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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
0x
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
0x 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
km n
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
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
high frequency waves are effectively stopped”
1
222
0
2
0
2
0
2
010
4
4log10
mc
cLT
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
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