Lecture 3
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 3/2

Solution of Homework 2: Phase Differences f

500 Hz, v

350 m s
(a)
  v f x


2



350
500

0.7 m
2


 

2

3
(0.7)

0.117

11.7 cm
(b) T

1 f
T t


2


1
500

0.002 s

2 ms
T t
2
 
1 ms
2 ms
2
  
rad
Erwin Sitompul University Physics: Wave and Electricity 3/3

Example 1
A wave traveling along a string is described by y ( x , t ) = 0.00327sin(72.1
x –2.72
t ), 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
( , )

(3.27 mm) sin(72.1
x
 t

 y x t
 t
  x
 t
  x
 t u (0.225 m,18.9 s)
 
 
7.197 mm s

Erwin Sitompul University Physics: Wave and Electricity 3/4

Example 1
A wave traveling along a string is described by y ( x , t ) = 0.00327sin(72.1
x –2.72
t ), in which the numerical constants are in SI units.
(b) What is the transverse acceleration a y of the spring at that time? x

22.5 cm, t

18.9 s of the same element y
( , )
  x
 t

 u x t
 
 t
   sin(72.1
x
 t
  2
(24.192 mm s ) sin(72.1
x
 t
 a y
(0.225 m,18.9 s)
 
2

 
14.21mm s
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 y
1
( x , t ) and y
2
( x , t ) be two waves travel simultaneously along the same stretched string, then the displacement of the string is given by:
( , )
 y x t
1
( , )
 y x t
2
( , )
 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.
Erwin Sitompul University Physics: Wave and Electricity 3/7

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 .
y x t
1
( , )
 y m y x t
2
( , )
 y m sin( sin( kx
  t ) kx
  
)
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Interference of Waves
 The resultant wave as the superposition of y
1 of the two interfering waves is:
( x , t ) and y
2
( x , t )
( , )
 y x t
1
( , )
 y x t
2
( , )
 y m sin( kx
  t )
 y m sin( kx
  
)

2 y m sin( kx
  t

1
2

) cos( 1
2

) y x t


2 y m cos( 1
2
  kx
  t

1
2

)
 The resultant sinusoidal wave – which is the result of an interference – travels in the same direction as the two original waves.
Erwin Sitompul sin
  sin
 
2sin (
2 2
University Physics: Wave and Electricity
)
3/9

Interference of Waves
Fully constructive interference
Erwin Sitompul
Fully destructive interference
Intermediate interference
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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
   
360







0.2
    
0.45
   
162



0.6
   
216

0.8
   
288

Erwin Sitompul
Amplitude
 y
 m

2 y m cos( 1
2

)



 cos( 1 72 ) 0.809
2
   cos( 1 162 ) 0.156
2
   cos( 1 216 ) 0.309
2
    cos( 1 288 ) 0.809
2
   
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 y m of each wave is 9.8 mm, and the phase difference
Φ between them is 100 °.
(a) What is the amplitude y m
’ of the resultant wave due to the interference, and what is the type of this interference?
y
  m
2 y m cos( 1
2

)

2(9.8 mm) cos( 1
2
  
12.599 mm
The interference is intermediate, which can be deducted in two ways:
1. The phase difference is between 0 and π radians.
2. The amplitude y m
’ is between 0 and 2 y m
.
Erwin Sitompul University Physics: Wave and Electricity 3/13

Example 2
Two identical sinusoidal waves, moving in the same direction along a stretched string, interfere with each other. The amplitude y m 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( 1
2

)
4.9 mm

2(9.8 mm) cos( 1
2

) cos( 1
2

)
 
4.9 mm
2(9.8 mm) cos( 1
2

)
 
0.25
1
2
 
1.3181 or 1.8235
 
2.6362 or 3.6470
  
2.636 rad x


2

 x
 
2.636
2


 
0.420

 
0.420 wavelength
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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: y x t
1
( , )
 y m y x t
2
( , )
 y m sin( kx
  t ) sin( kx
  t )
 The principle of superposition gives:
( , )
 y x t
1
( , )
 y x t
2
( , )
 y m sin( kx
  t )
 y m sin( kx
  t )
( , )
 
2 y m sin kx
 cos
 t
Erwin Sitompul sin
  sin
 
2sin (
2 2
University Physics: Wave and Electricity
)
3/17

Standing Waves
 For a standing wave, the amplitude 2y m position.
sin kx varies with
 For a traveling wave, the amplitude y m position.
is the same for all
Erwin Sitompul
0
N
AN
N N N x
AN AN
University Physics: Wave and Electricity 3/18

Standing Waves
( , )
 
2 y m sin kx
 cos
 t
 In the standing wave, the amplitude is zero for values of kx that give sin kx = 0.
kx
 n
 x
 n

2
, for n

0,1, 2,
, for n

0,1, 2,
• Nodes
 In the standing wave, the amplitude is zero for values of kx that give sin kx = ±1 kx
 1
2
  
2 2
, kx

( n
 1
2
 n

0,1, 2, x


 n

1
 
, for n

0,1, 2,
• Antinodes
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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(5 x –4 t )
(b) y ’( x , t ) = 4sin(5 x )cos(4 t )
(c) y ’( x , t ) = 4sin(5 x+ 4 t )
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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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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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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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:
 
2 L
, for n

1, 2, 3, n
 The resonant frequencies that correspond to these wavelengths are: f
 v

 n v
2 L
, for n

1, 2, 3,
 The last equation tells us that the resonant frequencies are integer multiples of the lowest resonant frequency, f = v /2 L , for n = 1.
 The oscillation mode with the lowest frequency is called the fundamental mode or the first harmonic .
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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
2 L
, for n

1, 2, 3, f h
 f h

1
 h v
2 L
 
1) v
2 L
 v
2 L
• 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
 
1 cos (40 min ) 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?
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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 y
 
(0.8 m) sin 

 
1
 x 

1
8

1 cos ( 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