Lecture 28, Dec. 8

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Lecture 28, Dec. 8

• Goals

• Chapter 20
• Work with a few important characteristics of
  sound waves. (e.g., Doppler effect)
• Chapter 21
• Recognize standing waves are the superposition
  of two traveling waves of same frequency
• Study the basic properties of standing waves
• Model interference occurs in one and two
  dimensions
• Understand beats as the superposition of two
  waves of unequal frequency.

• Assignment
• HW12, Due Friday, Dec. 12th
• For Wednesday, Review for final, Evaluations

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Doppler effect, moving sources/receivers
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Doppler effect, moving sources/receivers

• If the source of sound is moving
• Toward the observer ?
• ? seems smaller
• Away from observer ?
• ? seems larger

• If the observer is moving
• Toward the source ?
• ? seems smaller
• Away from source ?
• ? seems larger

Doppler Example Audio Doppler Example Visual
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Exercise Plane Waves

• A You are driving along the highway at 65 mph,
  and behind you a police car, also traveling at 65
  mph, has its siren turned on.
• B You and the police car have both pulled over
  to the side of the road, but the siren is still
  turned on.
• In which case does the frequency of the siren
  seem higher to you?
• (A) Case A
• (B) Case B
• (C) same

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Superposition

• Q What happens when two waves collide ?
• A They ADD together!
• We say the waves are superimposed.

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Interference of Waves

• 2D Surface Waves on Water

In phase sources separated by a distance d
d
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Principle of superposition

• The superposition of 2 or more waves is called
  interference

Destructive interference These two waves are out
of phase. The crests of one are aligned with the
troughs of the other.
Constructive interference These two waves are in
phase. Their crests are aligned.
Their superposition produces a wave with
amplitude 2a
Their superposition produces a wave with zero
amplitude
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Interference space and time

• Is this a point of constructive
• or destructive interference?

What do we need to do to make the sound from
these two speakers interfere constructively?
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Interference of Sound
Sound waves interfere, just like transverse waves
do. The resulting wave (displacement, pressure)
is the sum of the two (or more) waves you started
with.
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Example Interference

• A speaker sits on a pedestal 2 m tall and emits a
  sine wave at 343 Hz (the speed of sound in air is
  343 m/s, so l 1m ). Only the direct sound wave
  and that which reflects off the ground at a
  position half-way between the speaker and the
  person (also 2 m tall) makes it to the persons
  ear.
• How close to the speaker can the person stand (A
  to D) so they hear a maximum sound intensity
  assuming there is no phase change at the ground
  (this is a bad assumption)?

The distances AD and BCD have equal transit times
so the sound waves will be in phase. The only
need is for AB l
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Example Interference

• The geometry dictates everything else.
• AB l AD BCCD BC (h2 (d/2)2)½ d
• AC ABBC l BC (h2 d/22)½
• Eliminating BC gives ld 2 (h2 d2/4)½
• l 2ld d2 4 h2 d2
• 1 2d 4 h2 / l ? d 2 h2 / l
  ½
• 7.5 m

t1
t0
7.5
t0
D
A
A
4.25
3.25
B
C
Because the ground is more dense than air there
will be a phase change of p and so we really
should set AB to l/2 or 0.5 m.
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Exercise Superposition

• Two continuous harmonic waves with the same
  frequency and amplitude but, at a certain time,
  have a phase difference of 170 are superimposed.
  Which of the following best represents the
  resultant wave at this moment?

Original wave (the other has a different phase)
(A)
(B)
(D)
(C)
(E)
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Wave motion at interfacesReflection of a Wave,
Fixed End

• When the pulse reaches the support, the pulse
  moves back along the string in the opposite
  direction
• This is the reflection of the pulse
• The pulse is inverted

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Reflection of a Wave, Fixed End
Animation
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Reflection of a Wave, Free End
Animation
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Transmission of a Wave, Case 1

• When the boundary is intermediate between the
  last two extremes ( The right hand rope is
  massive or massless.) then part of the energy in
  the incident pulse is reflected and part is
  transmitted
• Some energy passes
• through the boundary
• Here mrhs gt mlhs

Animation
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Transmission of a Wave, Case 2

• Now assume a heavier string is attached to a
  light string
• Part of the pulse is reflected and part is
  transmitted
• The reflected part is not inverted

Animation
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Standing waves

• Two waves traveling in opposite direction
  interfere with each other.
• If the conditions are right, same k w,
  their interference generates a standing wave
• DRight(x,t) a sin(kx-wt) DLeft(x,t) a
  sin(kxwt)
• A standing wave does not propagate in space, it
  stands in place.
• A standing wave has nodes and antinodes

Anti-nodes
D(x,t) DL(x,t) DR(x,t) D(x,t) 2a sin(kx)
cos(wt) The outer curve is the amplitude
function A(x) 2a sin(kx) when wt 2pn n
0,1,2, k wave number 2p/?
Nodes
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Standing waves on a string

• Longest wavelength allowed is one half of a wave
• Fundamental l/2 L ? l 2 L

Recall v f l
Overtones m gt 1
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Vibrating Strings- Superposition Principle
D(x,0)

• Violin, viola, cello, string bass
• Guitars
• Ukuleles
• Mandolins
• Banjos

Antinode D(0,t)
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Standing waves in a pipe

• Open end Must be a displacement antinode
  (pressure minimum)
• Closed end Must be a displacement node (pressure
  maximum)
• Blue curves are displacement oscillations. Red
  curves, pressure.
• Fundamental l/2 l/2 l/4

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Standing waves in a pipe
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Combining Waves
Fourier Synthesis
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Organ Pipe Example

• A 0.9 m organ pipe (open at both ends) is
  measured to have its first harmonic (i.e., its
  fundamental) at a frequency of 382 Hz. What is
  the speed of sound (refers to energy transfer) in
  this pipe?

L0.9 m
f 382 Hz and f l v with l 2 L / m (m
1) v 382 x 2(0.9) m ? v 687 m/s
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Standing Waves

• What happens to the fundamental frequency of a
  pipe, if the air (v 300 m/s) is replaced by
  helium (v 900 m/s)?
• Recall f l v
• (A) Increases (B) Same (C) Decreases

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Superposition Interference

• Consider two harmonic waves A and B meet at t0.
• They have same amplitudes and phase, but
• ?2 1.15 x ?1.
• The displacement versus time for each is shown
  below

Beat Superposition
A(?1t)
B(?2t)
C(t) A(t) B(t)
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Superposition Interference

• Consider A B,
• yA(x,t)A cos(k1xw1t) yB(x,t)A
  cos(k2xw2t)
• And let x0, yyAyB 2A cos2p (f1 f2)t/2
  cos2p (f1 f2)t/2
• and f1 f2 fbeat 1 / Tbeat

A(?1t)
B(?2t)
t
Tbeat
C(t) A(t) B(t)
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Exercise Superposition

• The traces below show beats that occur when two
  different pairs of waves are added (the time axes
  are the same).
• For which of the two is the difference in
  frequency of the original waves greater?

• Pair 1
• Pair 2
• The frequency difference was the samefor both
  pairs of waves.
• Need more information.

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Interference of Waves, Splitting and Guiding

• Controlling wave sources is exploited in numerous
  applications

Optical Y Splitter
A Crystal with Line Defect Acting as a
Waveguide Si (n3.4) Period A 0.58mm
Filling Factor 5/16 Excitation l 1.55mm
Light turning a corner
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Lecture 28, Dec. 8

• Assignment
• HW12, Due Friday, Dec. 12th