Traveling Waves: Superposition

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Traveling Waves Superposition

• Wave Superposition

Add the two waves together (superposition of wave
1 and wave 2) as follows
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Traveling Waves Superposition

• Wave Superposition

Wave 2
Superposition!
Wave 1
The intensity of the new wave is proportional to
A12 squared!
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Traveling Waves Superposition

• Wave Superposition Consider two waves with the
  same amplitude, frequency, and wavelength but
  with an overall phase difference of DF f.

sinAsinB 2sin(AB)/2cos(A-B)/2
Superposition!
New intensity!
New amplitude!
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Traveling Waves Interference

• Maximal Constructive Interference Consider two
  waves with the same amplitude, frequency, and
  wavelength but with an overall phase difference
  of DF 2pn, where n 0, 1, 2,

Max Constructive!

• Maximal Destructive Interference Consider two
  waves with the same amplitude, frequency, and
  wavelength but with an overall phase difference
  of DF p2pn, where n 0, 1, 2,

Max Destructive!
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Example Problem Superposition

• Two traveling pressure waves (wave A and wave B)
  have the same frequency and wavelength. The waves
  are superimposed upon each other. The amplitude
  of the resulting wave (wave C) is 13 kPa. If the
  amplitude of wave A is 12 kPa and the phase
  difference between wave B and wave A is fB fA
  90o, what is the amplitude of wave B and the
  magnitude of the phase difference between wave A
  and wave C, respectively?

Answer 5 kPa, 22.62o
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Traveling Waves Superposition

• Lateral Phase Shift Consider two waves with the
  same amplitude, frequency, and wavelength that
  are in phase at x 0.

Wave 1 distance d1
Wave 2 distance d2
Max Constructive
Max Destructive
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Examples Superposition
Dd l/2 max destructive
Dd l max constructive
Dd l/4
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Example Problem Superposition
x

• The figure shows four isotropic point sources of
  sound that are uniformly spaced on the x-axis.
  The sources emit sound at the same wavelength l
  and the same amplitude A, and they emit in phase.
  A point P is shown on the x-axis. Assume that
  as the sound waves travel to the point P, the
  decrease in their amplitude is negligible. What
  is the amplitude of the net wave at P if d l/4?

Answer Zero
Max Destructive
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Example Problem Superposition

• Sound with a 40 cm wavelength travels rightward
  from a source and through a tube that consists of
  a straight portion and a half-circle as shown in
  the figure. Part of the sound wave travels
  through the half-circle and then rejoins the rest
  of the wave, which goes directly through the
  straight portion. This rejoining results in
  interference. What is the smallest radius r that
  results in an intensity minimum at the detector?

Point B
Point A
Answer 17.5 cm
At point A the waves have the same amplitude,
wavelength, and frequency and are in phase.
Wave 1 travels a distance d1 2r to reach the
point B, while wave 2 travels a distance d2 pr
to reach the point B.
Max Destructive