Transcript Standing Waves - Erwin Sitompul

Lecture 3
Ch16. Transverse Waves
University Physics: Waves and Electricity
Dr.-Ing. Erwin Sitompul
http://zitompul.wordpress.com
2013
Homework 2: Phase Differences
A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s.
(a) How far apart are two points that differ in phase by π/3
rad?
(b) What is the phase difference between two displacements
at a certain point at times 1 ms apart?
Erwin Sitompul
University Physics: Wave and Electricity
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Solution of Homework 2: Phase Differences
f  500 H z, v  3 5 0 m s
(a)  
v




(b) T 
2
1


T
Erwin Sitompul
 x
1

2
 
 3
2
(0.7 )  0 .1 1 7  11.7 cm
 0 .0 0 2 s  2 m s
500
f
t
 0 .7 m
500
f
x
350

2
 
t
T
2 
1 ms
2    rad
2 ms
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Example 1
A wave traveling along a string is described by
y(x,t) = 0.00327sin(72.1x–2.72t),
in which the numerical constants are in SI units.
(a) What is u, the transverse velocity of the element of the
string, at x = 22.5 cm and t = 18.9 s?
x  22.5 cm , t  18.9 s
y ( x , t )  (3.27 m m ) sin(72.1 x  2.72 t )
u ( x, t ) 
y ( x, t )
t
 (  2 .7 2 rad s )(3 .2 7 m m ) co s(7 2 .1 x  2 .7 2 t )
 (  8 .8 9 4 m m s) co s(7 2 .1 x  2 .7 2 t )
u (0 .2 2 5 m ,1 8 .9 s )  (  8 .8 9 4 m m s ) co s(  3 5 .1 8 5 5 r ad )
  7.197 m m s
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Example 1
A wave traveling along a string is described by
y(x,t) = 0.00327sin(72.1x–2.72t),
in which the numerical constants are in SI units.
(b) What is the transverse acceleration ay of the same element
of the spring at that time?
x  22.5 cm , t  18.9 s
u ( x , t )  (  8 .8 9 4 m m s ) co s(7 2 .1 x  2 .7 2 t )
a y ( x, t ) 
u ( x, t )
t
 (  2 .7 2 rad s )(  8 .8 9 4 m m s )   sin (7 2 .1 x  2 .7 2 t ) 
  (24.192 mm s ) sin(72.1 x  2.72 t )
2
a y (0 .2 2 5 m ,1 8 .9 s)   (2 4 .1 9 2 m m s ) sin (  3 5 .1 8 5 5 rad )
2
  1 4 .2 1 m m s
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2
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The Principle of Superposition for Waves
 It often happens that two or more
waves pass simultaneously through
the same region (sound waves in a
concert, electromagnetic waves
received by the antennas).
 Suppose that two waves travel
simultaneously along the same
stretched string, the displacement of
the string when the waves overlap is
then the algebraic sum.
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The Principle of Superposition for Waves
 Let y1(x,t) and y2(x,t) be two waves travel simultaneously
along the same stretched string, then the displacement of the
string is given by:
y ( x , t )  y1 ( x , t )  y 2 ( x , t )
 Overlapping waves algebraically add to produce a resultant
wave (or net wave).
 Overlapping waves do not in any way alter the travel of each
other.
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Interference of Waves
 Suppose there are two sinusoidal waves of the same
wavelength and the same amplitude, and they are moving
in the same direction, along a stretched string.
 The resultant wave depends on the extent to which one wave
is shifted from the other.
 We call this phenomenon of combining waves as
interference.
y1 ( x , t )  y m sin( kx   t )
y 2 ( x , t )  y m sin( kx   t   )
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Interference of Waves
 The resultant wave as the superposition of y1(x,t) and y2(x,t)
of the two interfering waves is:
y ( x , t )  y1 ( x , t )  y 2 ( x , t )
 y m sin( kx   t )  y m sin( kx   t   )
 2 y m sin( kx   t  12  ) cos( 12  )
y  ( x , t )   2 y m co s( 12  )  sin ( kx   t  12  )
 The resultant sinusoidal wave – which is the result of an
interference – travels in the same direction as the two original
waves.
sin   sin   2 sin 12 (   ) cos 12 (   )
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University Physics: Wave and Electricity
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Interference of Waves
Fully constructive
interference
Erwin Sitompul
Fully destructive
interference
Intermediate
interference
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Phase Difference and Resulting Interference Types
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3/11
Checkpoint
Here are four possible phase differences between two identical
waves, expressed in wavelengths: 0.2, 0.45, 0.6, and 0.8.
Rank them according to the amplitude of the resultant wave,
greatest first.
Rank: 0.2 and 0.8 tie, 0.6, 0.45
1  2  rad ian s  3 6 0 
A m plitude  y m  2 y m cos( 12  )
  0.2   0.4  radians  72 
 cos( 12  72  )  0.809
  0.45   0.9  radians  162 
 cos( 12  162  )  0.156
  0.6   1.2  radians  216 
 cos( 12  216  )   0.309
  0.8   1.6  radians  288 
 cos( 12  288  )   0.809
Erwin Sitompul
University Physics: Wave and Electricity
3/12
Example 2
Two identical sinusoidal waves, moving in the same direction
along a stretched string, interfere with each other. The
amplitude ym of each wave is 9.8 mm, and the phase difference
Φ between them is 100°.
(a) What is the amplitude ym’ of the resultant wave due to the
interference, and what is the type of this interference?
y m  2 y m cos( 12  )  2(9.8 m m ) cos( 12  100  )  12.599 m m
The interference is intermediate, which can be deducted in
two ways:
1. The phase difference is between 0 and π radians.
2. The amplitude ym’ is between 0 and 2ym.
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3/13
Example 2
Two identical sinusoidal waves, moving in the same direction
along a stretched string, interfere with each other. The
amplitude ym of each wave is 9.8 mm, and the phase difference
Φ between them is 100°.
(b) What phase difference, in radians and wavelengths, will
give the resultant wave an amplitude of 4.9 mm?
y m  2 y m cos( 12  )
4.9 m m  2(9.8 m m ) cos(  )
1
2
cos(  )  
1
2
4.9 m m
2(9.8 m m )
cos(  )   0.25
1
2
1
2
  1.3181 or 1.8235
  2.6362 or 3.6470
x



2
x
2.636
2

  0 .4 2 0 
  0.420 w avelength
   2.636 rad
Erwin Sitompul
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3/14
Standing Waves
 The following figures shows the superposition of two waves
of the same wavelength and amplitude, traveling in opposite
direction.
• Where?
 There are places along the string, called nodes, where the
string never moves. Halfway between adjacent nodes, we
can see the antinodes, where the amplitude of the resultant
wave is a maximum.
• Where?
 The resultant wave is called standing waves because the
wave pattern do not move left or right.
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Standing Waves
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Standing Waves
 To analyse a standing wave, we represent the two combining
waves with the equations:
y1 ( x , t )  y m sin( kx   t )
y 2 ( x , t )  y m sin( kx   t )
 The principle of superposition gives:
y ( x , t )  y1 ( x , t )  y 2 ( x , t )
 y m sin( kx   t )  y m sin( kx   t )
y  ( x , t )   2 y m sin kx  co s  t
sin   sin   2 sin 12 (   ) cos 12 (   )
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Standing Waves
 For a standing wave, the amplitude 2ymsinkx varies with
position.
 For a traveling wave, the amplitude ym is the same for all
position.
0
N
x
N
AN
Erwin Sitompul
N
AN
N
AN
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Standing Waves
y  ( x , t )   2 y m sin kx  co s  t
 In the standing wave, the amplitude is zero for values of kx
that give sinkx = 0.
kx  n ,

xn
for n  0,1, 2,
, for n  0,1, 2,
•Nodes
2
 In the standing wave, the amplitude is zero for values of kx
that give sinkx = ±1
kx 
,2,2,
2
1
3
5
kx  ( n  2 ) ,
1
for n  0,1, 2,
•Antinodes
1

x   n   , for n  0,1, 2,
22

Erwin Sitompul
University Physics: Wave and Electricity
3/19
Checkpoint
Two waves with the same amplitude and wavelength interfere
in three different situations to produce resultant waves with the
following equations:
(a) y’(x,t) = 4sin(5x–4t)
(b) y’(x,t) = 4sin(5x)cos(4t)
(c) y’(x,t) = 4sin(5x+4t)
In which situation are the two combining waves traveling (i)
toward positive x, (ii) toward negative x, and (ii) in opposite
directions?
• Toward positive x: (a), the sign before t is negative
• Toward negative x: (c), the sign before t is positive
• In opposite directions: (b), resulting standing wave
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3/20
Standing Waves and Resonance
 Consider a string, such as a guitar
string, that is stretched between two
clamps.
 If we send a continuous sinusoidal
wave of a certain frequency along the
string, the reflection and interference
will produce a standing wave pattern
with nodes and antinodes like those in
the figure.
 Such a standing wave is said to be
produced at resonance. The string is
said to resonate at a certain resonant
frequencies.
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3/21
Standing Waves and Resonance
 For a string stretched between
two clamps, we note that a node
must exist at each of its end,
because each end is fixed and
cannot oscillate.
 The simplest patterns that meets
this requirement is a single-loop
standing wave, with two nodes
and one antinode.
 A second simple pattern is the
two loop pattern. This pattern
has three nodes and two
antinodes.
 A third pattern has four nodes,
three antinodes, and three loops
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3/22
Standing Waves and Resonance
 Thus, a standing wave can be set up on a string of length L
by a wave with a wavelength equal to one of the values:
 
2L
,
for n  1, 2, 3,
n
 The resonant frequencies that correspond to these
wavelengths are:
f 
v

n
v
,
for n  1, 2, 3,
2L
 The last equation tells us that the resonant frequencies are
integer multiples of the lowest resonant frequency, f = v/2L,
for n = 1.
 The oscillation mode with the lowest frequency is called the
fundamental mode or the first harmonic.
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3/23
Standing Waves and Resonance
 The second harmonic is the oscillation mode with n = 2, the
third harmonic is that with n = 3, and so on.
 The collection of all possible
oscillation modes is called
the harmonic series.
 n is called the harmonic
number.
Erwin Sitompul
University Physics: Wave and Electricity
3/24
Checkpoint
In the following series of resonant frequencies, one frequency
(lower than 400 Hz) is missing: 150, 225, 300, 375 Hz. (a)
What is the missing frequency? (b) What is the frequency of the
seventh harmonic?
f 
v

n
v
,
for n  1, 2, 3,
2L
f h  f h 1  h
v
2L
 ( h  1)
v
2L

v
2L
• The most possible value for v/2L from the above
series is 75 Hz?
• The missing frequency below 400 Hz is thus 75 Hz.
• The seventh harmonic has the frequency of f5 + 2
v/2L = 375 + 2·75 Hz = 520 Hz.
Erwin Sitompul
University Physics: Wave and Electricity
3/25
Homework 3: Standing Waves
Two identical waves (except for direction of travel) oscillate
through a spring and yield a superposition according to the
equation
y   (0.50 cm ) sin 



3
mm
1

x  cos  (40  m in

1
) t 
(a) What are the amplitude and speed of the two waves?
(b) What is the distance between nodes?
(c) What is the transverse speed of a particle of the string at
the position x = 1.5 cm when t = 9/8 s?
Erwin Sitompul
University Physics: Wave and Electricity
3/26
Homework 3A: Standing Waves
1. Two waves propagate in one direction on a stretched rope. The frequency
of the waves is 120 Hz. Both have the same amplitude of 4 cm and
wavelength of 0.04 m. (a) Determine the amplitude of the resultant wave if
the two original waves differ in phase by π/3? (b) What is the phase
difference between the two waves if the amplitude of the resultant wave is
0.05 cm?
2. Two identical waves (except for direction of travel) oscillate through a
spring and yield a superposition according to the equation
1 1
1
y   (0.8 m ) sin   3 cm  x  cos  ( 8 s ) t 




(a) What are the amplitude and speed of the two waves?
(b) What is the distance between nodes?
(c) What is the transverse speed of a particle of the string at the
position x = 2.70 m when t = 0.25 min?
Erwin Sitompul
University Physics: Wave and Electricity
3/27