Wave Superposition and Interference
chap. 17
Book web site: kineticbooks
Reading assignments:
chap.17
(skip 17.6)
(17.5 is the most fun!)
end of chapter practic problems
4.2, 4.6, 5.1, 7.1, 7.2, 8.4, 8.5
Alert:
Most of the simulations in this chapter use transverse (light) waves instead of
longitudinal (sound) waves. But all the wave properties in this chapter also apply to
sound waves. Transverse waves are just easier to visualize interfering with another.
Prior knowledge:
wave speed, parts of waves, sound concepts
VOCAB
interference: overlap of waves
constructive interference: 2 or more waves add to a new bigger wave
Destructive interference: 2 or more waves overlap creating a smaller wave
Node: location where particles are not oscillating
Anti-node: location where particles are oscillating at their maximum amplitude
Standing wave: larger wave created when 2 identical waves interfere
Resonance: standing wave using the initial wave and its inverted reflection
Fundamental frequency: lowest energy, most common frequency of a standing wave
Harmonic: higher than the fundamental, by a factor of a whole number: 2, 3, 4 etc.
Overtones: frequencies either whole number or fractions larger than the fundamental
Timbre: richness of sound which varies due to the resonant frequencies of instruments
octave: double or half the frequency
Equations
Velocity of a wave in a string:
V= √ (F/u) where F = force due to tension, u=string mass/length
harmonic frequencies: (simpler if just draw a picture and count waves!)
f = V/ λ where λ= L/n and n= # of waves in the string
beat frequency:
Fbeat = f2 – f1
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Section 17.0 Introduction
Simulation questions: wave interactions
17.0.01
Hit reset and then change the amplitude and width of each pulse until you can get a line
when they intersect (pause and time step frame by frame to make sure you created a horizontal line)
This is called total destructive interference.
Write down the 4 values of amplitude and width you used below:
left wave: amp=_1.0__, width=_1.0__
17.0.02
right wave: amp= _-1.0____, width=_1.0___
Reset and find the values which make a total peak of 2.0 m when the 2 waves meet.
This is called total constructive interference since you are creating one larger wave.
left wave: amp=__1.0_, width=_1.0__
right wave: amp= __1.0___, width=__1.0__
17.0.03
In the second simulation, waves are used instead of a single wave pulse. Set one wave to a
very short wavelength and small amplitude and the other wave just the opposite- long wave and large
amplitude. You should be able to see the small wave riding over the larger wave. Sketch what you see:
17.0.04
Set the two waves so that they create what is called a standing wave: one wave which seems
to just move up and down instead of also moving horizontally. Record the values below:
left wave: amp=_____, wavelength=______
right wave: amp= ________, wavelength=_______
As long as the two settings are the same you will get a standing wave!
Section 17.1 Combining waves: the principle of superposition
Simulation questions:
superposition
17.1.01
Do the two traveling waves pass through or bounce off another?
(think about all the sound waves in a typical classroom!)
They pass through each other
17.1.02
If a crest of one wave overlaps with a trough of another wave,
is the resulting wave bigger or smaller than the original waves?
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Smaller
17.1.03
Where might sound engineers use destructive interference?
17.1.04
The new wave is formed by adding the “displacements” of the other waves.
What other word could the book authors use instead of “displacements” ?
(hint: what part of the wave is being added or subtracted?)
Section 17.2 Standing Waves
Simulation questions: standing waves
17.2.01
Step frame by frame with the components marked on so you can see the two individual
waves. What do notice about those 2 waves where they create a node?
They are equal and opposite.
17.2.02
Standing waves are due to what type of interference, constructive, destructive, or both?
Both
17.2.03
Where do anti-nodes form, at crests, troughs, both, or at the middle equilibrium line?
At both the crests and the nodes.
17.2.04
Standing waves require waves to be exactly the same frequency, same amplitude, opposite
directions, and to be “in phase”. What does in phase mean? Use words or picture to explain
For two waves to be in phase, they have to have the same “starting” point, (either in time or space)
Section 17.3 Reflected Waves & Resonance
Simulation questions: string waves reflect off a wall
17.3.01
Watching the simulations you can see how there are two waves, the initial wave
and the ___________________ wave.
17.3.02
If a crest hits a solid barrier like a wall, then what bounces off- crest, trough, flat?
Trough
17.3.03
What if shook a rope on one end and it was not tied down at the other end.
Would you expect the reflected wave to be inverted or the same?
(this is what happens when you create vibrations at one end of a flue or organ pipe)
17.3.04
Resonance means re-sound. Explain why the new sound is louder and lasts a while
17.3.05
The reflected wave is always shown to have the same amplitude as before. Why is this not so
realistic and how will it affect the standing waves created between the initial and reflected wave?
The wave will be damped through various frictional forces and the standing wave will be
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MRI- what does the “R” stand for?
RESONANCE!!!!
Section 17.4 Harmonics
Simulation questions: fundamental and higher frequencies
** this section is heavy in math. Focus on the pictures and count # waves in the string **
17.4.01
Look at the pictures of the different harmonics. As you play higher harmonics, what happens
to the wavelength? What happens to the frequency?
The wavelength gets shorter and the frequency get higher
17.4.02
The frequency of a piano string depends on its mass, length, and tension.
How would you change each of these variables to increase the frequency?
(no math, just use your physics insight and wave knowledge)
Increase the Tension, Decrease the Length, or lessen the mass.
17.4.03
Assume the fundamental frequency of a French horn is 120 Hz.
Which of the following frequencies could not be a harmonic:
60 Hz, 240 Hz, 300 Hz, 360 Hz?
300 Hz
17.4.04
Draw a picture of a vibrating string which is 3m long and contains 1.5 waves:
(the wavelength is ___2m____m)
17.4.05
You could play the same note and hence frequency on a flute and a piccolo,
yet their sound or timbre will be different. This difference is due to what?
The amount of each harmonic that you are able to hear.
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Section 17.5 Interactive problem: tune the string
Simulation questions: this one is actually quite fun !
17.5.01
Adjust both the harmonic number and the string length to get 392 Hz
harmonic number=____2_______, string length=___1.5 m____________
17.5.02
For the above exercise, what is the wavelength___1.5 m_________?
(hint: you should have see the finger in the middle of the string)
17.5.03
The computer doesn’t let you change the values very much.
Can you finger out any other combinations of values that would work?
(hint: the velocity stay the same, so you just need the same wavelength)
harmonic number=____________ , string length=____________
17.5.04
As you play around with the simulation, you should notice that the harmonic number counts not the
number of whole wavelengths, but the number of _half wavelengths____.
This is the same distance between any node and the next nearest anti-node!
Section 17.6 Sample problem: string tension
This is a practice problem and the math is challenging but doable. Try it!
Section 17.7 wave interference & path length
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* they made this overly complicated, focus on the pictures!
Destructive interference happens if crest meets trough, node meets anti-node which means one wave is a ½
wavelength or some multiple of ½ (3/2, 4/2, 5/2, etc). In circular units, this is the same as multiples of pi radians.
Simulation: wave interference
17.8.01
If two identical waves start at different times but later meet so that their crests overlap, this is
called constructive interference. What is the distance called between one crest to the next?
A Whole Wavelength
17.8.02
For the above question, the same constructive interference occurs if the trough of one wave
meets the trough of another wave. If the waves have a wavelength of 2m, which separation
distance is possible for constructive interfence?
1m, 3m, 4m, 6m, 8m (more than 1 answer here!)
4m, 6m, and 8m
17.8.03
Destructive interference occurs when the crest of one wave meets the trough of another.
Why does this only happen if the two waves are out of sync by half-wavelengths?
Because a trough is half a wavelength away from a crest on a normal wave
17.8.04
The simulation uses longitudinal waves. Draw a picture using two identical transverse waves
which are separated by a half wavelength to produce destructive interference.
17.8.05
What do you think are dead spots in a concert theatre and how do you get rid of them?
Dead spots are places where sound waves interact destructively; change the position of the speakers.
Section 17.8 Beats
….not beets!
** ignore the sample math problem using the sine function **
Simulation: beats effect
17.8.06
Up to now we have studied waves that are identical so that when they overlap they also
produce identical repeating patterns of constructive and destructive interference. So what
has to be different about the two waves for beats to be produced?
The frequency
17.8.07
Is it possible to have a negative beat frequency?
Hint: if you hear two different tuning forks, could you tell which
has a higher frequency and hitting your ear more often, or just that they are different?
No, beat frequency is never negative
17.8.08
Cicada insects can create sounds of different frequencies which blend as to make a very loud
hum every 5 seconds. What is the corresponding beat frequency?
5 seconds
17.8.09
What beat frequency do you want to hear when you’re guitar is perfectly tuned?
You don’t want to hear a beat frequency, if you do then you’re not tuned!
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17.8.10
A tuning fork of 440 Hz is played at the same time a middle A note is struck on the piano.
A beat frequency of 3 Hz is heard. What possible frequency is the A note before being tuned?
Either 443 Hz or 337 Hz.
Section 17.9 Gotchas
no assignment!
Section 17.10 Summary
Directions: Add some of your own notes or questions in your notebook
Quizboard:
take the quiz!
End of chapter practice problems: do 4.2, 4.6, 5.1, 7.1, 7.2, 8.4, 8.5
(do on separate paper & show all work for full credit!-4 steps)
Links:
physics classroom notes
Technical paper on fluorescent lamps
x-rays, microwaves, cat scans, more
Glenbrook classroom practice questions
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