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Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
•  Superposition
•  Constructive and
destructive interference
•  Standing waves
•  Resonant modes of
systems
•  Beats
Chapter 16
Superposition and Standing Waves
Topics:
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Principle of Superposition
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
An example of superposition
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
An example of destructive interference
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Checking Understanding
Two waves on a string are moving toward each other. A picture
at t = 0 s appears as follows:
How does the string appear at t = 2 s?
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Two waves on a string are moving toward each other. A picture
at t = 0 s appears as follows:
How does the string appear at t = 2 s?
Answer
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Standing waves
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Standing waves on a string
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Standing Wave Modes on a string: wavelengths
In this case we have one anti-node (max
amplitude), and two nodes.The wavelength is
= 2L
Here we have two-antinodes (and three
nodes).The wavelength is L (=2L/2)
And here three anti-nodes (and four nodes).
The wavelength is 2L/3
In general, if we have m anti-nodes, the
wavelength is 2L/m.
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
•  The case m=1 is called the fundamental mode, or first harmonic
(with fundamental or primary frequency and wavelength). 
•  The case m=2 is called the second harmonic. 
•  And so on.
•  ‘m’ is called the mode number.
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Imagine that the string where we have a standing wave is that one of a guitar
or violin. 
The standing wave produces vibrations in the air surrounding the string,
creating a sound wave. 
Does the pitch of the tone become higher or lower as go to higher
harmonics?
Notice that here v is the speed of propagation of waves on the string, not
the speed of sound in air.Why?
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
String instruments
Recall that the velocity of sound
in a string is given by:
T= tension;
m = linear mass density
For a fixed speed, the frequency (pitch) gets higher with shorter strings.
For a fixed length, the higher the velocity the higher the frequency (pitch).
In that case, the pitch changes with:
1) Mass density m:The heavier the string the lower the note/pitch.
2) Tension T: the tighter the string, the higher the note/pitch. For a fixed
length, the higher the velocity the higher the frequency (pitch).
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Standing Sound Waves
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Physics of Speech and Hearing
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Interference of Spherical Waves
Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
Beats
fbeat = |f1 - f2|

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Ch16

  • 1. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. •  Superposition •  Constructive and destructive interference •  Standing waves •  Resonant modes of systems •  Beats Chapter 16 Superposition and Standing Waves Topics:
  • 2. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Principle of Superposition
  • 3. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. An example of superposition
  • 4. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. An example of destructive interference
  • 5. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Checking Understanding Two waves on a string are moving toward each other. A picture at t = 0 s appears as follows: How does the string appear at t = 2 s?
  • 6. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Two waves on a string are moving toward each other. A picture at t = 0 s appears as follows: How does the string appear at t = 2 s? Answer
  • 7. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Standing waves
  • 8. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
  • 9. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.
  • 10. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Standing waves on a string
  • 11. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Standing Wave Modes on a string: wavelengths In this case we have one anti-node (max amplitude), and two nodes.The wavelength is = 2L Here we have two-antinodes (and three nodes).The wavelength is L (=2L/2) And here three anti-nodes (and four nodes). The wavelength is 2L/3 In general, if we have m anti-nodes, the wavelength is 2L/m.
  • 12. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. •  The case m=1 is called the fundamental mode, or first harmonic (with fundamental or primary frequency and wavelength). •  The case m=2 is called the second harmonic. •  And so on. •  ‘m’ is called the mode number.
  • 13. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Imagine that the string where we have a standing wave is that one of a guitar or violin. The standing wave produces vibrations in the air surrounding the string, creating a sound wave. Does the pitch of the tone become higher or lower as go to higher harmonics? Notice that here v is the speed of propagation of waves on the string, not the speed of sound in air.Why?
  • 14. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. String instruments Recall that the velocity of sound in a string is given by: T= tension; m = linear mass density For a fixed speed, the frequency (pitch) gets higher with shorter strings. For a fixed length, the higher the velocity the higher the frequency (pitch). In that case, the pitch changes with: 1) Mass density m:The heavier the string the lower the note/pitch. 2) Tension T: the tighter the string, the higher the note/pitch. For a fixed length, the higher the velocity the higher the frequency (pitch).
  • 15. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Standing Sound Waves
  • 16. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Physics of Speech and Hearing
  • 17. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Interference of Spherical Waves
  • 18. Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Beats fbeat = |f1 - f2|