Homework Assignment #12 Solutions  1
ECE 473/TAM 413
Homework Assignment #12 Solutions
Final Exam
: Monday, May 5, 2008; 1:30  4:30 pm, 163 EL
1. A ventilation duct propagates a strong and very annoying signal at 250 Hz. It is desired to use
a Helmholtz resonator to eliminate that signal. One engineer has already drilled a hole of
diameter 1 cm in the thin walled duct. What volume must you make for the resonator in order to
eliminate the annoying signal?
L'
=
1.6a
=
1.6 0.005m
(
)
=
0.008m
,
S
=
!
a
2
=
0.00007854m
2
, and c = 343 m/s
V
=
c
2
S
!
o
2
L'
=
468 cm
3
2. Pipe 1 branches into pipes 2 & 3 at x = 0, where each pipe has the cross sectional areas:
S
1
=
0.04m
2
,
S
2
=
0.02m
2
and
S
3
=
0.01m
2
. Pipes 2 & 3 can then be considered to extend to
infinity. If the incident wave has a pressure amplitude of 10 Pa, then (a) determine the pressure
amplitude of the reflected wave and (b) determine the pressure amplitudes of the waves
transmitted down pipes 2 & 3.
Let
Z
o
be the acoustic impedance that the incident wave sees at x = 0. Then
a)
Z
o
=
Z
2
Z
3
Z
2
+
Z
3
=
!
o
c
S
2
!
o
c
S
3
!
o
c
S
2
+
!
o
c
S
3
=
!
o
c
S
2
+
S
3
R
=
p
r
p
i
=
Z
o
!
"
o
c
S
1
Z
o
+
"
o
c
S
1
=
"
o
c
S
2
+
S
3
!
"
o
c
S
1
"
o
c
S
2
+
S
3
+
"
o
c
S
1
=
S
1
!
S
2
!
S
3
S
1
+
S
2
+
S
3
=
0.143
Therefore,
p
r
=
Rp
i
=
1.43Pa
b)
p
2
=
p
3
=
p
i
+
p
r
=
10
+
1.43
=
11.43Pa
3. Problem 10.8.1 in Kinsler et al. Use flanged approach. For part (d), it may be useful to use the
arguments on pp 284285 of the text for estimating effective mass and effective stiffness.
a)
V
=
4
3
!
0.1 m
2
"
#
$
%
&
’
3
=
0.0005236 m
3
L'
=
1.7a
,
S
=
!
a
2
, and c = 343 m/s
!
o
=
c
S
L' V
=
343 m / s
"
a
2
1.7a
(
)
0.0005236 m
3
(
)
=
20,377
a
!
o
=
2
"
f
=
2
"
320 / s
(
)
=
2,010.6
=
20,377
a
#
a
=
0.00974 m
Diameter of opening
=
2a
=
1.95 cm
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Homework Assignment #12 Solutions  2
b) From Eqs. 10.8.12 and 10.8.13,
Q
=
P
c
P
=
2
!
V
L'
S
"
#
$
%
&
’
3
Thus,
P
+
20
μ
bar
P
=
2
!
0.0005236
1.7a
!
a
2
"
#
$
%
&
’
3
where
a
=
0.00974 m
P
+
20
μ
bar
P
=
59.536
!
P
=
0.342
μ
bar
c)
S
new
=
2S
part(a)
;
Radius
new
=
2
Radius (from part (a))
!
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 Fall '08
 OBRIAN
 Gate, Pallavolo Modena, Sisley Volley Treviso, Associazione Sportiva Volley Lube, Piemonte Volley, Kinsler

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