44.2 Application of Sound Insulation

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44.2.1 Acoustic Enclosure

Performance of an enclosure may be represented by the insertion loss (IL), which is the difference of

acoustic power level before and after installation of the enclosure. When we assume a noise source and

also an enclosure with one-dimensional model as shown in Figure 44.14a, the insertion loss through

frequency is shown in Figure 44.14b. It is divided into the following four regions:

Mass law

18 log[(m1 + m2) f ]− 44

√2 fr 2 fr 1/3 min ( fc1, fc2)

Frequency (Hz)

Transmission loss TL (dB)

Double wall with

sandwiched porous

material

(18 dB/octave)

Double wall with air space

(12 dB/octave)

6 dB/octave

D TL

FIGURE 44.13 Design chart for estimating the transmission loss of a double wall with sound bridges.

Design of Sound Insulation 44-13

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1. Region I ð f , frÞ is controlled by the stiffness of the enclosure plate and air space.

2. Region II ðf ø frÞ is the resonance region for a vibrating system consisting of the mass, the stiffness

of the enclosure plate, and the capacitance of air space.

3. Region III ðfr , f , c=2dÞ is controlled by the mass of the enclosure plate.

4. Region IV ðf . c=2dÞ is controlled by the diffused sound field. This region cannot be represented

by a one-dimensional model.

44.2.1.1 Near Field Type [4]

When the distance between the noise source and the enclosure is less than half of a wavelength of the

emitted sound from the source, insertion loss corresponds to the characteristics of the regions I to III, and

we can represent them with a one-dimensional acoustic model, as shown in Figure 44.14a.

Consider an infinite flat enclosure plate with distance d from a noise source of plane sound wave, as

shown in Figure 44.14a. The insertion loss of the plate is given by

IL ¼ 10 log

v0

v1

􀀏 􀀐2

¼ 10 log 1 2

2 sin uðX cos u 2 R sin uÞ

rc þ

sin2uðX2 þ R2Þ

r2c2

" #

u ¼ kd ¼ vd=c; R ¼ hvm; X ¼ ðvm 2 K=vÞ ¼ vm½1 2 ðv11=vÞ2􀀉; v11 ¼

ffiffiffiffiffiffi

K=m p

ð44:65Þ

where m and K are the density and equivalent stiffness of the plate per unit area, respectively, and h is the

loss factor of the plate. If the enclosure is a rectangular plate of size a £ b and is simply supported at its

edges, the equivalent stiffness is given by Equation 44.22, and v11 is the natural (angular) frequency of the

first mode.

In Equation 44.65, the conditions in which the brackets of the right-hand side are equal to zero or

IL ¼ 21 are satisfied by following frequencies:

(1) u ,, p

fr ¼

1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K þ rc2=d

m

s

¼

1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2

11 þ

rc2

md

s

ð44:66Þ

This is the natural frequency of vibration of the one-degree-of-freedom (one-DoF) system determined by

the stiffness of the plate, the spring constant of the air space, and the surface density of the plate, as shown

in Figure 44.15.

FIGURE 44.14 One-dimensional model for calculating the insertion loss characteristics of an acoustic enclosure:

(a) One-dimensional model; (b) insertion loss of an enclosure.

44-14 Vibration and Shock Handbook

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(2) u ¼ np ðn ¼ 1; 2; …Þ

fn ¼

nc

2d ðn ¼ 1; 2; …Þ ð44:67Þ

These are the resonant frequencies of the air space.

The frequency characteristics of the IL given by

Equation 44.65 are shown in Figure 44.16, where

the normal incidence transmission losses are

shown by broken lines, as a reference. Equation

44.65 is approximated by

1. f , fr

IL ø 10 log 1 þ

2Kd

rc2

􀀏 􀀐

ð44:68Þ

2. fr # f , f1

IL ø 20 logðmdf 2Þ þ 20 log

4p2

rc2

􀁻 !

¼ 20 logðmdf 2Þ 2 71 ð44:69Þ

44.2.1.2 Far Field-Type (Absorption Type)

Enclosure [1]

When the distance between the noise source and

the enclosure is larger than half of a wavelength of

the emitted sound, insertion loss may be represented

by the characteristics of Region IV, and it

can be analyzed using the theory of room or hall

acoustics.

Consider the enclosure shown in Figure 44.17,

with a noise source of power level LW0:

From the theory of room acoustics, the average

sound pressure level, LP0; on the inner surface of

the enclosure plate is obtained as the sum of the

direct and reverberant sound pressures:

LP0 ¼ LW0 þ10 log

1

S þ

4

R

􀀏 􀀐

¼ LW0 2 10 log S þ10 log 1 þ

4S

R

􀀏 􀀐

ð44:70Þ

R ¼

a􀀊S

ð12a􀀊Þ ð44:71Þ

where S is the inner surface area of the enclosure

and a􀀊 is the average absorption coefficient on the

inner surface of the enclosure. In Equation 44.70, the first and the second terms of the right-hand side

represent the influence of the direct sound field, and the third term represents the buildup caused by the

covering.

When we use SP for the area of the enclosure plate and SOð¼ S 2 SPÞ for the area of the enclosure

opening, and assume diffusing condition for the sound field in the enclosure, acoustic power levels

10

−30

−20

−10

0

10

20

30

40

50

60

100

Frequency (Hz)

Insertion loss (dB)

1000

Mass law TL

m = 16 Kg/m2, d = 0.19 m

Curve A: f11 = 0 Hz, h =~ 0.033 (at 33 Hz)

Curve B: f11 = 100 Hz, h =~ 0.033 (at 105 Hz)

Curve C: f11 = 475 Hz, h =~ 0.033 (at 475 Hz)

A

B

C

Resonance in air space

FIGURE 44.16 Example of theoretical insertion loss of

a near field-type enclosure.

FIGURE 44.17 Calculation model of a far field type

enclosure.

d

K +

m

V0

V1

rc2

FIGURE 44.15 One-degree-of-freedom vibrating

system.

Design of Sound Insulation 44-15

© 2005 by Taylor & Francis Group, LLC

radiated from the plate and the opening are given,

respectively, by

LWP ¼ LP0 2 ðTL þ 6Þ þ 10 log SP;

LW0 ¼ LP0 2 6 þ 10 log SO

ð44:72Þ

where TL is the random incident transmission loss

of the enclosure plate. Then, the insertion loss of

the enclosure is

IL ¼ LW0 2 10 log 10LWP =10 þ 10LW0 =10

􀀍 􀀎

ð44:73Þ

In the design of an acoustic enclosure, special

attention should be paid to the following points:

1. Buildup increases the sound pressure in the

enclosure and also the power level radiated

from the enclosure. We must treat the inner

surface of the enclosure with sound absorbing materials to reduce the absorption coefficient and

to decrease the buildup.

2. Opening radiates more acoustic power than the enclosure plate by TL, as is clear from Equation

44.72. It is desirable to make the opening as small as possible within the range given by

SO

SP

# 102TL=10 ¼ t ð44:74Þ

This equation means that the acoustic power from the opening is less than that from the enclosure

plate. If the relation in Equation 44.74 cannot be satisfied because of ventilation requirements, and

so on, some type of silencers should be provided at the opening, as shown in Figure 44.18.

3. Structure-borne noise, which is caused by the vibration propagating from the base of the machine

(noise source) to the enclosure plate, significantly decreases the insertion loss of the enclosure.

In this case, some means of noise/vibration suppression should be provided, for example, the

following:

* Place supporting structures of the enclosure at the points of lowest vibration level, and the

vibrations of the machine should be prevented from propagating to the enclosure plate, using

vibration isolation materials.

* Add damping materials to the enclosure plate so as to reduce the vibration level of the plate.

44.2.2 Sound Insulation Lagging

In electric power plants and chemical plants, for example, piping for high-pressure water or steam, and

ducts for air or gas flow form major noise sources. For controlling these noise sources, we usually use

sound insulation laggings, which cover the noise sources with heavy and impermeable plates or sheets

with sound absorbing materials, as shown in Figure 44.19.

44.2.2.1 Pipe Lagging [2]

Approximate the cylindrical piping and pipe lagging with a one-dimensional model as shown in

Figure 44.20. The insertion loss of one-layered lagging approximated by a one-DoF system is given by

Equation 44.69 in the frequency region fr # f , f1; as mentioned before. It is not practical, however, to

directly apply Equation 44.69 to actual laggings, and we approximate the insertion loss of actual laggings

by

IL ¼ a logðmdf 2Þ þ b

where a and b are constants.

FIGURE 44.18 Example of a silencer at the opening.

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By taking mdf 2 as the horizontal axis and plotting the insertion loss data from laboratory tests and

field tests (the vertical coordinates), we obtain Figure 44.21. Apply regression analysis to the data in

Figure 44.21 to obtain the insertion loss of one layered lagging as

IL ¼ 11:7 logðmdf 2Þ 2 43:3 ð5 £ 103 # mdf 2 # 108Þ ð44:75Þ

Applying the same method to double layered lagging, approximated by a two-DoF vibrating system,

we get Figure 44.22, and the insertion loss

IL ¼ 6:9 logðm1m2d1d2 f 4Þ 2 40:3 ð106 # m1m2d1d2 f 4 # 1015Þ ð44:76Þ

where the subscripts “1” and “2” denote the first layer and the second layer, respectively.

FIGURE 44.19 Examples of typical sound insulation laggings: (a) pipe lagging, (b) duct lagging.

FIGURE 44.20 Examples of pipe laggings and calculation model: (a) one-layered lagging; (b) double-layered

lagging.

Design of Sound Insulation 44-17

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44.2.2.2 Duct Lagging [4]

Various types of the duct laggings are used according to the need, for example, as shown in Figure 44.19.

A simpler and more practical approach is to place a thin plate on the duct casing through absorbing

materials, as shown in Figure 44.23. In this case, assuming that vibration of the duct casing is not affected

by the placed plate, the insertion loss of the duct lagging is obtained by following equations, which can be

deduced from the method used in the transmission loss of double wall with sound bridges.

FIGURE 44.21 Measured insertion loss data obtained from laboratory and field tests for one-layered laggings and

regression analysis.

FIGURE 44.22 Measured data of insertion loss obtained from laboratory and field tests for double-layered laggings

and regression analysis.

44-18 Vibration and Shock Handbook

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1. Point connection

IL ¼ 210 log b2 8

p3 nP

c2

f 2

c þ

fr

f

􀀏 􀀐4

" #

þ 10 log s ð44:77Þ

2. Line connection

IL ¼ 210 log 0:64nL

c

fc þ

fr

f

􀀏 􀀐4

" #

þ 10 log s ð44:78Þ

where

b ¼ vibration isolation factor of the flexible support (b ¼ 1 for rigid support)

nP ¼ number of attachment points per unit area

nL ¼ number of studs per unit length

FIGURE 44.23 Examples of duct laggings and connection types of thin plate to the duct casing: (a) point

connection; (b) line connection.

FIGURE 44.24 Plateau height for point connection as a function of the number of bridges per unit area [4]. (Source:

Beranek, L. L. 1988. Noise and Vibration Control, INCE/USA. With permission.)

Design of Sound Insulation 44-19

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fc ¼ critical frequency of the thin plate given by Equation 44.12

fr ¼ resonant frequency given by Equation 44.66

s ¼ sound radiation efficiency of the thin plate

Knowing the critical frequency, fc; and the resonant frequency, fr; we can obtain the insertion loss from

the charts given in Figure 44.24 and Figure 44.25 instead of using Equation 44.77 and Equation 44.78. In

Figure 44.24 and Figure 44.25, it is assumed that s ¼ 1: Note from Equation 44.77 and Equation 44.78

that we must consider the following measures to obtain a higher insertion loss.

1. Make the distance between attachment points or studs as large as possible (decrease nP and nL).

2. Make the air space as large as possible (decrease fr).

References

1. Shiraki, K., ed. 1987. From Designing of Noise Reduction to Simulation (in Japanese), Ouyou-gijutsu

Shuppan, Chiyoda-ku, Tokyo.

2. Tokita, Y., ed. 2000. Sound Environment and Control Technology, Vol. I, Basic Engineering

(in Japanese), Fuji-techno-system, Bunkyo-ku, Tokyo.

3. Okura, K. and Saito, Y., Transmission loss of multiple panels containing sound absorbing materials

in a random incidence field, Inter-noise, 78, 637, 1978.

4. Beranek, L.L., ed. 1988. Noise and Vibration Control, INCE/USA, Ames, IA.

5. Sharp, B.H. 1973. A study of techniques to increase the sound insulation of building elements,

Wyle Laboratory Report WR 73-5, El Segundo, CA.

FIGURE 44.25 Plateau height for line connection as a function of the number of studs per unit length [4].

(Source: Beranek, L. L. 1988. Noise and Vibration Control, INCE/USA. With permission.)

44-20 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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