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001
10.0 points
Two similar tuning forks produce sound waves
of the same 2
.
9 mPa amplitude. One tuning
fork
y
1
has a frequency 936 s
−
1
and the other
y
2
has a frequency of 1040 s
−
1
.
These two
tuning forks produce interference.
+
A
+
A

A
A
0
0
T
beat
T
wave
y
1
y
2
y
1
+
y
2
+
A

A
0
0
3
T
beat
6
T
beat
What is the number of beats per second
that is heard?
Correct answer: 104 Hz.
Explanation:
Let :
A
= 2
.
9 mPa
,
f
1
= 936 s
−
1
,
and
f
2
= 1040 s
−
1
.
+5
.
5

5
.
5
A
(mPa)
t
(milliseconds)
9
.
6 ms
19
.
2 ms
y
1
y
2
y
1
+
y
2
+5
.
5

5
.
5
A
(mPa)
t
(milliseconds)
28
.
8 ms
57
.
6 ms
Two harmonic waves are described by
y
1
=
A
cos(
k x

ω
1
t
)
and
y
2
=
A
cos(
k x

ω
2
t
)
.
f
2

f
1
2
= 52 s
−
1
,
f
1
+
f
2
2
= 988 s
−
1
,
T
wave
=
1
f
2

f
1
2
= 19
.
2 ms
,
ω
1
= 2
π f
1
= 2
π
(936 s
−
1
)
= 5881
.
06 rad
/
s
,
ω
2
= 2
π f
2
= 2
π
(1040 s
−
1
)
= 6534
.
51 rad
/
s
,
ω
2

ω
1
2
= 326
.
726 rad
/
s
,
ω
2
+
ω
1
2
= 6207
.
79 rad
/
s
,
oldhomework 39 – Turner – (58185)
1
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hinojosa (jlh3938) – oldhomework 39 – Turner – (58185)
2
and
y
1
+
y
2
= 2
A
cos
bracketleftbigg parenleftbigg
ω
1

ω
2
2
parenrightbigg
t
bracketrightbigg
×
cos
bracketleftbigg parenleftbigg
ω
1
+
ω
2
2
parenrightbigg
t
bracketrightbigg
A
= 2
A
cos
bracketleftbigg parenleftbigg
ω
1

ω
2
2
parenrightbigg
t
bracketrightbiggvextendsingle
vextendsingle
vextendsingle
vextendsingle
max
= 2 (2
.
9 mPa)
·
cos
bracketleftbigg
(

326
.
726 rad
/
s)
t
bracketrightbiggvextendsingle
vextendsingle
vextendsingle
vextendsingle
max
= 5
.
8 mPa
,
where
A
is the amplitude of the sum of the
two waves
y
1
+
y
2
,
and
A
is the amplitude of
either wave
y
1
or
y
2
.
The wave produced by the interference of
two waves of frequencies
f
2
and
f
1
, respec
tively, has frequency
f
wave
≡
f
2

f
1
2
,
where
f
2
> f
1
=
1040 s
−
1

936 s
−
1
2
= 52 s
−
1
,
and the period
T
wave
≡
1
f
wave
=
1
52 s
−
1
= 19
.
2 ms
.
Since we hear two beats per cycle of the re
sultant wave (
T
wave
= 2
T
beat
), the beat fre
quency is
f
beat
= 2
f
wave
= 2 (52 s
−
1
)
=
104 s
−
1
,
twice the frequency of the wave.
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 Spring '08
 Turner
 Work, Correct Answer, sound level, Hinojosa

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