Homework 3_rev
Acoustics I
due date: November 19
2007
(1) An expansion chamber muffler is shown below.
If,
2
S
a
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 0-3,000 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
A
B
T
T
T
T
B
a
T
T
T
T
A
B
A
B
A
B
Z
C
D
C
D
⎧
⎫
⎧
⎫
⎧
⎫
⎡
⎤
⎡
⎤
=
=
⎨
⎬
⎨
⎬
⎨
⎬
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦ ⎩
⎭
⎩
⎭
⎩
⎭
Q
Q
P
P
P
Q
.
(1)
where, the particle velocity
B
o
c
ρ
=
P
U
at
B
,
B
B
o
S
S
c
ρ
=
=
Q
U
P
; therefore
o
B
a
B
c
Z
S
ρ
=
=
P
Q
in this particular case.
There is only one element between the input and output ports; therefore,
1
1
1
1
1
1
sin
cos
sin
cos
o
T
T
T
T
o
jS
kL
kL
c
A
B
C
D
j
c
kL
kL
S
ρ
ρ
⎡
⎤
⎢
⎥
⎡
⎤
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣
⎦
⎢
⎥
⎣
⎦
.
(2)
Because
P
P
and
A
i
=
+
r
P
(
A
i
o
S
c
ρ
=
−
)
r
P
P
Q
, Eq. (1) can be re-written;
(
o
i
r
T
T
a
c
A
B Z
S
)
B
ρ
−
=
+
P
P
Q
(3)
(
)
i
r
T
T
a
C
D Z
+
=
+
P
P
Q
B
(4)
Dividing Eq. (3) by Eq. (4), we obtain
1
1
p
i
r
o
T
T
a
i
r
T
T
a
c A
B Z
S
C
D Z
ρ
p
−
−
+
=
=
+
+
R
P
P
P
P
R
+
.
(5)
1

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