Attenuation of a resonant dissipative silencer in a rigid duct

ATTENUATION OF A RESONANT DISSIPATIVE SILENCER IN A RIGID DUCT* THOMASJ. TRELLA

US Department of Transportation, TransportationSystems Center, 55 Broadway, Cambridge, Massachusetts02142 (United States) PETER K. KASPER

The Wile Companies, Hampton Facility, 3200 Magruder Boulevard, Hampton, Virginia23366 (United States) (Received: 3 December, 1973)

SUMMA R Y Soundimenuation characteristics of a resonant-type dissipative silencer consisting of a reactive chamber with a porous facing have been considered. Such a silencer provides a high degree of attenuation within a narrow frequency range. Predicted attenuation values are compared with experiment for plane waves propagating in a rigid duct containing the dissipative silencer. The sound fieM is described by one-dimensional acoustical expressions taking into account the effect of boundary conditions and the presence of the silencer. The theoretical model incorporates the acoustical properties of porous materials and inertance of the sound field in the duct adjacent to the silencer. Good agreement was achieved between theoretical predictions and actual measurements. Results presented indicate the dependence of the attenuation spectrum upon flow resistivity and thickness of the porous material.

INTRODUCTION There are m a n y approaches to the control of noise in ducted systems. In certain cases the use of a silencer with high attenuation over a narrow frequency range is particularly effective. One case would be the circumstance where a strong discrete frequency component exists in the source spectrum. Another cireurnstance would * This work was conducted by the authors while employed by the Koppcrs Company, Sound

Control Department, Baltimore, Maryland 21203 (United States). 127

Applied Acoustics (7) (1974)--© Applied Science Publishers Ltd, England, 1974 Printed in Great Britain

128

THOMAS J. TRELLA, PETER K. KASPER

be the use of the tuned silencer as a low-frequency attenuator as part of a total silencing system. Presented in this paper are the results of an investigation of a particularly effective type of frequency selective dissipative silencer. The silencer utilises an internal acoustic resonance to emphasise the dissipation provided by a porous facing material. Maximum attenuation occurs within a band of frequencies centred at approximately the quarter wave resonance frequency of the resonator chamber. The shape of the attenuation spectrum is influenced by the thickness and flow resistance of the porous facing. The specific case considered consisted of a resonant silencer mounted on one side of a 2 ft 2 cross-section rigid walled duct. A mathematical model of the sound field in the duct was developed to predict the attenuation provided by the silencer. The model incorporates expressions for the acoustical impedance of the porous liner and the propagation constant as determined by experimental flow resistance data and power law expressions. The effect of duct cross-modes and presence of a mean air flow were not considered, t.2 Of particular importance in the analyses is the inclusion of the inertance of the sound field in the duct adjacent to the silencer. The predicted and measured attenuation of the silencer and the influence of the pertinent parameters are shown in the following sections.

THEORETICAL CONSIDERATIONS

An acoustic filter attached to the side wall of a duct is not uniquely specified from the acoustical viewpoint. As shown in Fig. I, the acoustic impedance boundary discontinuity appears in the duct at the silencer which transforms a plane propagating sound wave, forming a standing wave between the source and silencer. ', 3 Higher energy non-propagating duct modes appear in the duct but are present only a short distance on either side of the silencer. Considering the source plane at x = 0, and the sound pressure wave to propagate in the positive x-direction, four separate regions were considered. Regions I and II consist of the portion of duct between the source plane and silencer, and the silencer and anechoic termination, respectively. Regions III and IV consist of the reactive chamber and the region of the duct adjacent to the silencer, respectively. Assuming sinusoidal acoustic pressures and velocities, the one-dimensional acoustic continuity and momentum balance equations provide the following expressions for Regions I, II and III: d2u*" + k2u*. = 0

dx2n

du* n

P*" = iZ°/k dx,

n = 1, 2, 3

(I)

(2)

A T T E N U A T I O N OF A R E S O N A N T DISSIPATIVE SILENCER IN A RIGID D U C T

129

RESONANT DISSIPATIVE MUFFLER

_1REG'°N '-

b

lr L POROUS ACOUSTICAL LINER I

REGION Z

z SOURCE ( CONE SPEAKER )

REGION ~o"I REGION Tr

x

Fig. 1.

Duct-silencer arrangement.

where P*.(x., t) = P*.e -i~', and u*.(x., t) = u*.e - ~ ' . The subscript n denotes the region under consideration. The acoustic continuity and momentum equation for Region IV can be obtained by considering a plane sheet of thickness dx and area S = LyL,. When a uniform perturbated density is assumed and the non-linear coupling terms are neglected, the following acoustic continuity equation results: p.

= iZo/k (du*. v*/L,) \ dx4 +

(3)

Here, the velocity v* at the entrance of' the silencer is assumed to be uniformly distributed over the infinitesimal area Ly dx. Similarly, the two components of the momentum equation become Ly dx at z = 0 and z = L,: (axial component) (z-component)

u*4 = i(Zok) dP*4 dx4

0* 4 dz = i/(Zok)(P* , - P*4)

(4)

Delaney and Bazley 4 showed that the characteristic normal impedance and acoustic wave propagation for semi-rigid materials can be determined from the specific flow resistance: (characteristic impedance) (propagation constant)

Z I = R I + iXs ~z = ~t + i/~j.

(5)

130

T H O M A S J. T R E L L A , PETER K. KASPER

where R¢, X.r, ~¢ and/~1" are defined by:

[

(:1-°"'1

R/ = Zo 1 + 9.08 ~ a /

j

-°'' 8I =co/c

[1 +

10.8

0.01 _<

;

(:)

_< 1

Applying eqns. (1) and (2) to the cavity of the silencer and combining them with the expressions relating the acoustic pressure and velocity response in the porous liner, we have for an assumed locally reactive liner: v* = (P'"~ 0

(6)

\z:/

Here 0 is defined by: 0 = 1+ 2

/{[(

i sin (kb) +

]/

cos (kb) ert"

[(isin(kb)-

(-~i))cos(kb)e-'L]

- I)

(7)

The first ordered differential acoustic equation for Region IV was obtained by combining eqns. (3), (4), (5) and (6) along with the assumed velocity distribution: v*4 = v *

[ 1 + e o s ~ (2.=q ]/2 d2u,, L, ] J / dxZ,~ +k22u*4

=0

(8)

The propagation constant k 2 in Region IV is defined by:

k22 = {k2/l + iZo/(kL,)[l/(Z:/O ) - (iZo/(kL,) - ioopoLJ4)]}

(9)

For an infinite array of dissipative silencers mounted on one side of a rigid walled duct, the attenuation can be calculated directly from eqn. (9): Attenuation = 8.686 Imag (Ik2J) (10) The general solution for the transformed acoustic velocity and pressure in Regions !, II and IV are: U* 1 =

A l e -ikxl "1- B l elkxx

P*I = Zo[AI e-ikxx - BI eik'l]

(0 < xl < L1)

(I1)

u*2 = a z e -i~xn

P*2 = ZoAz e-ikx:'

(0 < x 3 < oo)

(12)

A T T E N U A T I O N OF A R E S O N A N T

DISSIPATIVE SILENCER IN A RIGID D U C T

131

u*4 = A4e -ik'~x'~ + B4e i~'~x'~ P*4 = Zo/Ol[A4e -s~2*'~ - B4e ~'~x'`]

(0 ~ x 2 ~ L2)

(13)

where O, = k / k 2 { l + iZo/[kLz/(Zs/O - iO~poLJ4)] }. The complex coupling coefficients A 2, A 4, B ~and B4 were determined by specifying acoustic velocity and pressure continuity at the interface of adjoining regions, namely between Regions I and IV and Regions IV and I[. Finally, an infinite sound source impedance P* (source) = Z0A 1 was assumed.

TEST APPARATUS AND MEASUREMENTS

A diagram of the test apparatus is shown in Fig. 1. A 15 in cone speaker was used and discrete frequency sound was passed through a 4 ft impedance source tube, 15 in diameter, packed with steel wool. Sound measurements were made in a 2 f t x 2 ft, ¼ in reinforced steel rectangular duct. Three fibreglass wedges 4 ft long were positioned at the exit of the duct. The silencer was constructed from ~ in thick plywood. The side panels of the silencer were reinforced with steel angle brackets. Three separate silencers were attached to the top side of the duct at a location 12 ft from the source. The length and width of each silencer was 2 ft and 1 ft respectively and the depth was 19 in. The depth of the silencer corresponded to a one-quarter wavelength of sound at a frequency of 178 Hz. This frequency lies below the.first cross-mode frequency of the duct which for a 2 f t x 2 ft duct occurs at 282 Hz. A ¼ in thick plywood baffle was positioned in each silencer at the mid-width location, creating two reactive cavities each of a cross-sectional area of I ft z. The side of the silencer which was attached to the duct consisted of a porous acoustic liner material. The liner thickness and flow resistivity were varied during the experiment. The flow resistivity of the porous material was determined experimentally for a range of densities. Sound pressure level measurements were made with a ½ in Bruel and Kjaer microphone. The intensity and frequency of sound was controlled by a sine-wave generator and frequencies between 70 and 250 Hz were considered for the experiment. For IL measurements, SPLs were taken in the duct 4 ft downstream from the silencer. The three dissipative silencers were removed and a ¼ in thick plywood panel was attached to the duct and the measurements were repeated. PREDICTED AND EXPERIMENTAL RESULTS AND DISCUSSION

Some predicted and measured IL spectra of 3 ft resonant dissipative silencers are shown in Fig. 2. The silencer IL (attenuation) is defined here as the SPL difference between the incident and transmitted sound wave per foot of the silencer length.

THOMAS J. TRELLA, PETER K. KASPER

132

12-

IO-

/f~ +..-~"~'.+'° ~,.

CALC. MEAS. x ~ L : 0.5 INCHES t. O" 10.5(¢g/) RAYLS/IN. r L = I.O INCHES -o .((.o"= IO.0(cgs) RAYLSIIN. Zl ,~ L = 1.5 INCHES , ) RAYLS/IN.

o 6.

=~4-z 2 .

I

70

I

90

I

I10

130

Fig. 2.

I

I

i

150 170 190 FREQUENCY, HZ

I

ZIO

I

I

230

250

Measured and predicted IL.

DISSIPATIVE SILENCER

IIII

5040-

17"'

ITION 2

,\ /Io o\

DUCT CENTER LINE

30--

-

li It

T POSITION 3

rn

"'

CALC. MEAS. POSN. I x POSN. 2 - - - o POSN. 3 - - -

5m ]= .u 20(n I

I

-J

C

jl,

. . . . . .

"~

-I0 I

-

2

90

0

ttO

~ Fig. 3.

130

-'--~==

I **l-"[ .... l"'t*'*t

150 170 FREQUENCY, Hz

190

Dissipative silencer SPL.

210

230

250

ATTENUATION OF A RESONANT DISSIPATIVE SILENCER IN A RIGID DUCT

133

Three cases were considered. The first case consisted of a 1-25 in thick acoustical liner of flow resistivity a = 10.0 (cgs) Rayls/in. As observed, the attenuation versus frequency characteristics are different for the three silencer systems considered. It is apparent that the theoretical model predicts reasonable values of IL over the entire range of frequencies tested. For the system under investigation it was found that when a low flow resistivity liner was used, a high degree of sound attenuation occurs in a narrow frequency band centred around the silencer resonant frequency. For Case IH, the resonance frequency was approximately equal to 138 Hz. When a high flow resistance liner was considered, a broader IL spectrum resulted with increased sound attenuation at the lower and higher frequencies. However, unlike a silencer with a low flow resistivity liner, the maximum value &attenuation decreased accordingly, dependent on the flow resistance of the liner. As an example (Fig. 2), a maximum attenuation of 11.0 dB]ft resulted when using an R~ = 5-0 (cgs) Rayls flow resistance liner, while when R~ was increased to 10"0 and 21.6 (cgs) Rayls the maximum attenuation decreased to 7.1 and 3.2 dB/ft respectively. As a further verification of the modelling technique, SPL measurements were taken in the silencer. The measurement locations are shown in Fig. 3. Results of the predicted and measured SPLs are presented in a plot of the difference in SPL without and with the silencer (SPLwo - SPLw) versus frequency (Fig. 3). The presence of the reactive cavity is illustrated by the reduced SPL at the surface of the liner adjacent to the cavity (position 2) at approximately 182 Hz (where the reactive depth is equal to one-quarter wavelength). The SPL at the back wall of the silencer here was found to be equal to the SPL of the incident sound wave. The effect of the inertance of the sound field in the duct adjacent to the silencer is illustrated by the lower SPLs at approximately 138 Hz (resonance frequency of the silencer system). The predicted resonance frequency was slightly lower than the measured value since the magnitude of the inertance term was based on arbitrarily chosen acoustic velocity and pressure profiles spatially lumped in a one-dimensional approximation. It should be noted that the SPLs on both sides of the liner (positions 2 and 3) are equal when the reactance at the silencer-duct intersection, (P*,/v*), becomes zero (166 Hz). Figure 4 shows the predicted and measured IL spectrum for a 1 ft long silencer. For the frequency range considered, these IL values did not differ from the IL values obtained for a 3 ft long silencer (compare Figs. 2 and 4). Some calculated magnitude and phase values of the 1 ft silencer system's characteristic reflection and transmission factors 1 are also presented in Figs. 5 and 6. As expected, the transmission factor assumes a maximum at 138 Hz with phase /~o = 0.0 tad. However, unlike a loss-free filter system, the maximum value of the reflection factor occurred at a frequency slightly higher than 138 Hz with phase c(o = 7t rad. The calculated spectra in Figs. 7 and 8 illustrate the change in levels and shape due to the liner thickness and flow resistivity. When the liner flow resistivity was

THOMAS J. TRELLA, PETER K. KASPER

134

IOO-

CALCULATED x

MEASURED

m

¢r) x

_•SDI--

X

I1¢

z

X

CUD

i 140

,do

80

'1 160

FREQUENCY,

Fig. 4, I,O- ~ . . = O t

t

i 180

~) 2 0

! 220

Hz

I L (! ft silencer).

L • I INCH

"" 0 . 8 0 I...

UV- 0.6~ 0,4U

~0.2O 70

Fig, 5.

I 90

I I10

~ I I O 150 FREQUENCY, H z

I 170

I 190

Characteristic factor magnitude us, frequency (I ft silencer).

I 210

ATTENUATIONOF A KESONANTDISSIPATIVESILENCF__EIN & RIGID DUCT 135

0.6 0.4

'-1°

a: O.Z

•

0-

,, 0 2 ~ 0.4-

J

L = I INCH 6 = 10.5(¢9=) RAYLS/IN.

~50 II0

150

Q6~0.8-

190

230 270 FREQUENCY, Hz

310

350

390

1.0 Fig. 6. Characteristic factor phase angle v s . frequency (1 ft silencer).

I0~ mo 8 1 ~.

~ L=0.5 INCHES

/

~

0" , IO.O (CQS)RAYLS/ IN. ~

=

70

I

90

I

I10

I

I

130 150 FREQUENCY, Hz Fig. 7. Calculated IL.

I

I"r0

.

/IN.

I

190

I

210

THOMAS J. TRELLA, PETER K. KASPER

136 12-

,o'

o" • IO.O(cos) RAYLS/IN. ~

0

.

5

INCHES

IM. ,% 8-

ul (/) 0,,J

o71.

6-

4-

2-

.jJ

0

i

70

i

90

IlO

i

i

130 150 FREQUENCY, Hz

I

i

170

190

Fig. 8. Calculated IL, a = 10"0 (cgs) Rayls/in. 40-

L • 1.5 INCHES } Rn, IS RAYLS (~ = I0.0 (cOil) RAYLS/IN. L : 0.5 INCHES "~ O" 30.0(¢gi} RAYLS/IN.J R0" 15 RAYL.S

30u) " 20-

RESISTANCE

I00

z

_z ~n ~. Ix

u

86 -IO-

102 118 134//,~,,,~FREQUENCY , Hz / ~ - i * T / / / 150

I i 166

-20-30-

-40-

f

f~~E

/

-50-60

Fig. 9. Silencer characteristic impedance us. frequency.

I

i82

i

210

ATTENUATIONOF A RESONANTDISSIPATIVESILENCERIN A RIGID DUCT

137

increased from 10 to 30 (egs) Rayls/in the maximum attenuation decreased, as expected, and the resonance frequency of the silencer system shifted from 138 Hz to 142 Hz. Conversely, when the liner thickness was varied from 0.5 to 1.5 in, increasing the flow resistance from 5 to 15 (cgs) Rayls, ~ e ~ resonance frequency decreased from 138 Hz to 130 Hz. Such shifts in resonance frequencies can best be explained in a frequency plot of the magnitude of the reactance and resistance at the surface of the liner intersecting the duct (Fig. 9). For a 1.5 in thick liner with a = 10-0 (cgs) Rayls/in, the zero reactance (i.e. minimum impedance) point was calculated at 158 Hz, while for a liner 0.5 in thick with a = 30.0 (cgs) Rayls/in the point of zero reactance occurs at 171 Hz. This difference in frequency is due to the inertance of the liner.

CONCLUSIONS A mathematical model was developed to predict the sound field contained within a resonant silencer--duct system. The resulting attenuation predicted compared favourably with measured values. The liner thickness and flow resistance were found to be important in determining the shape and magnitude of the IL spectrum. Silencers with thin or low flow resistivity liners exhibit a high degree of attenuation within a narrow frequency range. Broader IL spectra result when thick or high flow resistivity liners are used. From results of calculated IL spectra it was found that two liners of equal flow resistance do not exhibit identical IL spectra. Also, except possibly for small duet area cross-sections, the presence of the inertance of the induct sound field adjacent to the silencer shifts the frequency of maximum sound attenuation from the frequency where the silencer reactive cavity depth is equal to one-quarter wavelength. The results presented here provide some insight into the development of dissipative silencer systems. In addition, the theoretical modelling technique is considered to be useful as a design tool. Although flow was not considered in this paper, it is felt that an extension of the present techniques will provide good results.

REFERENCES 1. W. K. R. LIPPERT,Sound propagation in ducts with side branches, Acustiea, 7 (1957)pp. 138-45.

2. K. F. L^MBERT, Side branch insertion loss in a moving medium, J. Acoust. See. Amer., 28

0956) pp. 1059--62. 3. W. K. R. LIPPERT,A method of measuring discontinuity effects in ducts, Acustica, 4 0954) pp. 307-12. 4. M. E. DELANEY and E. N. BAZLEY, Acoustical properties of fibrous absorbent materials, Applied Acoustics, 3 (1970) pp. 105-16.

138

THOMAS J. TRELLA, PETER K. KASPER NOTATION

A1, A2, A3, A,, Bt, B2, B3, B4

b c f i k k2 L Lt L2 L, P*,, P*t R1

R/ U* n I)*

x: Xn z

Zo or/

0 Ol 3': Po ~y o)

acoustic velocity coupling coefficients acoustic pressure coupling coefficients dissipative silencers depth speed of sound of air frequency of sound, Hz =~'-1 wave number in-duct acoustic propagation constant adjacent to silencer porous liner thickness distance between source plane and dissipative silencer dissipative silencer length duct width, normal to dissipative silencer transformed acoustic pressure in region n = i . . . . . 4 in-duct transformed acoustic pressure adjacent to dissipative silencer porous materials specific flow resistance resistive component of porous materials characteristic impedance transformed acoustic velocity in region n = 1. . . . . 4 transformed acoustic velocity at entrance of dissipative silencer reactive component of porous materials characteristic impedance axial particle displacement co-ordinates, n = 1. . . . . 4 in-duct particle displacement co-ordinate normal to the silencer characteristic impedance of porous material characteristic acoustic impedance of air media attenuation constant in the porous material wavelength constant in the porous material dissipative silencer characteristic impedance factor dissipative silencers wave number factor complex propagation constant in the porous material mass density of air media material flow resistivity sound frequency, rad/s