duct acoustics - aancl.snu.ac.kraancl.snu.ac.kr/aancl/lecture/up_file/_1554091631_chap.3 duct...

Aeroacoustics 2008 - 1 -

Chap.4 Duct Acoustics

❖Duct Acoustics

⚫ Plane wave

• A sound propagation in pipes with different cross-sectional area

• If the wavelength of sound is large in comparison with the

diameter of the pipe the sound propagates as an one-dimensional

wave ( λ>>d → 1-d wave)

( ) ( )

( ) 0in

0in

/

//

=

+=

−

+−

xTe

xeRIep

cxti

cxticxti

0=x

0x0x

I

R

T

1A Area 2A Area

투과반사입사 :,:, : TRI


Transmission & Reflection of Plane Waves

❖Duct Acoustics• The mass flux into the junction must equal the mass flux out

• The velocity must equal at both sides of the junction

• Energy flux)in = Energy flux)out

• The pressure of both sides of junction is continuous

220110 uAuA =

( ) Tc

ARI

c

A

0

2

0

1

=−

TRI =+

2

'

221

'

11 upAupA =



❖Duct Acoustics• The amplitudes of other wave, R and T ,are can be solve from

above the relations

• The transmission loss , LT is symmetric in A1 and A2

( )

+=

=

=

21

2

21102

2

2

110

10

4log10 log10

power dtransmitte

powerincident log10

AA

AA

TA

IA

LT

IAA

ATI

AA

AAR

21

1

21

21 2 ,

+=

+

−=



❖Duct Acoustics

⚫A single expansion-chamber ‘silencer’

• The simple muffler that is a used in car ‘silencer’ consists of inlet

and outlet pipes with cross-sectional area A1, and expansion

chamber between them of cross-sectional area A2 and length l

0=x lx =

0x

T

1A Area

2A AreaI

R

B

C

1A Area

l



❖Duct Acoustics• The first area change occurs at x=0 and the second occurs at x=l.

• The condition of continuity of mass flux,

• The condition of continuity of pressure

( ) ( )

( ) ( )

( ) xlTe

lxCeBe

xeRIep

cxti

cxticxti

cxticxti

=

+=

+=

−

+−

+−

in

0in

0in

/

//

//

( ) ( )

( ) lxCeBeATeA

xCBARIA

cliclicli =−=

=−=−

−−− at

0at

//

2

/

1

21

lxCeBeTe

xI

cliclicli =+=

=+=+

−−− at

0at CBR

///



❖Duct Acoustics• The algebraic equation when solved for R and T

• However, the simple ‘silencer’ does not reduce the total energy of

sound in the system.

• Reducing the acoustic energy of transmitted wave

→ Increasing in the reflected wave

• Sound absorbing material

→ reduce the acoustic energy by converting it into heat or vibration

c

l

A

A

A

Aicl

c

lIi

A

A

A

A

R

sin/cos2

sin

1

2

2

1

1

2

2

1

++

−

=

c

l

A

A

A

Aicl

IeT

cli

sin/cos2

2

1

2

2

1

/

++

=

222ITR =+



❖Duct Acoustics• The transmission loss , LT is

• The transmission loss is maximum at frequencies for which

• The effect of expansion ratio

=

2

2

10log10T

ILT

sin4

11log10

2

1

2

2

110

−+=

c

l

A

A

A

ALT

etc...2

5,

2

3,

2 i.e. 1/sin

l

c

l

c

l

ccl

==

1

2

AA

m =

LT

(dB)

frequency

m increase



❖Duct Acoustics

⚫Note• ‘Tuning’ for dominant frequencies of noise

• Theory work for only λ≫d “Low frequency wave only”

• High frequency waves behave like 3-D

• Also, the geometrical shape of the duct is not important (provided

the area change occurs in a distance short in comparison with the

wavelength



❖Duct Acoustics

Effect of expansion chamber ratio Effect of expansion chamber shape

Ref. “Theoretical and experimental investigation of

mufflers with comments on engine-exhaust muffler

design”, Davis et al. NACA 1192(1954)



❖Higher order modes

• As an illustration, the sound of frequency ω in a rigid walled duct

of square cross-section with sides of length a is considered

• With substitution for p′ into the wave equation,

a

a

2x

1x

3x

( ) ( ) ( ) ( ) tiexhxgxftp

321, = x

2

2

2

−=−

−

−=

ch

h

g

g

f

f



❖Higher order modes• Since a wall boundary condition is applied, function f is derived

like this

• Similarly function g is derived like this

• Finally, function h is derived to the propagation form

• The axial phase speed, cp=ω/kmn is now a function of the mode

number and the propagation of a group of waves will cause them to

disperse.

( ) ma

xmAxf integer somefor , cos 1

11

=

( ) na

xnAxg integer somefor , cos 1

22

=

( ) 33

3

xik

mn

xik

mnmnmn eBeAxh +=

− ( )22

2

2

2

2

nmac

kmn +−=



❖Higher order modes• The pressure perturbation in the (m,n) mode has the form

• When kmn is real, the pressure perturbation equation represents that

waves are propagating down the x3 axis with phase speed.

• When kmn is purely imaginary, i.e. exceeds the cut-off frequency,

the strength of mode varies exponentially with distance along the

pipe. Such disturbances are evanescent

( ) tixik

mn

xik

mn eeBeAa

xn

a

xmtp mnmn

3321 coscos, +

= −

x



❖Pipes of varying cross-section

⚫Wave equation

• If the pipe diameter is small in comparison with both the acoustic

wavelength and the length scale over which the cross-sectional area

change, most particle motions are longitudinal.

• Conservation of mass

• Linearized momentum equation is

• Modified wave equation

( )xAx

( )uAxt

A

−=

0

x

p

t

u

−=

0

=

x

pA

xt

p

c

A2

2

2



❖Pipes of varying cross-section

⚫Application to the ‘exponential horn’

• Evaluation of the case of ‘exponential horn’ which cross-sectional

area defined as, A(x)=A0eαx

• For such an area variation of wave equation simplifies to

• The pressure perturbation in sound waves of frequency ω then has

the form

• Disturbance with ω > αc/2 propagates and the pressure but not the

energy flux attenuates during propagation, while lower frequency

modes are ‘cut-off’

x

p

x

p

t

p

c

+

=

2

2

2

2

2

1

( ) ( ) ( ) kxtikxtix BeAeetxp +−− += 2,



❖Normal transmission

⚫ Physics at the interface

• When a sound wave crosses an interface between two different

fluids some of the acoustic energy is usually reflected.

• There are two boundary conditions

◼The pressure on the two sides of the boundary must be equal

◼The particle velocities normal to the interface must be equal

( )0cxtiIe

−

( )0 cxti

eR+

( )1cxtiTe

−

00 , c11 , c( )0 interface =x

c

f

ccT

2===

0

0

2 c=

1

1

2 c=



❖Normal transmission• The pressure must be equal at the interface : I+R=T

• The particle velocities normal to the interface must be equal

• The result pressure coefficients, R and T , are determined with I

• Velocity Transmission Coefficient :

• The energy flux of the incident wave per unit cross sectional area is

equal to that of the reflected and transmitted waves

110000 c

T

c

R

c

I

=−

Icc

ccR

+

−=

0011

0011

I

cc

cT

+=

0011

112

( )( ) ( ) 00

2

11

2

0011

22

1

2

1

00

2

0011

22

0011

11

2

00

2 4

c

I

ccc

Ic

ccc

Icc

c

T

c

R

=

++

+

−=+

0011

00

00

11 2

/

/

cc

c

cI

cT

+=



⚫Reflection from a high and low impedance fluid

• A typical example is aerial sound waves incident onto a water

surface. (ρ0c0 ≪ ρ1c1 )

• Velocity transmission coefficient

so, the transmission wave carries negligible energy


ITIR 2 , ==

02

0011

00 +

=cc

c




⚫Reflection from a high and low impedance fluid

• In the opposite case, for sound in water incident onto a free surface

with air, the reflected and transmitted waves are

• The acoustic energy is totally reflected

0 , =−= TIR



❖Sound propagation through walls

⚫ Effect of a wall in transmission

• A sound wave normally incident on a plane material layer

partitioning a fluid which has uniform acoustic properties, ρ0c0

• Some sound will be reflected from the layer and some will be

transmitted through the wall



❖Sound propagation through walls• There are two boundary conditions that must be satisfied at all

times and points

◼The velocity of the wall must be equal to wave of each side

◼A pressure difference across the wall in order to provide the

force necessary to accelerate unit area of the surface of

material

• By continuity the velocity of wall,

• The pressure difference is the net force of mass per unit area of the

wall

( ) titi

ec

T

c

eRIu

0000

=−=

( ) titi ec

Tim

t

umeTRI

00

=

=−+

( )ti

ti

Tep

eRIp

=

+=

'

2

'

1



❖Sound propagation through walls• The result pressure coefficients, R and T , are determined with I

• Surface Impedance

• Energy transmitted

Imic

miR

+=

002I

mic

cT

+=

00

00

2

2

( )( )

( )mic

T

RIc

u

p

mi

mic

RI

RIc

u

p

+=+

=

−

+=

−

+=

0000

'

2

0000

'

1

1

1

00

2

222

0

2

0

2

0

2

0

00

2

4

4

c

I

mc

c

c

T

+=



❖Sound propagation through walls• The transmission loss is dependent on the frequency ω.

• For high frequency(ωm≫ρ0c0), the sound waves mostly reflected

• For low frequency(ωm≪ρ0c0), the sound waves mostly travels

through the wall with very little attenuation

• “Low frequency waves get through a massive wall easily, while

high frequency waves are effectively stopped”

1

222

0

2

0

2

0

2

010

4

4log10

−

+=

mc

cLT



❖Sound propagation through walls

⚫ Example) Attenuation by a wall

• Transmission loss

250m

kgm =

sec00 2410m

kgc =

Hzffor 10=

1

222

0

2

0

2

0

2

010

4

4log10

−

+=

mc

cLT

=

2

2

10log10T

ILT

dBLT 12

kHzffor 1= dBLT 50

I

duct acoustics - aancl.snu.ac.kraancl.snu.ac.kr/aancl/lecture/up_file/_1554091631_chap.3 duct...

Documents