Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics •...

23
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 ) 0 in 0 in / / / x Te x e R Ie p c x t i c x t i c x t i 0 x 0 x 0 x I R T 1 A Area 2 A Area 투과 반사 입사 : , : , : T R I

Transcript of Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics •...

Page 1: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

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

0x

0x0x

I

R

T

1A Area 2A Area

투과반사입사 :,:, : TRI

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Aeroacoustics 2008 - 2 -

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

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Aeroacoustics 2008 - 3 -

Transmission & Reflection of Plane Waves

❖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 ,

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Aeroacoustics 2008 - 4 -

Transmission & Reflection of Plane Waves

❖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

0x lx

0x

T

1A Area

2A AreaI

R

B

C

1A Area

l

Page 5: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 5 -

Transmission & Reflection of Plane Waves

❖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

///

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Aeroacoustics 2008 - 6 -

Transmission & Reflection of Plane Waves

❖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

Page 7: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 7 -

Transmission & Reflection of Plane Waves

❖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

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Aeroacoustics 2008 - 8 -

Transmission & Reflection of Plane Waves

❖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

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Aeroacoustics 2008 - 9 -

Transmission & Reflection of Plane Waves

❖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)

Page 10: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 10 -

Transmission & Reflection of Plane Waves

❖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

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Aeroacoustics 2008 - 11 -

Transmission & Reflection of Plane Waves

❖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

km n

Page 12: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 12 -

Transmission & Reflection of Plane Waves

❖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

m n

xik

m n eeBeAa

xn

a

xmtp mnmn

3321 coscos,

x

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Aeroacoustics 2008 - 13 -

Transmission & Reflection of Plane Waves

❖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

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Aeroacoustics 2008 - 14 -

Transmission & Reflection of Plane Waves

❖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,

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Aeroacoustics 2008 - 15 -

Transmission & Reflection of Plane Waves

❖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

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Aeroacoustics 2008 - 16 -

Transmission & Reflection of Plane Waves

❖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

Page 17: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 17 -

❖Normal transmission

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

Transmission & Reflection of Plane Waves

ITIR 2 ,

02

0011

00

cc

c

Page 18: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 18 -

Transmission & Reflection of Plane Waves

❖Normal transmission

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

Page 19: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 19 -

Transmission & Reflection of Plane Waves

❖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

Page 20: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 20 -

Transmission & Reflection of Plane Waves

❖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

Page 21: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 21 -

Transmission & Reflection of Plane Waves

❖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

Page 22: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 22 -

Transmission & Reflection of Plane Waves

❖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

Page 23: Duct Acoustics - AANCLaancl.snu.ac.kr/aancl/lecture/up_file/_1505879839_chap.3...Duct Acoustics • The algebraic equation when solved for R and T • However, the simple ‘silencer’does

Aeroacoustics 2008 - 23 -

Transmission & Reflection of Plane Waves

❖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