Standing waves' Doppler effect'

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Lecture 32

• Standing waves. Doppler effect.

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Reflected waves fixed end
A pulse travels through a rope towards the end
that is tied to a hook in the wall (ie, fixed end)
The force by the wall always pulls in the
direction opposite to the pulse.
The pulse is inverted (because of Newtons
3rd law!)
Another way Consider one wave going into the
wall and another coming out of the wall. The
superposition must give 0 at the wall. Virtual
wave must be inverted
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Reflected waves free end
A pulse travels through a rope towards the end
that is tied to a ring that can slide up and down
without friction along a vertical pole (ie, free
end)
No force exerted on the free end, it just keeps
going
Fixed boundary condition
Free boundary condition
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Standing waves
A harmonic wave traveling along the x direction
is reflected at a fixed point. What is the result
of the its superposition with the reflected wave?
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x
-x
Standing wave
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Standing waves and boundary conditions
We obtained
We need fixed ends to be nodes and free ends to
be antinodes!
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Normal modes
Which standing waves can I have for a string of
length L fixed at both ends?
I need nodes at x 0 and x L
Allowed standing waves (normal modes) between two
fixed ends
Mode n n-th harmonic
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1 fixed, 1 free
1 fixed, 1 free
2 free ends
2 free ends
2 free ends
2 fixed ends
Normal modes for fixed ends (lower row)
First harmonic
Second harmonic
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Normal modes 2D
For circular fixed boundary
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Resonance
To produce a wave, we need to apply an external
force (driving force). This driving force can be
periodic with frequency fD.
The amplitude of the perturbation is maximum when
the frequency of the driving force is equal to
one of the natural (or harmonic, or normal)
frequencies of the system.
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Doppler Shift
Even better http//www.lon-capa.org/mmp/applist
/doppler/d.htm
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Doppler math moving source

• Speed of sound v is constant.

t 0
Source moving with vS (vSgt0 from listener to
source) Stationary listener
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Doppler math moving listener
v L (listener)
v (sound)
vS
?
t 0
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Moving source and moving listener
vL, vS gt 0 in direction from listener to source
(v gt 0 always)

• To get signs correct
• sketch the situation, including a few wavefronts
• decide whether observed wavelength or period will
  be shorter or longer
• use this to guide whether frequency increases,
  decreases
• keep in mind speed of sound does not depend on
  what the source or observer is doing

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ACT Doppler
A train is approaching you as you stand on a
platform at a railway station. As the train
approaches, it slows down. All the while, the
engineer is sounding the horn at a constant
frequency of 500 Hz.

• Heard frequency is greater than 500 Hz and
  increases as train slows down
• Heard frequency is greater than 500 Hz and
  decreases as train slows down
• Heard frequency is less than 500 Hz and increases
  as train slows down
• Heard frequency is less than 500 Hz and decreases
  as train slows down

?? ? f?
Source approaching listener wavefronts are
squeezed together
Effect must be getting smaller (back to source
frequency) f decreases
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In-class example Doppler

• A source of sound has a characteristic frequency
  f. The speed of sound is v. Consider the
  following four scenarios
• Static source, vobserver v/2 toward source
• Static source, vobserver v/2 away from source
• Static observer, vsource v/2 toward observer
• Static observer, vsource v/2 away from observer
• Order f1, f2, f3, f4 from lowest to highest.

A. f1 f2 f3 f4 B. f2 f4 , f1 f3 C.
f1 , f2 , f3 , f4 D. f2 , f4 , f1 , f3 E. f4 ,
f3 , f2 , f1
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• A source of sound has a characteristic frequency
  f. The speed of sound is v. Consider the
  following four scenarios
• Static source, vobserver v/2 toward source
• Static source, vobserver v/2 away from source
• Static observer, vsource v/2 toward observer
• Static observer, vsource v/2 away from observer
• Order f1, f2, f3, f4 from lowest to highest.

A. f1 f2 f3 f4 B. f2 f4 , f1 f3 C.
f1 , f2 , f3 , f4 D. f2 , f4 , f1 , f3 E. f4 ,
f3 , f2 , f1
It is NOT option B 2 and 4 (or 1 and 3) are not
equivalent. You need to think about the motion
relative to air, too.
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Shock waves
What if the source (a plane, for instance) is
moving almost at the speed of sound?
http//www.lon-capa.org/mmp/applist/doppler/d.htm
High pressure and density in front of plane
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Supersonic speeds
And what if vsource gt v ?
http//www.lon-capa.org/mmp/applist/doppler/d.htm
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Mach number
vsource gt v
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Movie
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Appendix Reflections/standing waves in
pipesReflection of sound waves against a surface
Consider a sound pulse (air moves to the right
and back to initial position) traveling along a
pipe toward a closed end
A closed end is a fixed end
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Reflection of sound at an open end
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Boundary conditions for sound

• Open end
• gauge pressure 0
• maximum (absolute) air displacement

• Closed end
• air displacement 0
• maximum (absolute) gauge pressure

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Standing sound waves in pipe open at both ends
A harmonic wave and its reflection on an open end
Standing wave within pipe does not travel,
bounces back and forth. Amplitude will decrease
as energy is transported out of the pipe
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ACT Pipe open at both ends
This is the air displacement for a standing wave
inside this tube.
Sketch the gauge pressure vs position for this
wave.
Compare with your neighbor and discuss.
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Higher harmonics
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ACT Pipe closed at one end
Open end Max s, p 0
Closed end s 0, max p
First harmonic or fundamental frequency
What is the standing wave for the next harmonic?
A
s 0 at an open end? (No!)
And s max at a closed end? (No!)
B
C
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A little music
When you blow air into a pipe, all the harmonics
are present.
Example Blow into a tube of length 19.2 cm open
at one end
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ACT Wave motion
A boat is moored in a fixed location, and
waves make it move up and down. If the spacing
between wave crests is 20 meters and the speed of
the waves is 5 m/s, how long does it take the
boat to go from the top of a crest to the bottom
of a trough?
A) 2 sec B) 4 sec C) 8 sec
t
t Dt
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We know that v ?/ T, hence T ?/ v
In this case ? 20 m and v 5 m/s, so T 4
sec
The time to go from a crest to a trough is T/2
(half a period)
So ?t 2 sec
t
t ?t
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In-class example
A tube with both ends open has a fundamental
frequency f. What is the fundamental frequency of
the same tube if one end is closed?

• 4f
• 2f
• f
• f/4
• None of the above