meter (7) is held constant at the pressure indicated by the manometer (8). The air pressure within the flow meter was held at 310 kN/m 2 using the regulator (9). The pressure of the air supply (10) varied between 4000 and 6000 kN/m 2. For the 102 mm diameter test specimens tested using the apparatus illustrated in

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

361

--@ @

® 580

®

---®

Fig. 1. Flow resistance measuring apparatus. (1) Sample holder and cutter. (2) Porous material sample.

(3) O-ring seal. (4) Conical tube to ensure uniform air flow through the sample. (5) Tube. (6) Valve. (7) Flow meter. (8) Manometer. (9) Pressure regulator. (10) Air supply. (11) Barocell. (12) Electronic manometer. Fig. 1, the following relationship between measured quantities and sample flow resistivity holds:

R 1 = 1.18pA/ml

(8)

In eqn. (8) R 1 = flow resistivity ( M K S rayls/m), p = differential pressure (N/m2), m = air mass flow rate (kg/s), l = specimen thickness (m) and A = specimen crosssectional area (m2).

Measured values o f flow resistivity The range of fibrous materials available in Australia has been tested with the results shown in Figs 2, 3 and 4 and Table 1. Some measured values for f o a m e d

362

D . A . BIES, C. H. HANSEN

I

ACI

]

i

1

I

i

I

i

]

FI BREGLASS

105

E

o xX

•

O

10z,

6+ Z cZ X

o

10 3 10

1

Bulk

Fig. 2,

Symbol +

o []

!

V /x

X

I

I

density

I

I

I

I

I

I

100 D (kg/m 3 )

200

Measured flow resistivity for ACI fibreglass.

Material RA24 RA24 SRD HA25 ISB Pink building blanket Flexible LEI AT200 Special No. 1 Special No. 2 Special No. 3

Thickness (ram) 35 76 25 25 50 50 50 13 75 25 15

Fibre diameter (ram) 7.6 7"6 6-5 7-2 7.0 6.8 7.1 6-5-8.0 7.1 7.7 9"4

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

I

I

I

1

I

I

I

I

I

BRADFORD INSULATION

105

x.q~+ I A

E t~ O

10 4

_

Y yZ zZ

Y

I

10 3

I

I

I0

Bulk Fig. 3. Symbol X <>

!

+ A 0 V [] .Ye

® Y Z

density

I

I

I

1 I ]

100 D ( k g / m 3)

200

Measured flow resistivity for Bradford Insulation Rockwool. Material Type A Type A Type A Type B Type B Type B Type C Type C Type C Type D Type D Type D R/4 R/4 Tuff skin fibreglass--rigid Tuff skin fibreglass--semi-rigid

Thickness (ram) 25 37-5 50 25 37.5 50 25 37-5 50 25 37-5 50 35 76 50 50

363

364

D . A . BIES, C. I-I. I-IANSEN

i

10 5 m

i

i

i

i

i

f

I j

[

I

J I I

AUSTRALIAN GYPSUM INSULWOOL

E >,,, 0 v

10 4

rt

A

P n

103

I

I

I

10 Bulk Fig. 3,.

I

0 V [] <>

100 D ( k g l m 3)

200

Measured flow resistivity for Australian Gypsum lnsulwool.

Symbol

+

density

I

lnsulwool lnsulwool Insulwool Insulwool lnsulwool lnsulwool lnsulwool Insulwool lnsulwool Insulwool Insulwool

Material light industrial semi-rigid rigid board rigid board Attenuliner SRB Attenuliner LISR Attenuliner LISR semi-rigid board semi-rigid board Flexible Roll Flexible Roll Attenuliner SRB

Thickness (ram) 50 25 50 25 50 25 25 50 50 25 50

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

I

I

I

OLYMPIC

I

I

FOAMS

I

I

I i

50mm

105

E I/1

o

Y

10 4

M

x

E

•

10 3

x~

8

O

I

I

I

10 Bulk Fig. 5.

Symbol • • <> X + A © V [] ® l>~ cO I~1 Y

density

I

I I I l[

100 O ( k g / m 3)

200

Measured flow resistivity for Olympic Foams.

Material L230 PG20 E30 E50 S125 PG70 E71 PG40 E20 H80 E40 LBD PGI0 S115 S145

Thickness (mm) 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

365

366

D. A. BIES, C. H. HANSEN

TABLE 1 A D D I T I O N A L F L O W RESISTIVITY D A T A

Manufacturer Mineral Fibre Insulation Pty. Ltd (rockwool)

Sample type

Density (kg/m °)

Flow resistivity (MKS rayls/m)

1

73.3 68-0 69.0 36.5 36.5 40.1 31.4 31.4 67.3 68.1

8690 6890 5600 3380 3560 3860 4410 4410 7930 7390

32.4 32.6 23.9 23.6

5060 5150 4430 5650

2

Joubert & Joubert Pty. Ltd (foam)

1-9 4-5 2-1 1-45

10 6

10 5 E

0

10z,

10 3 10 0

101

10 2 D(kglm

Fig. 6.

10 3

3 )

Flow resistivity as a function of bulk density and fibre diameter. Courtesy: Owens Corning Fibreglass, USA.

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

367

materials are shown in Fig. 5 and also in Table 1. Measured values of flow resistivity are presented as a function of bulk density of the sample. For the two classes of fibrous material manufactured in Australia the fibre sizes varied slightly between material types. The material type and fibre size, where available, are recorded in the figure captions. We note the general upward trend of the data for the fibrous materials in Figs 2, 3 and 4 with increasing bulk density. In fact, if a fibrous material is characterised by a fairly uniform fibre diameter and any binder does not play too large a part, the flow resistivity may be estimated, based upon a knowledge of fibre diameter and bulk density alone, using eqn. (5) or Fig. 6. The latter information was given to one of the authors by Owens Corning of the United States. 2 SOUND PROPAGATION (UNSTEADY FLOW) IN POROUS MEDIA

Introduction Thus far the discussion has been concerned with steady-state properties of porous materials. A quantity called the flow resistance and a related quantity called the flow resistivity have been defined in terms of steady-state measurements. Experiments have shown, however, that these quantities defined for steady-state conditions have a very wide range of application for unsteady conditions in the pressure and flow rate regime characteristic of ordinary acoustic phenomena. Thus, in this section, it will be shown how sound propagation in porous media may be described in terms of the easily measured flow resistivity of the material.

Review and discussion of previous work Zwikker and Kosten 3 have attempted a detailed microscopic description of sound propagation in a porous material. They introduced the concept of a mean pore radius, and, in terms of assumed viscous and thermal effects, they described an effective complex density and complex compressibility for the gas contained in the material. They expressed the complex propagation constant and complex specific acoustic impedance for a sound wave travelling in the medium in terms of the complex density and complex compressibility. Their work suggested that the complex density should be a function of the dimensionless parameter (r2pf/l~), irrespective of whether or not their model was strictly correct. This dimensionless quantity, like an acoustic Reynold's number, is made up of the mean pore radius, r, the gas density, p, the frequency of a travelling sound wave, f, and the gas viscosity, #. Bies4 considered sound propagation in steel wool and investigated the dependence of the complex density upon the dimensionless frequency parameter (r2pf/it). He obtained a well defined empirical relation for the magnitude and phase with well defined limits for both large and small values of the frequency parameter.

368

D. A. BIES, C. H. HANSEN

However, his efforts to determine the functional form of the complex compressibility were not as successful but he did show that this quantity also should be a function of the same frequency parameter as well as a function of the ratio of specific heats, ~, of the contained gas. He also provided a model for estimating the expected limiting behaviour of the complex compressibility at very small and very large frequency values. More recently, Delaney and Bazley 5 have investigated the acoustical properties of a large range of porous materials. They have shown that their results may all be cast in dimensionless form in terms of the dimensionless frequency parameter ( p f / R ~ ) where p and f are as before but in this case the remaining variable is the flow resistivity, R~, discussed earlier. It will now be shown that Delaney and Bazley's frequency parameter is identical with that of Zwikker and Kosten 3 given earlier. Using eqn. (4) we may write: p f / R 1 = pfd2/lafl = pfrZ/p

(9)

where: r2 = dZ/fl(p,./pf)

(I0)

We see that the experimental results of Bies4 and of Delaney and Bazley 5 imply that the mean pore radius, r, of Zwikker and Kosten 3 is related to the mean fibre diameter, d, by the function, f l . Equation (9) may be used to calculate the squared effective radius, r z. When this is done it is found that Bies's experimental data are in good agreement with the data of Delaney and Bazley. 5 Considering the physical meaning of the function f~, described earlier, and the general agreement between the experimental data which results from the use ofeqns (9) and (10), we see that they are readily justified. In fact, the functionf~ is now sufficiently well defined that it should be possible to calculate it purely upon the basis of statistical geometric considerations for any fibrous material. S o u n d propagation in porous media C o m p l e x descriptors: Various authors have considered the problem of sound

propagation in porous media with various degrees of success. 3.6,v All of these earlier formulations, however, rely upon suitable adjustment of semi-empirical parameters. For example, the structure factor remains essentially empirical. 3'7 Thus, although the structure factor is readily understood, it is not so readily calculated from first principles and descriptions which make use of it are thus semi-empirical at best. In the following we will make no attempt to derive a description for sound propagation in porous materials based upon first principles but, instead, we will concentrate on an accurate description of existing data. To proceed we imagine that the porous gas-filled medium is replaced with an

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

369

effective medium characterised by a complex density, Pl, and complex compressibility, r. In these terms we may write, for the characteristic impedance, W, and the complex propagation constant, k: w = p,c,

=

(11)

and:

k

=

colc

=

=

co/c.

(12)

In the above equations c 1 is the complex velocity of sound in the porous medium, co is the circular frequency 2~f, c mis the phase speed and ~ is the attenuation constant in nepers per wavelength of a wave travelling through the porous medium. In eqns (1 l) and (12) the complex quantities, px and ~c, lead to real and imaginary parts of the characteristic impedance and the complex propagation constant. In the latter case the real part corresponds to the wave number of a propagating plane wave whilst the imaginary part corresponds to the attenuation constant of the travelling wave. As shown by Zwikker and Kosten a and demonstrated by Bies 4 and Delaney and Bazley, s these relations may be used as the basis for the experimental determination of ~c and p~. Alternatively, if r and pa can be estimated then eqns (1 l) and (12) can be used to calculate the complex quantities, W and k. Thus, we will proceed by determining suitable values for the complex quantities ~c and p~. Delaney and Bazley s have pl ~vided empirical expressions derived from a great many measurements for the mid-frequency range and Bies 4 has provided the proper limits for pl and K at low and high values of the frequency parameter pf/R~. For the present purpose we rewrite eqns (1 I) and (12) as follows:

p,/p - (Ip,I/p) expj~b = k W/pco

(13)

r/yP = ([~[/yP) expj0 = coW/kpc 2

(14)

In eqns (13) and (14) p, ), and P are, respectively, the gas density, ratio of specific heats and the static pressure of the gas in the porous medium. The empirical expressions of Delaney and Bazley s are as follows:

W = pc[1 + 0"0571X -°'754 - j 0 " 0 8 7 X -°'732]

(15)

k = (co/c)[(1 + 0.0978X -°'7°°) - jO.189X -°'s9s]

(16)

where the frequency parameter:

X

=

pf/R,

(17)

Delaney and Bazley suggest the following bounds for the validity of their empirical expressions in terms of bounds on the frequency parameter, X, as follows: 0-01 < X < 1-0

(18)

370

D. A. BIES, C. H. H A N S E N

The limits suggested by Bies2 for small and large values of the frequency parameter, X, are as follows: lim ([p~[/p) = R t / p ~

(19(a))

X~O

lim q~ = - ~ / 2

(19(b))

X~0

lim (IM/YP) = 1/y

(19(c))

X~O

lim 0 = 0.0

(19(d))

X~O

lim (IPaI/P)= 1.0

(20(a))

X~

lim ¢ = 0

(20(b))

X~ct3

lim (Ixl/?P) = 1.0

(20(c))

lim 0 = 0-0

(20(d))

The empirical expressions of Delaney and Bazley approach the correct upper limits and thus the upper bound which they have suggested (eqn. (18)) may be relaxed. Their equations may be used to determine pl and K for medium and large values of the frequency parameter. However, their empirical expressions do not approach the correct lower limits and an alternative means is necessary to calculate p~ and x for small values of the frequency parameter. The means used here is to derive further empirical relationships which fit the Delaney and Bazley data in the mid-frequency range and approach the correct limits for small and large values of the frequency parameter. Empirical relationships are derived for the magnitudes and phases of the complex density and complex compressibility which are used to plot the data shown in Figs 7 and 8. These figures are based upon measured properties of porous fibrous materials in which the gas was air 5 but in the form shown the plots should apply for any gas. 4 The empirical relationships used to calculate the four quantities, complex density modulus and phase and complex compressibility modulus and phase follow. (1) Modulus of the complex density (IPlI/P): A polynomial expression which approaches the correct limits for large and small values of the frequency parameter, X, is:

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

1000

I

I

371

I

Q

100

IPll P 10

1

I

10-/,

-90

10-3

i

10-2

10-1 /of /R1

100

101

I

102

]

-80 -70 -60 ~L,.- 5 0 IlJ

lo -40 v

-e- -30 -20 -10 0

I

10-t.

I III

I

10-3

I III

i

10-2

Jill

I

10-1 p f / R1

,,,I

100

,

,J,l I~"~

101

102

Fig. 7. Empirical estimates for the complex density of a gas contained in a semi-riNd porous material as a function of the flow resistance parameter p f / R I. (a) Modulus of the complex density. (b) Phase of the complex density.

372

D . A . BIES, C. H. HANSEN

1.0

~.=

7 pcos o-IKI

08

-1 )

0.6

g 0.4

=

10-3

10-2

/

\\

10 -1

100

R,

k97-5

101

102

)

Fig. 8. Empirical estimates ofthe modulus and phase for the complex compressibility of a gas contained in a semi-rigid porous material as a function of the flow resistance parameter pflR~[8xT/(97 - 5)].

p

= 2 +

2LI~J

+ 2~XL1 + s , J

(21)

where: 15

S 1 -~- ~ a i

X-i

i=l The coefficients a~ are determined from data in the mid-frequency range which are calculated using eqns (13), (15) and (16). We now define four frequency ranges where different means are used to determine

Ipll/p: (a) X < 0-009, eqn. (21) is used.

(b) 0-009 < X < 0.021. In this range, eqn. (21) is unstable and results in large errors. Thus, we draw a

F L O W RESISTANCE I N F O R M A T I O N FOR A C O U S T I C A L D E S I G N

373

smooth curve in this range such that, at the upper and lower bounds of this range, the slope and value of IPl [/P are equal to those calculated using eqn. (21). The expression used for this purpose is as follows:

IPlI/P

= 7-841 + 0.352A 1 + 0.0211 + 2 + 3 + -..AI]

(22)

where: A l = 49"1131og10 [0"02095/X] (c) For 0-021 < X < 0.09, we use eqn. (21). (d) For 0.09 < X, we use eqns (13), (15) and (16). (2) Phase of the complex density ~b: A polynomial expression which approaches the correct limits for large and small values of the frequency parameter, X, is: tk = -9011 - $2/(1 + $2) ] degrees

(23)

where: 15

S 2 = ~ biXi i=1

The coefficients b i are determined from data in the mid-frequency range which are calculated using eqns (13), (15) and (16). We define two frequency ranges: (a) For X < 0.7, we use eqn. (23). (b) For X _> 0.7, we use eqns (13), (15) and (16). (3) Modulus of the compressibility ([xl/vP): A polynomial expression which approaches the correct limits for large and small values of the frequency parameter, X, is:

IrI/yP

=

1"01 + 1"414S3 1"41411 + S3]

(24)

where: 15

S 3 = XciXi i=l The coefficients ci are determined using eqns (14), (15) and (16) in the midfrequency range for values of the frequency parameter, X, between 0.03 and 1-0. The lower portion of the range of the Delaney and Bazley equations is ignored as suggested by the expected limiting behaviour at low frequencies of the modulus of the complex compressibility (see eqn. (19(c)).

374

D.

A.

BIES,

C.

H.

H A N S E N

We now define two frequency ranges: (a) For X < 0.9, we use eqn. (24). (b) For X > 0.9, we use eqns (14), (15) and (16). (4) Phase o f the compressibility 0: We define two frequency ranges: (a) For X < 0.014, we set 0 = 0. (b) For X > 0.014, we use eqns (14), (15) and (16).

Wave propagation: For the purposes o f the remainder of this paper it will be sufficient to consider one-dimensional plane wave propagation in a porous medium. Furthermore, it will be sufficient to consider that the porous medium consists of a layer of material l thick backed by an air filled cavity L deep which, in turn, terminates at a rigid wall. Sound propagation normal to the surface will be considered• The proposed arrangement is illustrated in Fig. 9. oo

"." ', ' . ' . ".'. ,I :. '. :" .' .'." ." '1 , ,,.. ,..|

•

•

•

.

.

.

.

•

. . • .-i

•.. w : - : : 4 • . ' . k ".'. ".1

pc

Z m

-

I

l

l

"":1

".". P1 • ."

L-,I"

"

.I

. . . - . . . . •

-

.

.

•

•

•

.

.

:

• •

| .

• . . .

" . . ' .

.

. d | !

Fig. 9. The theoretical model; a porous material of thickness l backed by a cavity of depth L which is terminated by a rigid wall. The porous material is characterised by a complex characteristic impedance, W, and a complex propagation constant, k. It may readily be shown that the normal specific acoustic impedance at the front surface of the layered structure o f Fig. 9 is as follows: 3

Z =jW

tan k/tan (2~jL/c) - (pc/W) tan (2rtfL/c) + (pc/W) tan kl

(25)

If the porous material is rigidly backed so that L is zero or, equivalently, L is an integer multiple of half wavelengths, then eqn. (25) reduces to: Z = - j W / t a n kl

(26)

If the porous material is backed by zero impedance or, equivalently, L is an odd multiple o f quarter wavelengths, then eqn. (25) reduces to:

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

Z = j W t a n kl

375

(27)

The complex quantities Wand k in the above equations can be expressed in terms of the complex compressibility, x, and complex density, P l, using eqns (11 ) and (12) and, in turn, the latter quantities can be expressed in terms of the dimensionless frequency parameter (pf/R~) as shown in Figs 7 and 8. Thus we can write the specific acoustic impedance, Z, of eqn. (25) entirely in dimensionless form as a function of the frequency parameter. The equation is parametric in the following dimensionless quantities; ratio of specific heats, y, porous layer thickness,fl/c and backing cavity depth, fL/c. Equation (25) will be useful in the next section. APPLICATIONS OF FLOW RESISTIVITY DATA

Introduction The sound absorptive properties of porous media are used in various ways to control sound but they all depend upon the entrance of the sound into the porous medium. Thus, the specific acoustic impedance presented to the incident sound wave is an important parameter in determining the effectiveness of a porous medium for the purpose of sound absorption. The specific acoustic impedance presented by a layer of porous material in turn depends upon the properties and thickness (measured in sound wavelengths) of the porous layer, as well as the specific acoustic impedance at the opposite face of the porous layer. Thus, in all of the applications which will be considered, the specific acoustic impedance and the parameters which control it will be of paramount importance. Three separate applications of porous materials for the control of sound will be considered. We will first consider the use of porous materials for the control of reverberant sound in enclosures. Next we will consider the use of porous materials for the purpose of increasing sound transmission loss through wrappings on pipes and through the walls of enclosures. Finally, we will consider the use of porous materials for the attenuation of sound propagating in ducts. In all of the applications the porous layer will form part of the boundary. Reverberation control in rooms The reverberation of sound in an enclosure is quite often controlled by the introduction of sound absorbent materials at the boundaries. For example, acoustic tile may be placed on the ceiling and over parts of the walls of a room to achieve optimum reverberation for some special purpose. For design purposes such sound absorptive materials as acoustic tile are characterised by a Sabine absorption coefficient. The latter quantity is generally measured according to standard procedures in a reverberation chamber, a In principle, the Sabine absorption is a measure of the fraction of sound energy incident from all angles which is absorbed but, as values greater than unity are sometimes recorded, it is clear that the mechanism of absorption is more complicated than is implied by simple reflection averaged over all angles of incidence.

376

D.A.

BIES, C. H . H A N S E N

Morse and Bolt 9 have considered in detail the reverberant response of rooms from an analytical point of view. They have defined a statistical absorption coefficient which is the fraction of incident energy absorbed averaged over all angles of incidence and they have described the latter quantity in terms of an assumed locally reactive normal wall impedance. Their formalism has been the basis for a procedure for estimating the expected statistical absorption coefficient of a material from normal impedance tube measurements of its specific acoustic impedance. 1° According to Morse and Bolt the reverberant response of a room should be accurately described in terms of the boundary impedances and hence in terms of the statistical absorption coefficient. One could be satisfied that the problem was completely understood if the Sabine absorption coefficient and the statistical absorption coefficient were equal but, unfortunately, they are not. In fact, a recent 'Round Robin' conducted in Australia 11 gave, as a measured maximum value of Sabine absorption for a commercially available fibrous material, 1.12, whereas, as will be shown, the theory of Morse and Bolt would predict a value of 0.92 for the statistical absorption coefficient. This discrepancy has been a matter of controversy for thirty years or more and will not be resolved here. It is interesting to note that the discrepancy is consistent with a 20 to 30 per cent discrepancy found for a two-dimensional room using a ray tracing technique in a computer simulation study. ~2 However, it will be shown that the theory of Morse and Bolt does describe the parametric dependence of the measured Sabine absorption on the flow resistance, thickness and depth of any backing cavity so that the analysis to be described can be used to calculate expected absorption. Morse and Bolt give the following expression for the statistical sound absorption coefficient of a wall of finite normal specific acoustic impedance, Z: =

{8COSfl/~}{1

--

[ C O S f l / ~ ] l o g e [1 + 2~cosfl + ~2]

+ [cos 2fl/~ sin fl] tan - t [~sinfl/(1 + ~cosfl)]}

(28)

In eqn. (28): Z = pc~ exp ifl

(29)

If we set eqn. (29) equal to eqn. (25) we may determine the magnitude, ~, and the phase, fl, of the normal specific acoustic impedance of a wall lined with a porous material of flow resistance, R1, thickness, l, and backed by an air cavity of depth, L, which, in turn, is backed by a rigid wall. Finally, using eqns (11) and (12) and the expressions for ~ and fl, we may calculate ct in terms of the dimensionless quantities, fl/c and R t l/pc. The results of such calculations are shown in Figs 10, 11, 12, 13 and 14.

In Figs 10 and 11 the depth of the backing cavity has been taken as nil, or equivalently at particular frequencies as a multiple of a half wave deep. In these cases the porous liner is rigidly backed. We note the general behaviour of the family of

377

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN 1.0

'

'

''1

''

'1

'

'

''1

'

' '

0.9 0.8 0.7

10 2O

0.6 0.5

/

0.4

oooo 2HEV' Z "

/

/

/

/

0.3 0.2 0.1

10-3

10-2

10-I

10 0

101

f[/c Fig. 10. Calculated statistical~absorption coefficient for the left-hand surface of the model in Fig. 9 in a reverberant field, as a function of a frequency parameter for various indicated values of flow resistance. In this case the backing cavity depth, L, is zero.

curves in the two figures. At low values of the dimensionless flow resistance, Rt l/pc, the absorption is small and for values of dimensionless frequency,fl/c, greater than 0.1 the absorption is quite erratic, evidently due to standing waves in the porous liner. However, as the flow resistance increases so does the absorption and the standing wave effects grow smaller. Finally, at a value of flow resistance of 3pc the absorption coefficient has reached a maximum value after which a further increase in the flow resistance causes the absorption coefficient to diminish again. We note that the standing wave effects entirely disappear when the flow resistance is 4pc or greater, indicating that, in this case, propagation attenuation in the porous liner is so great that reflected waves from the rigid back wall are quite negligible. In such cases the backing cavity can have no effect. In Figs 12 and 13 the case of zero impedance backing has been considered. In practice this case could only be realised at discrete frequencies for which the cavity was an odd multiple of quarter waves deep. As expected, for a value of flow resistance greater than 4pc and for values of the frequency parameter greater than 0-1, the absorption is the same as that previously observed for the rigidly backed case. However, Fig. 13 shows that the general character of the absorption coefficient is also similar to the rigidly backed case for values of flow resistance less than 4pc and

378

D. A. BIES, C. H. HANSEN

1.Or--

I

I

IIm

0.9

RI,~ I p c

0.8

2.01.0--

0.7

0.5-0.1~

0.050.01 . 0.001 0.00C

0.6 OL

0.5 0.4 0.3 0.2 0.1

10-3

10-2

10-1

100

101

f~lc Fig. 11. Calculated statistical absorption coefficient for the left-hand surface of the model in Fig. 9 in a reverberant field, as a function of a frequency parameter for the various indicated values of flow resistance. In this case the backing cavity depth, L, is zero.

values of the frequency parameter greater than 0.1. This, too, is to be expected since the erratic behaviour is due to standing waves in the porous layer and for this to occur all that is required is an impedance mismatch at the second surface of the porous layer. The most obvious difference between the curves of Figs 10 and 11 and 12 and 13 occurs at values of the frequency parameter less than 0.1. In the case when the porous layer is rigidly backed the absorption coefficient drops rapidly to negligible values, whatever the flow resistance, as the frequency is decreased, but in the case of the zero impedance backed layer the absorption drops to some constant value at all frequencies well below a value of the frequency parameter of 0.1. Of course, in a

379

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN 1.0

~

0.7

~

' 'I

'

'

~ ~I

~

'

~ ~I

'

~

~ r

5.0

0.6 0.5 0.t, 0.3 0.2 0.1 0

10-3

10-2

10 -1 ft/c

10 0

101

Fig. 12. Calculated statistical absorption coetficient for the left-hand surface of the model in Fig. 9 in a reverberant field as a function of a frequency parameter and for various indicated values of flow resistance. In this case the backing cavity is one quarter of a wavelength of sound deep, representing a zero impedance load on the back side of the porous material.

practical case, realisation of zero impedance at the second surface of the porous layer is only possible at discrete frequencies for which the backing cavity is one quarter wave deep. Again, we observe that a value of flow resistance of about 3pc can be expected to give greatest absorption. Consideration of the Figs 10, 11, 12 and 13 suggests the following behaviour for a porous layer backed by a cavity of arbitrary depth, L. I f the cavity is deep relative to the thickness of the liner, l, then there may be several frequencies for which the cavity depth is alternately one quarter wave, then one half wave, deep. If all these frequencies occur below the frequency for which the frequency parameter is less than 0" 1, the absorption can be expected to fluctuate between very small values, as given in Figs 10 and 11, and very large values, as given in Figs 12 and 13. Clearly, then, there is no advantage in excessive cavity depth. For best results the cavity should admit quarter-wave response but not half-wave response. An optimum cavity depth is indicated. In Fig. 14 the effect of increasing the backing cavity is explored. It is to be seen on review of the figure that a cavity depth equal to the thickness of the liner is to be preferred and a flow resistance between 2.8 and 3-5 gives the greatest absorption.

380

D . A . BIES, C. H. HANSEN

1.0

i

i

III

I

i

I

i I

2.0 0.9

Rlt

I pc

;

i

i

_

i I

i

,

.

= 1.0

0.8 0.5 0.7

0.6 m

oc 0.5

0./-,

0.3 0.1 0.2

0.1 0.01

10-3

.

10-2

10 -1

10 0

101

ft/c Fig. 13. Calculated statistical absorption coefficient for the left-hand surface of the model in Fig. 9 m a reverberant field as a function of a frequency parameter and for various indicated values of flow resistance. Inthis case the backing cavity is one quarter of a wavelength of sound deep, representing a zero impedance load on the back side of the porous material.

As mentioned at the beginning of this discussion, the calculated values of the statistical absorption coefficient are in disagreement with the values measured following standard procedures. The situation is indicated in Fig. 15. In Fig. 15 the average experimental results of a 'Round Robin' conducted in Australia and New Zealand are compared with the calculated statistical absorption coefficient for the sample tested. ~ The material, known as Silan, is 5 0 m m thick and has a flow resistance of 4pc. We note that the measured values of absorption rise to 1.12 and then fluctuate closely about 1.09 as values of the frequency parameter are increased above 0-1. If we arbitrarily reduce the measured absorption value of 1.09

381

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

I.O[

,

I 'I

l

,,

i I

l

J

J i I

I

I

I

0.91 0.8 0.7 Lit. 0.6 -

~0.0 ~0.5 ~0.75

of., 0.5

1.0 0./.-

~ • ~ ' 1 . 5

o.3 -

-2.0 ~3.0

0.2

Rle I toc = 3 . 0

0.1

10-3

10-2

10-1

100

101

ft./c Fig. 14. Calculated statistical absorption coefficient for the left-hand surface of the model in Fig. 9 in a reverberant field as a function of a frequency parameter and for various depths, L, of backing cavity. The flow resistance parameter in this case is, Rll/pc = 3.0.

(corresponding to a frequency parameter value of 0-18) to the predicted value of 0.92, and all other values in proportion as shown in Fig. 15, the agreement is reasonable. Morse and Bolt 9 suggest that, for use in the design of large auditoria, their calculated value of statistical absorption is a more reliable indicator of expected performance than is the measured value. In this regard it should be mentioned that measurements made in a 600 m 3 reverberant r o o m are included in t h e ' R o u n d Robin' data and are in agreement with the results of the 'Round Robin'. 11 The work described here shows that reverberation room tests are probably not necessary for a large class of porous materials; all that is required is a knowledge of the flow

382

D . A . BIES, C . H . H A N S E N 1.2

'

'I

'

'

'

'I

f

;

'

~I

~

,

r

,

,

,

){

1.1

xx

x Xx X x x x

x 1.0

x

oo •

0.9 x°

0.8 0.7 06 x

0.5 OJ. 03 0.2 0.1 i---.-r-<~ 0 10-3

,I

j

~

10-2

~ ,I

10 -1

~

,

~,I

10 0

101

fl/c

Fig. 15. Absorptioncoefficients for the commercial rockwoolproduct, Silan, plotted as a function of a frequency parameter. - predicted statistical absorption coefficient; x measured Sabine absorption coefficient; • adjusted Sabine absorption coefficient. resistance and the flow resistance should be about 3pc for optimum attenuation.

Transmission control through enclosures Porous materials may sometimes be used to reduce sound radiated from structures such as a pipe or through the wall of an enclosure. In these applications the porous material may be wrapped around the radiating surface or it may be placed within a cavity between two non-porous surfaces comprising the wall of an enclosure. ~s In these cases sound transmission through the porous layer and associated attenuation and reflection at the air-porous layer interfaces are important. These losses and means for their calculation will be considered in this section. For a porous layer of finite thickness, the problem of calculating the expected attenuation may be divided into three frequency ranges dependent upon the ratio of the material thickness to the wavelength of the sound in the material. ~3 The three frequency ranges are illustrated in Fig. 16 as a function of the frequency parameter and material flow resistance parameter. The low frequency range lies below the curve l/2m = 0.1, the high frequency range lies above the curve l/2,~ = 1-0 and the midfrequency range lies between the two curves. The quantity, 2., is the wavelength of sound in the material and is calculated on the basis of the reduced phase speed of

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN 100

....

I

'

'

"f

'

'

',l~....:--~h

~--~

High frequency range

383

, .

.,y \'lp~

to-'

/ 10-2

10"3 10-4

%'/

10.3

Mid frequency

range

LOW frequency range

10.2 10-1 p f / RI

100

101

Fig. 16. Limits showing when low and high frequency models should be used for estimating the transmission lossthrough a porous layer. The low frequencymodel should be used when the design point lies below the I/A,. = 0.1 curve and the high frequency model should be used when the design point lies above the I/2,. = 1-0 curve.

sound in the porous medium using the real part of eqn. (12) and the appropriate expressions for the complex density, Pl, and complex compressibility, r. The low frequency range will be dealt with first. In this case the points in Fig. 16 defined by the appropriate values ofpf/R 1 andfl/c lie below the line 2,. = 0.1. In this range the internal inertia of the porous material is small so that it is expected that the material will move with the particle velocity of a sound wave passing through it. The porous material m a y be treated as a lumped element circuit in this frequency range, in which case the phase shift through the layer is negligible and the effect of the material is to present resistance and mass inertance to a sound wave passing through it. The transmission loss to be expected from the porous medium in the low frequency range may be estimated using Fig. 17. We note that the surface density, p,.l, is the important parameter. Further consideration shows that, besides making p"l as large as possible, the flow resistance, RI l/pc, must also be chosen large to give greatest attenuation. Reference to eqn. (3) then shows that the fibre diameter should be very small. We conclude that to wrap pipework for greatest noise reduction one should use a layer of very dense material of the finest fibres possible. Figure 17 will then provide an estimate of the reduction in noise to be expected from such wrapping. In the near field of the pipe the noise reduction will be equal to the transmission loss.

384

D . A . BIES, C. H. HANSEN 24

~12

8

/

4

0 0.05

......

~Z'_-- . . . . . . . . . . ~

, 0.1

,

.....

I 1.0

. . . . . . . .

Y .... ' 10

15

fpm ! / p c Fig. 17.

Transmission loss through a porous layer for a design point lying in the low frequency range of

Fig. 16. In the high frequency range the porous layer is considered to be many wavelengths thick. In this case the points in Fig. 16 defined by the appropriate values o f p f / R 1 and fl/c lie well above the line l/2 m --- 1.0. In this case reflection losses at the first and second air-material interfaces, as well as propagation loss through the material, are important. A two-step calculation procedure is thus indicated which takes account of propagation loss and reflection losses separately. The final estimate of transmission loss is then the sum of all such losses.l 3 Figure 18 provides a means for estimating propagation loss. It is based on the analysis given above under the section headed 'Sound propagation in porous media' and follows from eqn. (12). The imaginary part of eqn. (12) provides the basis for the calculation of Fig. 18. The quantity, ~, has been converted to decibels per metre by multiplying the latter quantity by 8.69. We note that Fig. 18 presents attenuation in decibels in a distance through the porous medium equal to a wavelength in air, not the wavelength calculated at the reduced wave speed in the porous medium but at the wave speed of the gas (air) in the porous medium. At an air-porous medium interface the magnitude of the reflection of a sound wave travelling initially in either medium is the same so that Fig. 19 may be used for the calculation of the loss on reflection at entry and exit for a sound wave passing through a layer of porous material. Figure 19 is based upon simple one-dimensional wave analysis which gives the following expression for the reflection loss shown: reflection loss = 101Oglo {1 - [(D z - 1) 2 4- 4D2sin2@]/[(1 + D c o s @ ) 2 + D 2 s i n 2 ~/)] 2 } (30)

FLOW RESISTANCE INFORMATION

1000

FOR ACOUSTICAL DESIGN

F

385

r

100

~J

~ 10

I

,

10-z.

,

I

,

10-3

,

,,I

,

pf

Fig. 18.

,

,,I

10-2

,

10-1

,

,,I

,

,

10 0

L

101

/ R~

Propagation loss (decibels per wavelength of thickness) through a porous layer for a design point lying in the high frequency range in Fig. 16.

10

i

i i

9 8 OD

7o 6

o 5

g ~ 4

~3 2 1

10-~.

10-3

10-2

10 -1

10 0

/of / R1

Fig. 19.

Reflection loss (dB) at a porous material-air interface for a design point in the high frequency range of Fig. 16.

386

D. A. BIES, C. H. HANSEN

In eqn. (30):

W = pcDexp i~p

(31)

Before using Fig. 19 one should first decide whether or not the calculation is appropriate. For example, if a porous material is wrapped directly on to a vibrating surface then there is no initial air-porous medium interface and only the reflection at the exit surface will be important. In this case, too, if the porous material fills a gap between two impervious surfaces--for example as a filler between two panels of a wall--then there will be no reflection loss at the exit, either. On the other hand, if the porous material is installed as a free hanging curtain, reflection will occur at both entry and exit and the loss due to reflection to be added to the propagation loss will be twice that given by Fig. 19. In the mid-frequency range the behaviour in the porous material conforms to neither of the clear cut models appropriate to the high and low frequency regions. Generally, it is sufficient to estimate the transmission loss in the mid-frequency range graphically with a faired smoothly varying curve connecting plotted estimates of the low and high frequency transmission loss.

Propagation control in ducts The use of porous materials to line the walls of ducts for the attenuation of unwanted sound propagation in them is well known. If the requirements for attenuation are generally broad band and not extreme then the design can be quite straightforward. Procedures for such a design are considered in detail elsewhere and will not be repeated here. 14 Rather, the aim here will be to offer a small supplement to the referenced work and to point out that the performance is once again controlled by the flow resistance of the porous material used in the construction of the duct lining. One of the considerations of importance in the design of a lined duct is the propagation loss provided by the duct lining. One scheme for predicting this loss is shown in Fig. 20. Initial consideration of Fig. 20 suggests that optimum attenuation in a duct of width 2h is achieved when the duct width parameter, 2h/2, lies between 0.2 and 1-6, depending upon the percentage of open area of the duct and the flow resistance of the lining material. For example, for a duct having 50 per cent open area, Fig. 20 shows that the optimum attenuation at a frequency offo is achieved when the corresponding frequency parameter 2h/2 = 0.55 and this attenuation is 3-2 dB in a length of duct equal to 2h. This, in turn, implies an attenuation of 5-8 dB per duct length equal to one wavelength, at the optimum frequency, fo. Suppose, however, that this attenuation rate were insufficient and instead the duct was designed with a much narrower width, 2h, so that the frequency parameter corresponding tOjo was now 0.2. In this case Fig. 20 predicts an attenuation in a duct length, 2h, of 2.45dB which, in turn, implies an attenuation of 12-3dB in a duct length equal to one wavelength at the frequencyfo. The attenuation atjo is now more

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN f

I

I

I

I I II

i

I

I

I

I

I

I II I

I

387

I

RII / pc ,.5

3.0

1.25 ( 6 7 % ) ~ /

,.,C

e

t

/

0.75 ( 8 0 ~ ~

2

)q,,,\\

, /

~k\\~

/ i ~ ~

r-

0

0.01

Fig. 20.

0.04

0.1

0.4

2hlX

1

L,

Attenuation of axial waves for rigid ducts lined on two sides with porous material. The ordinate scale is the attenuation in decibels through a duct length equal to one duct width, 2h.

,

,

i

,

,,,,

I

I

F(S0) : 4.5 (17%) "~ 30 ~:c-25 o ~

,

,

,

,i,,

i

i

J

35

Rlt/pc SO= [2h/(2t+2h)]2 F =

3.0 {33°/o)A \

E

I~'~| ~::!:.':J

20

-

.E c 15 .2

~

lO

<

0.01

0.04

0.1

2hi ~.

0.4

1

l.

Fig. 2 l. Attenuation of axial waves for rigid ducts lined on two sides with porous material. The ordinate scale is the attenuation in decibels through a duct length equal to one wavelength at the frequency corresponding to a design point on the abscissa scale. This figure may be used to optimise the duct width to maximise the attenuation achieved in a given duct length at a particular frequency.

388

D . A . BIES, C. H. HANSEN

than twice as great as before for the same duct length. This has been achieved by reducing the duct width to 36 per cent of its initially chosen width. One may reasonably ask: How far can this process be carried? If attenuation at a given frequency is important then a replot of Fig. 20, such as that shown in Fig. 21, is useful. In Fig. 21 the attenuation to be expected in a length of duct equal to one wavelength is shown. It is to be observed that the data of Fig. 20 present quite a different picture when plotted in this format. For example, we observe that for the 50 per cent open duct design considered above, a maximum attenuation in one wavelength of about 13 dB is predicted for a value of the frequency parameter equal to 0.15. We have now answered the question posed above. The replot of Fig. 20 shown in Fig. 21 is offered as an alternative way of looking at the data of Fig. 20 but not as a replacement for the latter figure. Thus, Fig. 21 may be used to estimate the optimum value of the frequency parameter which will give a maximum attenuation in a fixed duct length at a specified frequency. Then Fig. 20 may be used with this optimum value or any other value of the frequency parameter to calculate the expected attenuation in a duct length equal to one duct width, 2h. Note that for a duct lined on only one side the attenuation is approximately equal to the attenuation calculated for an equivalent duct which is lined on two sides and which is twice as wide as the actual duct. For a duct lined on all four sides the attenuation is the arithmetic sum of the two attenuations calculated by considering each of the two pairs of opposite sides separately. Thus far consideration has been given to the design of a fairly broad band attenuator. However, if very high attenuation in a narrow frequency range is required, an alternative approach, based upon wave analysis of sound propagation in a duct, must be taken. Design procedures for the case without flow are readily available.l 3.15 Unfortunately, such design is quite sensitive to flow speed, which complicates the problem, but design procedures can readily be developed for the cases with flow as well. ~'1~ However, all such design procedures merely prescribe what wall impedance is required; they do not suggest how the required wall impedance is to be achieved nor what wall impedance may be achieved with a practical design. If a porous liner is to be used to achieve high attenuation rates then eqn. (25) and the procedures outlined earlier in this paper are useful. In Figs 22 and 23 the predicted modulus and phase of the normal wall impedance of a fibrous porous layer, l thick, calculated according to eqn. (25), are presented. Since the condition for high attenuation rates generally requires low values of normalised wall impedance, IZWpc, the figures emphasise values of normalised liner flow resistance which provide low values of the modulus of the wall impedance. 13.15 17 The curves in Figs 22 and 23 have been calculated for the case where the backing cavity depth, L, is zero (see Fig. 9); however, it has been found that Fig. 22 is very little changed if the backing cavity is increased from zero to as large as twenty-five times the thickness, l, -

389

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

101

i

i

i

]

i

i

I

I

~

t

|

I

I

10 0 -

Rl~/pc

= 5.0 -/

_J

•

2.0--

Izl

__J

pc

1.0--

o.5 --Y 1 0 -1

O. 2 - - - ~

-

0 . 1 - 0 . 0 5 ~ 0 . 0 2 ~ 0 . 0 1 ~ 10 .2

I

10 .2

I

I

I

I

10-I

10 0

fl/c Fig. 22. Modulus of the normalised normal surface impedance of a fibrous porous layer, I thick, rigidly backed. For estimation purposes the frequency parameter,fl/c, may be replaced w i t h f ( L + l)/c where L is the depth of a backing cavity for values of L/! up to 25. In the figure Rll/pc is the normalised flow resistance of the porous layer.

of the liner. What does change significantly with backing cavity depth is the phase shown in Fig. 23. Increasing the cavity depth generally has the effect of shifting all curves to the left. Thus, for the design of a high performance dissipative liner, the following strategy is suggested. Use Fig. 22 to estimate initial value o f f ( L + l)/c and

390

D. A. BIES, C. H. HANSEN

90

I

I

I

I

@ ki

Rlt/pc

"o 0

= 5.0

\

2.0~

@

1.0~

O c" 13-

0.5

\

0.2. 0.1 .05 .02 .01 -90

r'--~ 10-2

I

I

I

I

I 10-1

I

I

I

I

I 100

f/./c Fig. 23.

Phase of the normal surface impedance of a fibrous porous layer, l thick, rigidly backed. If a backing cavity is introduced the curves in the figure are shifted to the left.

R~l/pc to obtain the required normalised impedance modulus, IZI/pc, then adjust the backing cavity depth, L, using eqn. (25), to obtain the required impedance phase. One or two iterations will generally be sufficient to achieve a satisfactory design. Figure 22 makes clear the practicality and limitations of such a design. For optimum attenuation values of the modulus of the wall impedance much less than

FLOW RESISTANCE INFORMATION FOR ACOUSTICAL DESIGN

391

one are required. ~3.~ s - ~7 Figure 22 shows that such values are only possible within a narrow frequency range about a value of the frequency parameter between 1-5 and 3.5. The second minimum in the figure and successively higher minima not shown are probably of little practical interest. Furthermore, as the required value of the modulus decreases, the frequency range over which it can be achieved grows narrower and the sensitivity of the phase, as shown by Fig. 23, to small changes in the frequency parameter greatly increases. Clearly, the frequency range of optimum attenuation grows narrower rapidly as the required optimum attenuation grows larger. ACKNOWLEDGEMENT S u p p o r t f o r t h i s w o r k as p a r t o f a g r a n t f r o m t h e D e p a r t m e n t I n d u s t r y o f S o u t h A u s t r a l i a is g r a t e f u l l y a c k n o w l e d g e d .

of Labour

and

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