Exam 3: Friday December 3rd, 8:20pm to 10:20pm

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• Exam 3 Friday December 3rd, 820pm to 1020pm
• You must go to the following locations based on
  the 1st letter of your last name
• Review session Thursday Dec. 2 (Woodard), 615
  to 810pm in NPB 1001 (HERE!)

• Final Exam (cumulative) Tuesday December 14th,
  1230pm to 230pm.
• Room assignments A to K in NPB1001 (in here)
  L to Z in Norman Hall 137.
• Two more review sessions Dec. 7 (Hill) and Dec.
  9 (Woodard), 615 to 810pm in NPB1001 (HERE!)

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Class 40 - Waves I Chapter 16 - Wednesday
December 1st

• QUICK review
• Wave interference
• Standing waves and resonance
• Sample exam problems
• HiTT (if time permits, otherwise Friday)

Reading pages 413 to 437 (chapter 16) in
HRW Read and understand the sample
problems Assigned problems from chapter 16 (due
Dec. 2nd!) 6, 20, 22, 24, 30, 34, 42, 44, 66,
70, 78, 82
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Review - traveling waves on a string
Velocity

• The tension in the string is t.
• The mass of the element dm is mdl, where m is the
  mass per unit length of the string.

Energy transfer rates
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The principle of superposition for waves

• It often happens that waves travel simultaneously
  through the same region, e.g.

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The principle of superposition for waves

• If two waves travel simultaneously along the same
  stretched string, the resultant displacement y'
  of the string is simply given by the summation

where y1 and y2 would have been the displacements
had the waves traveled alone.
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Interference of waves

• Suppose two sinusoidal waves with the same
  frequency and amplitude travel in the same
  direction along a string, such that

• The waves will add.

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Interference of waves
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Interference of waves

• Mathematical proof

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Interference of waves
If two sinusoidal waves of the same amplitude and
frequency travel in the same direction along a
stretched string, they interfere to produce a
resultant sinusoidal wave traveling in the same
direction.
Link
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Standing waves
If two sinusoidal waves of the same amplitude and
wavelength travel in opposite directions along a
stretched string, their interference with each
other produces a standing wave.

• This is clearly not a traveling wave, because it
  does not have the form f(kx - wt).
• In fact, it is a stationary wave, with a
  sinusoidal varying amplitude 2ymcos(wt).

Link
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Reflections at a boundary

• Waves reflect from boundaries.
• This is the reason for echoes - you hear sound
  reflecting back to you.
• However, the nature of the reflection depends on
  the boundary condition.
• For the two examples on the left, the nature of
  the reflection depends on whether the end of the
  string is fixed or loose.

Movies
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Standing waves and resonance

• At ordinary frequencies, waves travel backwards
  and forwards along the string.

• Each new reflected wave has a new phase.
• The interference is basically a mess, and no
  significant oscillations build up.

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Standing waves and resonance

• However, at certain special frequencies, the
  interference produces strong standing wave
  patterns.

• Such a standing wave is said to be produced at
  resonance.
• These certain frequencies are called resonant
  frequencies.

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Standing waves and resonance

• Standing waves occur whenever the phase of the
  wave returning to the oscillating end of the
  string is precisely in phase with the forced
  oscillations.

l determined by geometry
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Standing waves and resonance

• Here is an example of a two-dimensional vibrating
  diaphragm.
• The dark powder shows the positions of the nodes
  in the vibration.