Superposition Principle and Interference

Student Learning Objectives
Lessons / Lecture Notes
Important Equations
Example Problems
Applets and Animations
Videos


Student Learning Objectives



Lessons / Lecture Notes

The Physics Classroom (conceptual)

PY105 Notes from Boston University (algebra-based):

Introductory physics notes from University of Winnipeg (algebra-based):

HyperPhysics (calculus-based)

PHY2048 notes from Florida Atlantic University (calculus-based):

General Physics I notes from ETSU (calculus-based)



Important Equations

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pdf



Example Problems

Problem 1
Two loudspeakers are vibrating in phase. They are set up as in the figure below, and point C is located as shown there. The speed of sound is 343 m/s. The speakers play the same tone. What is the smallest frequency that will produce destructive interference at point C? (Solutions)



Problem 2
(a) A cello string has a fundamental frequency of 65.40 Hz. What beat frequency is heard when this cello string is bowed at the same time as a violin string with frequency of 196.0 Hz? (Hint: The beats occur between the third harmonic of the cello string and the fundamental of the violin.)

(b) A tube 1.20 m long is closed at one end. A stretched wire is placed near the open end. The wire is 0.330 m long and has a mass of 9.60 g. It is fixed at both ends and oscillates in its fundamental mode.  By resonance, it sets the air column in the tube into oscillation at the tubes fundamental frequency. Find that frequency and the tension in the wire. (Solutions)


Applets and Animations
Sound

This simulation lets you see sound waves. Adjust the frequency, volume, and harmonic content and you can see and hear how the wave changes. Move the listener around and hear what she hears.

Wave Interference

Make waves with a dripping faucet, audio speaker, or laser! Add a second source or a pair of slits to create an interference pattern.

Fourier: Making Waves

Learn how to make waves of all different shapes by adding up sines or cosines. Make waves in space and time and measure their wavelengths and periods. See how changing the amplitudes of different harmonics changes the waves. Compare different mathematical expressions for your waves.

Group Velocity

The Group Velocity model displays the time evolution for the superposition of two traveling waves of similar wave numbers and frequencies.  The simulation allows an arbitrarily superposition of two waves for the study of group and phase velocity.

Superposition of Waves
The applet plots the superposition of two traveling waves. The user can change the equation of either wave.
Two Source Interference

The Two Source Interference model displays the interference pattern on a screen due to two point sources.  The simulation allows an arbitrarily superposition of the two sources and shows both the current intensity and running average of the intensity on the screen. 

Optics Interference

The Optics Interference program simulates a ripple tank by showing the intensity of waves produced by a point source. Adding multiple point sources creates easily observable interference patterns showing constructive and destructive interference. Users can add point sources, move them around and change their wavelength.

Interference of Water Waves I
This applet shows the interference of water waves. Click any point on the pattern and the applet displays two wave trains coming from the sources to that point.
Interference of Water Waves II
This applet shows the interference of water waves. Click any point on the pattern and the applet displays the path length difference between the two sources.
Interference with Synchronous Sources

The Interference with Synchronous Sources model displays the interference pattern on a screen due to between one and twenty point sources.  The simulation allows an arbitrarily superposition of the sources and shows both the current intensity and running average of the intensity on the screen. 

Beats Illustrating beats between 2 oscillators of nearly identical frequencies.
Beats
This applets shows the beats formed by two waves of slightly different frequencies.
Beats

The Beats model displays the result of adding two waves with different frequencies.  The simulation displays the superposition of the two waves as well as a phasor diagram that shows how the waves add up at one point in space.

Reflection and Refraction between Taut Strings

The Reflection and Refraction between Taut Strings model displays the motion of a traveling pulse on a string when it is incident on a change of string density.

Waves on a String

Watch a string vibrate in slow motion. Wiggle the end of the string and make waves, or adjust the frequency and amplitude of an oscillator. Adjust the damping and tension. The end can be fixed, loose, or open.

Standing Waves on a String

The Standing Waves on a String model displays the motion of a standing wave on a string.  The standing wave can be augmented by adding the zero line and the maximum displacement of the string.

Resonance on a String
This applet demonstrates resonance on a string fixed at both ends. The user can control the frequency and amplitude of the mechanical vibrator.
Resonance in a Driven String

The Resonance in a Driven String model displays the displacement of taut string with its right end fixed while the left end is driven sinusoidally.

Standing Wave Explanation
The applet shows a transverse traveling wave reflecting off a rigid support. It gives a clear explanation of standing waves by the superposition of the incident wave with the reflected wave.
Standing Waves Explanation A wave is reflected from a barrier with a phase reversal, setting up a standing wave.
Standing Waves - Both Ends Fixed The first three standing waves for nodes at both ends. The frequencies of the waves are proportional to one over the wavelength.
Standing Waves - One Fixed End The first three standing waves for a node at one end and an antinode at the other. The frequencies are proportional to one over the wavelength.
Standing Waves and Traveling Waves
This applet shows either standing or traveling waves for both transverse and longitudinal waves.
Standing Waves in a Pipe

The Standing Waves in a Pipe model displays the displacement and pressure waves for a standing wave in a pipe.  The pipe can be closed on both ends, on one end, or open on both ends.

Standing Longitudinal Wave
This Java applet demonstrates the harmonics of the air in a tube as an example of standing longitudinal waves. It illustrates the movement of the molecules in the air during such an oscillation
Microwaves

How do microwaves heat up your coffee? Adjust the frequency and amplitude of microwaves. Watch water molecules rotating and bouncing around. View the microwave field as a wave, a single line of vectors, or the entire field.



Videos

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Slow-motion video of a glass that breaks because of resonance. The sound waves (which you can not hear in this video) match the natural frequency of the glass.
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Video of an old Memorex commercial where Ella Fitzgerald breaks a wineglass with her voice. The commercial then shows a recording of Ella's voice breaking a beaker. Note: if the recording was "perfect", they should have been able to break the same type of wine glass with the recorded voice. So the answer to the question "Is it live or is it Memorex?" is: It's Memorex!
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Video of a resonance test of a military helicopter. I am glad that they test for these kinds of things while still on the ground!
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High quality video of the Tacoma Narrow Bridge right before it collapsed.
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Small black and white video showing the collapse of the Tacoma Narrows Bridge.
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This "Mystery" video shows an application of resonance. See if you can figure out what the video is showing. Here is a hint: it is a medical procedure.