Superposition and Standing Waves

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Chapter 18

• Superposition and Standing Waves

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Outline

• Superposition and interference
• Superposition principle
• Constructive and destructive interference
• Superposition of sinusoidal waves
• Interference of sound waves

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Superposition principle

• Superposition principle If two or more traveling
  waves are moving through a medium, the resultant
  value of the wave function at any point is the
  algebraic sum of the values of the wave functions
  of the individual waves.
• Linear waves and nonlinear waves

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Superposition of two pulses

• Two traveling waves can pass through each other
  without being destroyed or even altered.
• Interference The combination of separate waves
  in the same region of space to produce a
  resultant wave is called interference.
• Constructive interference
• Destructive interference

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Superposition of sinusoidal waves

• Two sinusoidal waves traveling in the same linear
  medium
• y1 A sin(kx-?t)
• y2 A sin(kx-?t?)
• Resultant wave function
• y y1 y2 2A cos(?/2)sin(kx-?t?/2).
• If ? 0, 2?, 4?, y1 and y2 are in phase and
  interfere constructively y has an amplitude
  2A.
• If ? ?, 3?, y1 and y2 are ? rads out of phase
  and interfere destructively y has an amplitude
  0.
• If 0 lt?lt ?, y has an amplitude between 0 and 2A.

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Interference of sound waves

• A phase difference may arise between two waves
  generated by the same source when they travel
  along paths of unequal lengths.
• Because a path difference of one wavelength
  corresponds to a phase angle of 2? rad, ?/2?
  ?r/?, or ?r (?/2?)?.
• Conditions for constructive and destructive
  interference
• ?r (2n) (?/2) for constructive interference
• ?r (2n1) (?/2) for destructive interference

• Some definitions
• Path length r the distance along any path from
  speaker to receiver.
• Path difference ?r r2 r1

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Graphical example of interference of sound waves
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Example 18.1

• A pair of speakers placed 3.00 m apart are driven
  by the same oscillator. A listener is originally
  at point O, which is located 8.00 m from the
  center of the line connecting the two speakers.
  The listener then walks to point P, which is a
  perpendicular distance 0.350 m from O, before
  reaching the first minimum in sound intensity.
  What is the frequency of the oscillator?

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Homework

• Ch. 18, Problems 1, 5, 8, 11.