Interference of Waves Chapter 18

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Interference of Waves Chapter 18

• PHYS 2326-26

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Concepts to Know

• Boundary Conditions
• Principle of Superposition
• Standing Waves
• Traveling Waves
• Nodes
• Antinodes
• Interference
• Destructive Interference

3
Concepts to Know

• Constructive Interference
• Normal Modes
• Fundamental Frequency
• Overtones
• Harmonics
• Fourier Analysis (Harmonic Analysis)
• Open Pipe
• Closed Pipe
• Resonance

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Waves and Particles

• Are very different
• Particles 0 size, waves have a size
  wavelength
• Particles must be at different locations while
  multiple waves can be at the same place
• Bound waves (waves with boundaries) only allow
  discrete frequencies

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Superposition Principle

• If two or more 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
• Waves that obey this principle are linear waves.
  Those that dont are nonlinear waves. Usually
  the difference is the amplitude being large
  compared to their wavelength

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Superposition

• Consequence two traveling waves can pass
  through each other without being destroyed or
  altered

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Interference

• The combination of separate waves in the same
  region of space to produce a resultant wave is
  called INTERFERENCE

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Constructive Interference

• Occurs when waves or pulses combine to create a
  resultant pulse which is greater than either of
  the individual pulses
• Constructive interference occurs when path
  difference is an integer multiple of wavelength
  so the signal adds because the phase of the
  signals combining is 0

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Destructive Interference

• Occurs when the amplitude of the resultant wave
  is less than either wave
• Destructive interference occurs when the
  difference in pathlength for a signal is an odd
  integer multiplier of a half wavelength
• so that the waves arrive 180 degrees out of phase

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Phase

• The difference in phase between two tones of the
  same frequency arriving from slightly different
  paths is ?? 2p ?x/?
• If two speakers are driven by the same source
  there will be a difference in phase associated
  with the difference in distance and will depend
  upon the wavelength

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Standing WavesChapter 18.2

• Given two waves, one going left, the other going
  right, we have

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Standing Waves

• Note the equation for the standing wave doesnt
  have (kx-?t) so it is not a traveling wave
• It is an oscillation pattern with a stationary
  outline
• 2Asinkx is an amplitude that varies with position
• cos?t is simple harmonic motion oscillating at an
  angular frequency ?

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Nodes Antinodes

• Notice that has
  points where the amplitude is 0, kx0,p, 2p, 3p
  and remember that k 2 p/? For x0, ?/2, 3
  ?/2, n?/2, n 0,1,2,3,
• These are NODES
• In between these nodes are points where 2Asinkx
  are maximum( sine /- 1). These occur at kx
  p/2, 3 p/2, 5 p/2, so that x ?/4, 3 ?/4, 5
  ?/4, n ?/4 for n1,3,5,
• These are called ANTINODES

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Nodes Antinodes

• Distance between adjacent antinodes ?/2
• Distance between adjacent nodes ?/2
• Distance between a node and an antinode ?/4

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Standing Waves on Fixed String

• possible wavelengths ?n 2L/n
• fn v/ ?n n v/2L n1,2,3,
• these are quantized freq.
• since v sqrt(T/µ)
• fn (n/2L)sqrt(T/µ)
• Fundamental or first harmonic
• f1 (1/2L)sqrt(T/µ)
• fn n f1

A
N
N
L1? /2
n1
A
A
L2? /2
N
N
n2
N
A
A
A
N
N
N
n3
N
L3? /2
L
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Resonance

• A system can oscillate in one or more normal
  modes. If a periodic force is applied to the
  system, the amplitude of the resulting motion is
  greatest when the frequency of the applied force
  is equal to one of the natural frequencies of the
  system

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Periodic Sound WavesChapter 17.2

• Eqns 17.2 and 17.3 are for the displacement wave
  and the pressure wave
• These are 90 degrees out of phase with the
  displacement wave being a cosine function and the
  pressure wave being a sine function
• Pressure variations are a maximum when the
  displacement is 0 and the displacement wave is
  maximum where the pressure is a minimum

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Air Columns(Pipes)

• May be open or closed end
• Closed end pipe is rigid forms a pressure
  antinode
• An open ended pipe essentially forms a
  displacement antinode (maximum variation) and a
  pressure node

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• Open Pipe
• Fundamental or first harmonic is ½ wavelength
  resonance
• ?1 2L, f1 v/ ?1 v/2L
• Second harmonic
• ?2 L, f2 v/ ?2 v/L 2 f1
• Third harmonic
• ?3 2/3 L, f3 3v/2L 3 f1

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• Closed Pipe
• Fundamental or first harmonic is ¼ wave resonant
• ?1 4L, f1 v/ ?1 v/4L
• Second harmonic doesnt exist (no even ones)
• Third harmonic
• ?3 4/3 L, f3 3v/4L 3 f1
• Fifth harmonic
• ?5 4/5 L, f5 5v/4L 5 f1

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Harmonics Overtones

• Harmonics are multiples of the fundamental
  frequency
• Overtones are higher order frequencies above the
  fundamental frequency that are often harmonics
  but not always
• A harmonic (beyond first) is an overtone
• An overtone may not be a harmonic (integer
  multiple of the primary frequency)
• NOTE The first overtone is the 2nd harmonic

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Example 1

• A string of mass 100g and length 2m has a tension
  of 200N. What is a) its wave velocity, b) its
  fundamental frequency, c) its 3rd harmonic, d)
  its 5th overtone
• m0.1kg, µ m/L 0.1/2 0.05 kg/m
• v sqrt ( F/ µ) sqrt(200/0.05) 63.24m/s
• L 2 ¼ ? (twice the antinode distance)
• ? 4m, f1 v/ ? 63.25/4 15.81 Hz
• c) f3 3f1 3 15.81 47.4 Hz
• d) ?5o 2 L / 6 0.667 m

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Example 2

• Two omni directional speakers located at points
  (0,1) and (0,-2) meters, what is the minimum
  frequency of a sound wave that would produce
  constructive interference at the point (5,0)m.

r1
y1
x
y2
r2
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• y11m, y22m, x 5m v345m/s speed of sound in
  air, r1sqrt(x2y12), r2sqrt(x2y22),
  constructive interference occurs at multiples of
  ? so first is at 1 ? which is the difference in
  path lengths r2-r1 1 ?
• r1 5.099m, r2 5.385m, ? 0.286m
• f 1206 Hz

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Fouriers TheoremHarmonic Series

• Fouriers theorem states that for a periodic
  waveform, it can be represented as closely as
  desired by the combination of a sufficiently
  large number of sinusoidal waves that form a
  harmonic series.

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Waveforms

• Sine wave 1 frequency, a pure tone
• Square wave odd harmonics decreasing in
  amplitude
• Musical instruments combinations of even and
  odd harmonics that give a characteristic sound to
  the instrument