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Homework 3_rev
Acoustics I
due date: November 19
2007
(1) An expansion chamber muffler is shown below.
If,
2
Sa
π
=
,
a
is 2 cm,
, temp = 80
1
20
S
=
S
B
0
C,
calculate and plot the power transmission
coefficients
T
for the muffler with an anechoic termination shown below for the
frequency range 03,000 Hz.
Π
i
P
r
P
S
S
1
S
L
1
∞
A
B
Fig. 1.
Simple expansion chamber
L
1
was not given.
One can choose 0.3 m that was given in version 1 or 0.55 m, the total
length of the muffler given in problem 2.
The overall four pole equation of the system with an anechoic termination always
becomes
1
AB
TT
B
a
Z
CD
⎧⎫
⎧
⎫
⎡⎤
==
⎨⎬
⎨
⎬
⎢⎥
⎣⎦
⎩⎭
QQ
P
PP
Q
.
(1)
where, the particle velocity
B
o
c
ρ
=
P
U
at
B
,
BB
o
S
S
c
QU
P
; therefore
o
B
a
B
c
Z
S
P
Q
in this particular case.
There is only one element between the input and output ports; therefore,
11
1
1
1
1
sin
cos
sin
cos
o
o
j
Sk
L
kL
c
jc k
L
kL
S
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣
⎦
.
(2)
Because
and
Ai
=+
r
P
(
o
S
c
=−
)
r
P
P
Q
, Eq. (1) can be rewritten;
(
o
ir
T T
a
c
Z
S
)
B
−=
+
Q
(3)
( )
T
T
a
Z
+=
+
Q
B
(4)
Dividing Eq. (3) by Eq. (4), we obtain
1
1
p
oT T
a
T
T
a
cA BZ
SC DZ
p
−
−+
++
R
R
+
.
(5)
1
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View Full Document R
p
is obtained as
1
1
p
x
x
−
=
+
R
(6)
where,
oT
T
a
TT
cA BZ
x
SC DZ
a
ρ
+
=
+
.
Finally,
2
1
p
T
Π
=−
R
.
T
Π
with the given configuration is
obtained as in Fig. 2.
Each peak in Fig. 2 corresponds to the resonance frequency of the
bothend closed pipe of length
L
1
(30 cm).
0
500
1000
1500
2000
2500
3000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
frequency (Hz)
Power Trans. Coeff.
Fig. 2 Power transmission coefficient of the system in Fig. 1.
More often transmission loss (TL) is used, which is defined as
10
1
10log
TL
T
Π
⎛⎞
=
⎜
⎜
⎝⎠
⎟
⎟
.
(7)
TL is plotted in Fig. 3.
0
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This note was uploaded on 10/19/2009 for the course MECH 666 taught by Professor Kim during the Fall '07 term at University of Cincinnati.
 Fall '07
 Kim

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