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
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Duct Acoustics - aancl.snu.ac.kraancl.snu.ac.kr/aancl/lecture/up_file/_1554091631_chap.3 Duct Acou… · Duct Acoustics ⚫A single expansion-chamber ‘silencer’ • The simple
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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
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 =
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 ,
+=
+
−=
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
0=x lx =
0x
T
1A Area
2A AreaI
R
B
C
1A Area
l
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
///
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 =+
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
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
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)
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
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
kmn +−=
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
mn
xik
mn eeBeAa
xn
a
xmtp mnmn
3321 coscos, +
= −
x
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
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,
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=
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
+=
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
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
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
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
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
+=
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