42.4 Acoustic Characteristic Computation of Compound Wall

Back

42.4.1 Absorption Coefficient of Combined Plate with Porous Blanket

A common form of problem in noise control is the

need to reduce the sound radiated from a duct or

some other object. A way to achieve this is by lining

the duct with several centimeters of porous acoustic

material, and covering it with a solid plate of some

type, as indicated in Figure 42.5.

Consider the case of normal incidence with

the sound-absorbing structure of Figure 42.5.

Assume that the boundary conditions for the

sound pressure and the volume flow-rate are

identical. For plane wave incidence on the

hard wall, the magnitude of reflection coefficient

is 2 1 [1]. The following equation is obtained:

1 21 21 0

21 2m1 m1 0

0 e2gl1 egl1 2ð1 þ e22jkl2 Þ

0 m2e2gl1 m2egl1 2ð1 2 e22jkl2 Þ

2

66666664

3

77777775

B1

A1

B1

B2

2

66666664

3

77777775

¼

21

21

0

0

2

66666664

3

77777775

ð42:13Þ

FIGURE 42.3 Sound-absorption characteristics of a perforated plate structure: (a) cross sectional view;

(b) plan view.

FIGURE 42.4 Geometry of a Helmholtz resonator.

Volume, V ; is connected to an infinitely open area by a

neck tube of diameter d and length ln:

42-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

where

j ¼

ffiffiffiffi

21 p

m1 ¼ z0=z1; m2 ¼ z1=z2

z0; z1; z2: acoustic impedance of each medium

(Pa sec/m3)

g ¼ complex propagation constant (1/m)

The absorption coefficient for normal incidence is

given by the following equation:

a0 ¼ 1 2

B0

A0

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

2

¼ 1 2 lB0l2 ð42:14Þ

The absorption coefficient for random incidence

may be approximated by

a ¼

1

n

Xn

i¼1

aðuÞi ð42:15Þ

where u ¼ the incident angle of sound, 0 , u , p=2:

It is known that the propagation speed of the sound in fibrous materials changes with air, and the

following equation holds on the boundary surface:

sin u=sin u 0 ¼ c=cm ð42:16Þ

Here, cm is the sound speed in fibrous materials, which is calculated from the imaginary part of Equation

42.18, given later. The angle of reflection, u 0; in the boundary surface of the back air space is obtained in a

similar way. Hence, the following equation is substituted in Equation 42.13 instead of the thickness of the

absorber, l1; and the thickness of the air space, l2; to obtain the absorption coefficient in oblique

incidence:

l01

¼ l1=cos u 0; l02

¼ l2=cos u 00 ð42:17Þ

The complex propagation constant, g; is an important physical quantity in absorbing material of

propagated sound, which is given per unit length of acoustic attenuations, and phase changes. Between

the aeroelasticity rate, Ka; of absorbing material and the bulk modulus, Q; of absorbing material, g is

given by the following equation, for Ka . 20 Q [2,3]:

g ¼ jv

ffiffiffiffiffi

Y =K p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kr1l 2 jkR1l=v

q

ð42:18Þ

kR1l ¼

R1½1 2 r0ð1 2 Y Þ=rm􀀉

1 þ

r0ðk 2 1Þ

rm

􀀒 􀀓2

1 þ

R21

r2

mv2½1 þ r0ðk 2 1Þ=rm􀀉2

" #

kr1l ¼ r0k 2

R21

ðY =k þ rm=r0kÞ

r2

mv2½1 þ r0ðk 2 1Þ=rm􀀉2 þ

1 þ r0Y ðk 2 1Þ=rmk

1 þ r0ðk 2 1Þ=rm

1 þ

R21

r2

mv2½1 þ r0ðk 2 1Þ=rm􀀉2

where

rm ¼ density of acoustical material (kg/m3)

r0 ¼ density of air (kg/m3)

c0 ¼ speed of sound in air (m/sec)

FIGURE 42.5 Structure for sound absorption using a

blanket and an air space showing angles u1 in the air and

u2 in the blanket.

Design of Absorption 42-7

© 2005 by Taylor & Francis Group, LLC

K ¼ volume coefficient of elasticity of air (N/m2)

R1 ¼ alternating flow resistance for unit thickness of material due to the difference between the velocity

of the skeleton and the velocity of air in the interstices (Pa sec/m2). R1 values are given in Table 42.1

Y ¼ porosity ¼ the ratio of the volume of the voids in the material to the total volume; porosity equals

the total volume minus the fiber volume, all divided by total volume

k ¼ 5:5 2 4:5Y ; the structure factor of the interstices in the skeleton

v ¼ 2pf ; the angular frequency (radians/sec)

The acoustic impedance, z1; of absorbing material is given by

z1 ¼ R þ jX ¼ 2

jKg

vY ð42:19Þ

in which

R ¼ r0c0

n

1 þ 0:0571ðr0f =Rf Þ20:754

o

X ¼ 2r0c0

n

0:0870ðr0f =Rf Þ20:732

o

42.4.2 Transmission Loss through a Single Porous Board

Assume that a sound wave impinges on the left side of a porous board at normal incidence and emerges

with a reduced amplitude from the right side. The associated transmission loss of the porous board is

obtained from

TL0 ¼ 10 log10ðX þ Y Þ

X ¼ 1 þ

v2m2PRf

2r0c0ðv2m2P2 þ R2f

Þ

( )2

Y ¼

vmR2f

2r0c0ðv2m2P2 þ R2f

( )2

9>>>>>>>=

>>>>>>>;

ð42:20Þ

where

m ¼ surface density of the blanket (kg/m2)

P ¼ porosity of the blanket (porosity ¼ the total volume minus the fiber volume, all divided by the total

volume)

Rf ¼ specific flow resistance of material (Pa sec/m)

TABLE 42.1 Flow Resistance Values of Glass-Wool Board (Quality Regulation Range by JIS)

Board Type K value Gross Specific Gravity

(kg/m3)

Specific Flow Resistance

( £ 1023 N sec/m4)

Standard of JIS for Glass Wool

#1 Glass-wool board 8 8 ^2 1.5 , 7.0 JIS A 9505-A

12 12 ^2 2.5 , 12.0

16 16 ^2 4.7 , 17.0

20 20 ^3 5.0 , 22.0

24 24 ^3 6.5 , 27.0

#2 Glass-wool board 12 12 ^2 1.5 , 7.0 JIS A 9505-B

16 16 ^2 2.5 , 10.0

20 20 ^3 3.0 , 13.0

24 24 ^3 4.0 , 16.0

32 32 ^4 6.0 , 22.0

48 48 ^5 11.0 , 38.0

64 64 ^6 18.0 , 60.0

96 96 ^ 10 27.0 , 95.0

#3 Glass-wool board 96 96 ^ 10 15.0 , 40.0 JIS A 9505-C

42-8 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

r0 ¼ density of air (kg/m3)

c0 ¼ sound speed in air (m/sec)

42.4.3 Transmission Loss through a Sandwich Board

Consider a wide wall formed by two panels (sheets)

of infinite area separated with a homogeneous

filling of fibrous acoustical material, as shown in

Figure 42.6. Suppose that a plane wave impinges at

an angle u: As the pressure of both sides of the wall is

equal with regard to the amplitude of the progressing

wave and the reflected wave in each boundary

surface, the following result may be established [4]:

A0 þ B0 ¼ A1 þ B1

ðA0 2 B0Þ=z0 ¼ ðA1 2 B1Þ=z1

A1e2jkl01

þ B1ejkl01

¼ A2 þ B2

ðA1e2jkl01

2 B1ejkl01

Þ=z1 ¼ ðA2 2 B2Þ=z2

A2e2gl02

þ B2egl02

¼ A3 þ B3

ðA2e2gl02

2 B2egl02

Þ=z2 ¼ ðA3 2 B3Þ=z3

A3e2gl03

þ B3egl03

¼ A4 þ B4

ðA3e2gl02

2 B3egl02

Þ=z3 ¼ ðA4 2 B4Þ=z0

9>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>;

ð42:21Þ

where A and B are the amplitude of sound pressures.

From Equation 42.17, l01

¼ l1=cos u1; l02

¼ l2=cos u02

; and l01

and l02

may be calculated.

The speed of sound in the walls is given by the following equation in terms of the modulus of

longitudinal elasticity, Ei:

ci ¼

ffiffiffiffiffiffi

Ei=ri

p

ð42:22Þ

The real part of acoustic impedance, zi (i ¼ 1; 3), is given by Ri ¼ ri=cos u; and of the imaginary part is

given at Xi ¼ miv: The internal resistances, ri; are functions of such factors as the material, frequency,

temperature, and density. Some typical values are given in Table 42.2.

If the space of the transmission side is infinite, B4 in Equation 42.21 becomes equal to zero. Then, the

transmission loss is given is given by

TLðuÞ ¼ 10 log10

A4

A0

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

2

ð42:23Þ

FIGURE 42.6 Cross-sectional view of a sandwich

panel.

TABLE 42.2 Internal Resistance Values of Several Useful Materials

Material Thickness (mm) Internal Resistance (Pa sec/m3)

Aluminum 0.4 3.0

Plywood 3.0 7.5

Plaster board 7.0 15.0

Design of Absorption 42-9

© 2005 by Taylor & Francis Group, LLC

Навигация