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Physics 1710 Chapter 18 Superposition and Standing Waves

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A jet plane is flying at an altitude of 13,000 meters above your head. ... At the boundary of two media with different velocities Snell's Law obtains: ... – PowerPoint PPT presentation

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Title: Physics 1710 Chapter 18 Superposition and Standing Waves


1
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?

2
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.

3
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.

4
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

5
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

6
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.

7
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.

8
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 )

9
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,

10
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

11
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)
12
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

13
Physics 1710Chapter 18 Superposition and
Standing Waves
  • Resonance
  • Amplitude ? 1/?o2 - ?2 2

A
?
?o
14
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,

15
Physics 1710Chapter 18 Superposition and
Standing Waves
  • Use Mattesons Rule 49

16
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.

17
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
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