Physics 1710 Chapter 18 Superposition and Standing Waves

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Quiz
• A jet plane is flying at an altitude of 13,000
  meters above your head. You observe that the
  sound of the plane appears to be coming from
  behind the plane by an angle that you that you
  estimate to be about 30. How fast is the plane
  flying? Faster than the speed of sound?

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Physics 1710Chapter 18 Superposition and
Standing Waves

• 1' Lesson
• The propagation of waves is characterized by
• Reflection the rebound of the wave.
• Refraction the bending of a waves
  direction due to a velocity gradient
• Diffraction the bending of a wave around
  obstacles.
• Interference the combination of two or more
  waves in space.
• Beats the combination of two waves in time.

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Physics 1710Chapter 18 Superposition and
Standing Waves

• 1' Lesson
• Standing waves are the result of coherent
  combination of two or more waves of the same
  frequency.
• In a one dimensional arrangement, traveling
  waves whose frequencies are harmonic ratios add
  constructively to produce standing waves on a
  string, in a pipe or in a rod.

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Reflection
• Perpendicular component of wave reversed
• vx ? -vx
• While parallel component is unchanged
• vy ? vy
• Results is Rule of Reflection
• Angle of incidence angle of reflection
• ? i ? r

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Refraction
• Bending due to a velocity gradient across
  wavefront.
• Wavelength changes, since frequency is unchanged
  and ? v/ f .
• At the boundary of two media with different
  velocities Snells Law obtains
• sin ? 1 /v1 sin ? 2 / v2

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Diffraction
• A wave disturbance will bend around and
  obstacle.
• Huygens Principle applies
• Every point on a wave acts as if it were the
  origin of a new wave.

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Interference
• Coherent combination of two or more waves.
• If the waves are in phase they interfere
  constructively.
• If the waves are out of phase they interfere
  destructively.
• Under the correct conditions, two or more
  traveling waves can combine to produce a standing
  wave.
• Where the waves cancel, one observes a node.
• Where the waves reinforce each other one observes
  an anti-node.

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Beats
• The coherent addition of two waves of nearly the
  same frequency.
• The combination is a wave oscillating a the
  average of the two
• fave ( f1 f2 )/2
• Modulated by a beat frequency equal to the
  difference in the frequencies
• fbeat ( f1 - f2 )

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Physics 1710Chapter 18 Superposition and
Standing Waves

• One Dimensional Standing Wave
• String
• Node at ends
• Mattesons rule 49
• Distance from node to anti-node ?/4.
• N-A ?/4
• Thus
• L n (2 ? /4) n ? /2, n 1,2,3,4,5
• fn n v /2L,

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Physics 1710Chapter 18 Superposition and
Standing Waves

• fn n v /2L,
• fn n /(2L) v(T/µ)
• Marsenne Equation
• A harmonic series n 1,2,3,4,5

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Resonance
• The sympathetic oscillation of a driven SHO.
• Fext cos(?t) k x m d2x/dt 2

Trial Solution x A cos(?t F) Fext
cos(?t) k A cos(?t F)
m d2 A cos(?t F) /dt
2 RHS m A (- ?2) cos(?t F)
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Physics 1710Chapter 18 Superposition and
Standing Waves

• Resonance
• Trial solution works only if F 0. And if
• Fext k A m A (- ?2)
• A (Fext /m)/ k/m - ?2
• A (Fext /m)/ ?0 2 - ?2

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Resonance
• Amplitude ? 1/?o2 - ?2 2

A
?
?o
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Physics 1710Chapter 18 Superposition and
Standing Waves

• Standing Waves in a Tube
• Open ends Pressure nodes at ends
• L 2n (?/4) n ? /2, n 1,2,3,4
• fn 2n v /4L n v /2L
• One closed end NodeAnti-node combo
• L (2n1)(? /4) n 1,2,3,4,5
• fn (2n1) v /4L,

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Use Mattesons Rule 49

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Summary
• The propagation of waves is characterized by
• Reflection the rebound of the wave.
• Refraction the bending of a waves
  direction due to a velocity gradient
• Diffraction the bending of a wave around
  obstacles.
• Interference the combination of two or more
  waves in space.
• Beats the combination of two waves in time.

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Physics 1710Chapter 18 Superposition and
Standing Waves

• Summary
• Angle of incidence angle of reflection
  ? i ? r
• sin ? 1 /v1 sin ? 2 / v2
• fave ( f1 f2 )/2 fbeat ( f1 - f2 )
• fn n /(2L) v(T/µ)
• fnopen n v /2L fnstopped (2n1) v /4L
• A (Fext /m)/ ?0 2 - ?2