Speed of a wave on a string?

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Speed of a wave on a string?
Which of the following determines the wave speed
of a wave on a string? 1. the frequency at which
the end of the string is shaken up and down 2.
the coupling between neighboring parts of the
string, as measured by the tension in the
string 3. the mass of each little piece of
string, as characterized by the mass per unit
length of the string. 4. Both 1 and 2 5. Both 1
and 3 6. Both 2 and 3 7. All three.
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A wave on a string

• What parameters determine the speed of a wave on
  a string?
• Properties of the medium the tension in the
  string, and how heavy the string is.
• where µ is the mass per unit length of the string.

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Adding waves the principle of superposition

• When more than one wave is traveling in a medium,
  the waves simply add.
• The principle of superposition the net
  displacement of any point in the medium is the
  sum of the displacements at that point due to
  each individual wave.

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Constructive interference

• When the displacements of individual waves go in
  the same direction at a point, the result is a
  large amplitude there, because the displacements
  add. This is known as constructive interference.
  Simulation.

A neat feature of waves is that, after passing
through one another, waves (or pulses) travel as
if they had never met.
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Destructive interference

• When the displacements of individual waves are in
  opposite directions at a point, the waves cancel
  (at least partly). This is known as destructive
  interference. Simulation.

How is it possible for the two pulses to
re-emerge from the flat string? Where is the
energy to do this?
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Reflections (fixed end)

• How waves reflect at the ends of a medium, or at
  the interface between two media, is critical to
  understanding things like musical instruments.
• When a wave encounters a fixed end, for instance,
  it comes back upside down.
  Simulation

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Reflections (free end)

• When a wave encounters a free end, it comes back
  upright.

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Standing waves

• When two waves of the same frequency and
  amplitude travel in opposite directions in a
  medium, the result is a standing wave - a wave
  that does not travel one way or the other.
• If the waves are identical except from their
  direction of propagation, they can be described
  by the equations
• y1 A sin(kx - ?t) and y2 A sin(kx ?t)
  
• The resultant wave is their
• sum, and can be written as
• y 2A sin(kx) cos(?t)

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Standing waves

• The resultant wave is their sum,
• and can be written as
• y 2A sin(kx) cos(?t)
• This is quite different from the equation for a
  traveling wave, because the spatial part is
  separated from the time part. It tells us that
  the string is totally flat at certain points in
  time, and that there are certain positions where
  the amplitude is always zero - these points are
  called nodes. There are other points halfway
  between the nodes where the amplitude is maximum
  - these are the anti-nodes.

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Standing waves a string fixed at both ends

• A wave traveling in one direction on the string
  reflects off the end, and returns inverted
  because the end is fixed. This gives two
  identical waves traveling in opposite directions
  on the string, just what is needed for a standing
  wave. Simulation
• The waves reflect from both ends of the string.
  Completely constructive interference takes place
  only when the wavelength is related to the length
  L of the string by
• Using , the corresponding
  frequencies are
• , where n 1, 2, 3, ...

where n 1, 2, 3, ...
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Standing waves a string fixed at both ends

• The lowest resonance frequency (n 1) is known
  as the fundamental frequency for the string. All
  the higher frequencies are known as harmonics -
  these are integer multiples of the fundamental
  frequency.

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Standing waves a string fixed at both ends

• fundamental (n 1)

second harmonic (n 2)
third harmonic (n 3)
fourth harmonic (n 4)
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Standing waves a string fixed at both ends

• All stringed musical instruments have strings
  fixed at both ends. When they are played, the
  sound you hear is some combination of the
  fundamental frequency and the different harmonics
  - it's because the harmonics are included that
  the sound sounds musical. A pure sine wave does
  not sound nearly so nice.

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Standing waves a tube open at both ends

• For a tube open at both ends, reflections of the
  sound at both ends produce a large-amplitude wave
  for particular resonance frequencies. For the
  standing waves, an open end is an anti-node
  (maximum amplitude point) for displacement.

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Standing waves a tube open at both ends

• Simulation transverse representation
• Simulation longitudinal representation
• The resonance frequencies are given by the same
  equation we used for the string

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Standing waves a tube closed at one end only

• For a tube closed at one end, the closed end is a
  node (zero displacement) while the open end is an
  anti-node (maximum displacement). This leads to a
  different equation for the resonance frequencies.
  
• , where n can only be odd integers
• Simulation transverse Simulation longitudinal

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Standing waves a tube closed at one end only

• For a tube closed at one end, the closed end is a
  node (zero displacement) while the open end is an
  anti-node (maximum displacement). This leads to a
  different equation for the resonance frequencies.
  
• , where n can only be odd integers
• Simulation transverse representation
• Simulation longitudinal representation

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Whiteboard