### Standing Wave - tenaya.physics.lsa.umich.edu

```Physics 140 – Winter 2014
April 21
Wave
Interference
and
Standing
Waves
1
Questions concerning
3
Reflections
A sinusoidal wave is generated by shaking one end (x = L)
of a fixed string so that the following traveling wave is
created:
y1 (x,t)  Acos(kx  t)
What is a possible equation for a wave y2(x, t) reflected from the
other (fixed) end of the string (x = 0)?
A)  A cos(kx   t)
C)  A cos(kx   t)
E)  Asin(kx   t)
String fixed at x = 0 
B) A cos(kx   t)
D) Asin(kx   t)
y(0, t) = 0 for all t
 Need rightward-moving wave and y2 (0,t)  y1 (0,t)
 y2 (0,t)  Acos(t)
Standing Wave – part 1
• A string is clamped at both ends and then plucked so that
it vibrates in a standing mode between two extreme
positions (a) and (c). Let upward motion correspond to
positive velocities. When the string is in position (b), the
instantaneous velocity of points along the string is...
a
A.
B.
C.
D.
zero everywhere
negative everywhere
positive everywhere
depends on position
b
c
node: never moves.
Standing Wave – part 2
• A string is clamped at both ends and then plucked so that
it vibrates in a standing mode between two extreme
positions (a) and (c). Let upward motion correspond to
positive velocities. When the string is in position (c), the
instantaneous velocity of points along the string is...
a
A.
B.
C.
D.
zero everywhere
negative everywhere
positive everywhere
depends on position
b
c
Plucking a string
• A string on a string instrument plays an A (440 Hz)
when plucked. If you put your finger down in the
middle of the string, and then pluck, you are mostly
likely to hear:
A.
B.
C.
D.
E.
Note A an octave higher (880 Hz)
Note A an octave lower (220 Hz)
Same note/tone
Another note [different from (A), (B), and (C)]
Nothing
f = 1/T = v/λ
 Reducing λ by 2 increases f by 2
Two Piano Strings
• Two piano strings have the same tension, are made from
the same material, and have the same length, but string
B is thicker (has larger diameter) than string A. How
will the pitch from string B compare to that of string A?
A.
B.
C.
D.
Lower than that of string A
Higher than that of string A
The same as that of string A
Not enough information
1 Ft
f1 
2L 
μ is greater for B
 Lower f
A driven string oscillating between two posts (fixed
boundaries) exhibits a standing wave pattern with
three nodes between the posts. If the tension in the
string is increased by a factor 4 with all else held
constant, including driving frequency, how many
nodes will there be between the posts?
1.
2.
3.
4.
5.
Since v 
F

One
Three
Five
Seven
Nine
Before:
After:
, increasing F by 4 increases v   f by 2
  increased by 2  2 antinodes  1 node
9
A Guitar String
A nylon guitar string of mass 6.48 g and length 0.9 m is
supposed to resonate at the fundamental frequency of 80
Hz. What tension should be applied to the string?
A.
B.
C.
D.
E.
59 N
83 N
149 N
317 N
550 N
v
v
1
f1 



2L
2L
Ft

m
where  
L
 Ft  2Lf1    4 f Lm
2
2
1
Ft  4(80.0 Hz) (0.9 m)(0.00648 kg) = 149 N
2
Resonant rod
A metal rod can be excited to resonate in its audible
fundamental mode by rubbing it at the right frequency.
What is the nature of this normal mode?
A) Transverse
B) Longitudinal
C) Torsional
The vibration is a sound wave (longitudinal) traveling back and
11
forth along the rod
y(x,t)  Asin(kx)sin( t)
Varying amplitude
A string with both ends held fixed is vibrating in its third harmonic.
The waves have a speed of 192 m/s and a frequency of 240. Hz.
The amplitude of the standing wave at an antinode is 0.400 cm.
What is the amplitude at a point on the string 10.0 cm from its left
end?
1)
2)
3)
4)
5)
0.141 cm
0.212 cm
0.283 cm
0.354 cm
0.425 cm
k
2

f
240. Hz
 2  2
 7.85 m 1
v
192 m/s
Amplitude at x is Asin(kx)
 (0.400 cm)sin[(7.85 m1 )(0.100 m) = 0.283 cm
12
Elapsed time
A string with both ends held fixed is vibrating in its third harmonic.
The waves have a speed of 192 m/s and a frequency of 240 Hz. The
amplitude of the standing wave at an antinode is 0.400 cm. How
much time does it take the string to go from its largest upward
displacement at x = 10.0 cm to its largest downward displacement?
1) 1.08 × 10-3 s
2) 1.48 × 10-3 s
3) 2.08 × 10-3 s
4) 2.48 × 10-3 s
5) 3.08 × 10-3 s
Need half the period:
1
1
1
T

 2.08  10 3 s
2
2 f 2(240. Hz)
13
Acceleration
Amp(10 cm) = 0.283 cm
A string with both ends held fixed is vibrating in its third harmonic.
The waves have a speed of 192 m/s and a frequency of 240 Hz. The
amplitude of the standing wave at an antinode is 0.400 cm. What is
the maximum transverse acceleration of the string at x = 10.0 cm?
1)
2)
3)
4)
5)
6.43 × 10−2 m/s2
6.43 × 10−1 m/s2
6.43 × 10+1 m/s2
6.43 × 10+2 m/s2
6.43 × 10+3 m/s2
Each point on the string behaves as a S.H.O.:
 Maximum acceleration amax (x)   2 A(x)
 amax (10.0 cm) = [2 (240.0 Hz)]2 (0.283 cm)
= 6.43  10 3 m/s2
14
Changing frequency
kg
kg
W  1000 3 and  A  2700 3
m
m
When a massive aluminum sculpture is hung from a steel wire, the
fundamental frequency for transverse standing waves on the wire is
250.0 Hz. The sculpture (but not the wire) is then completely
submerged in water. What is the new fundamental frequency?
1)
2)
3)
4)
5)
Submerged:
Ft

mg
198 Hz
224 Hz
250 Hz
279 Hz
302 Hz
1 Ft
f1 
2L 
f1submerged

original
f1
Ft
Fbuoy
Ft submerged
Ft original
Ft  mg  Fbuoy  AlumVg  WaterVg
Alum  Water
2700  1000

 0.793
Alum
2700
mg
15
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