superposition and wave interference
DESCRIPTION
What happens when two waves are present at the same place at the same time? Web Link: Wave InterferenceTRANSCRIPT
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IB Physics
Superposition and
Wave Interference
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What happens when two waves are present at the same place at the same time?
Web Link: Wave Interference
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The Principle of SuperpositionThe net effect = The sum of the individual effects
For waves:The resulting wave = the sum of the individual waves
This applies to all waves: water, light, sound, etc.
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Interference of Sound Waves
Imagine two speakers, each playing a pure tone of wavelength 1 meter:
3 m 3 m
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This is called Constructive Interference
We also say that these two waves are In Phase
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Now suppose the listener moves:
3 m
5 m
What does he hear now??
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3.5 m
6 m
He moves again:
Path length difference = 2.5 m = 2.5
off by ½ wavelength
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This is called Destructive Interference
We also say that these two waves are Exactly Out of Phase
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Ex: Noise canceling headphones
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If you’re standing in a place where destructive interference is occurring, where did the energy of the sound waves go? Is energy still conserved in this case??
Web Link: Interference patterns
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Interference Summary
path 1 path 2
If the difference in path lengths is………
0, 1, 2, 3, etc…… Constructive
½ , 1½ , 2½ , etc…… Destructive
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Ex:
If these two speakers are each playing a 412 Hz tone, and the listener is standing 3.75 m away from one and 5.00 m away from the other, what does he hear?
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Diffraction –The bending of a wave around an obstacle
with diffraction without diffraction
Web Link: Diffraction
Why does a wave bend??
Huygen’s Principle – Every point on a wavefront acts as a new spherical source
Web Link: Huygen’s Principle
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All waves exhibit diffraction, including light
So why can’t you see around corners?
The extent of diffraction is determined by this ratio:
D
wavelength
size of obstacle
tiny for lightlarger for sound (better dispersion)
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Huygen’s principle + math = …………
For a single slit (or doorway) of width D :
s in
D
Angle of 1st diffraction minimum
D
For a circular opening of diameter D :
s in .
1 22D
Angle of 1st diffraction minimum
Web Links: Diffraction of lightSun diffraction
D
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Remember Constructive and Destructive Interference?
So far, we’ve only looked at interference between waves of the same frequency. What if the frequencies are slightly different?
We can still use Superposition to add them
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fbeat = f1 – f2
The beat frequency of an additional loudness wave
Web Links: Sound Beats, Beats
Ex: Piano Tuning
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Transverse Standing Waves
Hits the wall and bounces back
Web Link: Transverse Standing Wave
There are actually a number of different frequencies that will result in a standing wave
If the frequency is just right, an integral number of these fit on the string, and we have Resonance
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nodes (no vibration)
antinodes (max. vibration)
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In the previous example, the string was fastened to the wall:
If it had been loose instead:
Hard Reflection: inverts the wave
This creates a node at the end
Soft Reflection: the wave returns upright
This creates an antinode at the end
Web Link: Hard & soft reflections
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back to…… Harmonics-
Natural frequencies of the system(f1, f2, f3, etc.)
fundamental frequency
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Ex: The Cello
The C-string on a cello plays a fundamental frequency of 65.4 Hz. If the tension in the string is 171 N, and the linear density of the string is 1.56 x 10-2 kg/m, find the length of the string.
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We can derive a formula to calculate all of the harmonic frequencies for any string:
f n v2Ln
n = 1,2,3,4,…
Web Link: String Harmonics
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Longitudinal Standing Waves
antinodes (max. vibration) nodes (no vibration)
Web Link: Longitudinal standing wave
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Remember, this is a longitudinal wave even though we draw it like this to visualize the shape.
When air is blown over a bottle, it creates a standing longitudinal (sound) wave
open end: antinode
vibrating air molecules
closed end: node
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You can also ring a tuning fork over a bottle or tube, and if it creates wavelengths of just the right length, you’ll get a standing wave (loud sound).
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Just like we did for strings, we can also derive a formula to calculate……
f n v4Ln
n = 1,3,5,…
The Harmonic Frequencies for a tube open at one end
speed of sound
odd harmonics only
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Standing waves can also occur in a tube that is open at both ends
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f n v2Ln
n = 1,2,3,4,…
Harmonic Frequencies for a tube open at both ends
Web Link: Flute
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Find the range in length of organ pipes that play all frequencies humans can hear. Assume that the organ pipes are open at both ends, and they each play their fundamental frequency.
Ex:
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Complex Sound Waves
Musical instruments play different harmonics at the same time
Web Links: String Harmonics, Flute
f1
f2 f3
=
Shape identifies the instrument
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The shape of a vocal sound wave tells us who’s singing (or who’s on the other end of the phone)
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Fourier AnalysisAny periodic wave form can be represented as the sum of sine waves.
f1
f2 f3
=
Now imagine starting with the complex sound wave, and trying to separate it into sine waves:
Web Link: Fourier series