Standing Waves 1

1
Standing Waves 1

• Part 1 Strings (Transverse Standing Waves)
• 05/03/08

2
Standing Waves

• A standing (or stationary) wave occurs when two
  waves of the same frequency pass through each
  other in opposite directions.

3
Formation of a Standing Wave
4
Waves Travelling in Opposite Directions

• The diagram shows two waves (in green and blue)
  travelling in opposite directions on a string,
  setting up a standing wave (in black).
• Note that the standing wave is the sum of the two
  travelling waves.

5
Energy in a Standing Wave

• No energy is transferred by a standing wave.
• Energy is trapped in the wave, changing between
  kinetic and potential energy as particles vibrate
  back and forth.
• Standing waves is a resonance effect, because the
  superposition of the two waves of identical
  frequency results in an increased amplitude of
  oscillation.

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Properties II

• Nodes are points on the standing wave that remain
  stationary at all times.
• Antinodes are points on the standing wave that
  have the greatest displacement.

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Harmonics in a String

• The 1st harmonic in a string has a wavelength of
  twice the length of the string.
• The 2nd harmonic has a wavelength equal to the
  length of the string.
• The 3rd harmonic has a wavelength equal to 2/3
  the length of the string.

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Problem 1

• Telephone wires often resonate and hum in high
  winds. When the wind blows in a particular
  direction, some wires are heard to resonate at 50
  Hz. The telephone poles are 25 m apart.
• Sketch such a telephone wire vibrating at its
  fundamental frequency.
• Calculate the wavelength of the fundamental.
• 50 m
• Calculate the wave speed.
• 2500 ms-1
• Sketch the wave pattern of the second harmonic.

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Problem 2

• A guitar string is 75.0 cm long. It is made to
  vibrate at its fundamental frequency.
• What is the wavelength of the vibration?
• 1.50 m
• The frequency of the note is 465 Hz. Calculate
  the speed of the wave in the guitar string.
• 698 m s-1
• What is the frequency of the third harmonic?
• 1395 Hz
• Calculate the wavelength of the third harmonic.
• 0.50 m
• The string is shortened to 35.0 cm as the
  players fingers move down the fret board.
  Calculate the new fundamental frequency.
• 996 Hz

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Demonstrations

• Slinky transverse waves
• Guitar identify the first, second, third,
  fourth harmonics by forcing nodes.

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Standing Waves

• Part 2 Pipes
• (Longitudinal Standing Waves)

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Longitudinal Standing Waves

• When we blow across a bottle, we set up a
  standing wave that is at the natural frequency of
  the bottle.
• If we add water to the bottle, we have reduced
  the amount of space available for the wave to be
  set up in. The wavelength of the fundamental is
  also reduced.

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Longitudinal Waves

• A column of air inside a pipe can vibrate these
  vibrations are longitudinal they behaves like
  compressions in a spring.

14
Open and Closed Pipes

• As demonstrated in the diagram below, the closed
  end of a pipe acts much like the fixed point of a
  string here a node forms.
• The open end of a pipe, however, forms an
  antinode.

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Problem 3

• An organ pipe is closed at one end and open at
  the other.
• Sketch the wave pattern of the fundamental
  resonance inside the pipe.
• The pipe is tuned to 50.0 Hz for its fundamental
  note. Calculate the wavelength of this note,
  given that the speed of sound is 320 m s-1.
• 6.4 m
• Calculate the length of the organ pipe.
• 1.6 m
• Sketch the wave pattern when the third harmonic
  is set up in the pipe.
• Calculate the frequency of the third harmonic.
• 150 Hz

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Problem 2

• Some physics students use a speaker (loudspeaker)
  to resonate a pipe that is open at both ends.
  The pipe is 1.20 m long.
• Calculate the wavelength of the fundamental
  resonance.
• 2.40 m
• One end of the pipe is now closed off. Does the
  speaker need to be changed to a higher or lower
  pitch in order to re-tune the pipe to the new
  fundamental? Explain.
• Assuming that the speed of sound is 320 m s-1,
  calculate the new resonant frequency.
• 67 Hz

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Practice

• Complete problems 1, 3, 4, 5
• Complete worksheet 3 on standing waves.