16.10 Superposition and Interference

The waves result from the superposition of several waves from different sources.

Waves do not look simple.

The waves look formidable, even though they are more interesting.
Waves result from several simple waves adding together.
The rules for adding waves are easy to understand.

They superimpose themselves on one another when two or more waves arrive at the same point.

There are forces that add and there are forces that subtract.
If the individual waves are along the same line, the resulting wave is a simple addition of the individual waves.

There are two waves that arrive at the same point.

The troughs and crests of the waves are the same.
Pure constructive interference produces a wave that has twice the amplitude of the individual waves, but the same wavelength.

The waves completely cancel because the disturbances are in the opposite direction.

One of the waves has twice the amplitude but the same wavelength.

Zero amplitude or complete cancellation can be produced by pure destructive interference of two waves.

They require precisely aligned identical waves when they occur.

The interference can vary from place to place and time to time.
Sound from a stereo can be loud in one spot and quiet in another.
Sound waves can add and subtract at different locations.
Waves can reflect from walls and sound waves can be created by two speakers in a stereo.
The waves are superimpose.
An example of sounds that vary over time is found in the combined whine of airplane jets heard by a passenger.
The sound from the two engines can vary in volume from constructive to destructive.

There are examples of waves that are similar.

A more complicated looking wave is produced by the add and subtract of the disturbances.

Non-identical waves exhibit both constructive and destructive interference.

Waves don't seem to move, they just vibrate in place.

Waves can be seen on the surface of a glass of milk in a refrigerator.
The milk does not move across the surface because of the waves created by the refrigerator motor.
The waves move through each other.
If the two waves have the same wavelength and amplitude, they alternate between constructive and destructive interference.
Waves on the glass of milk are an example of standing waves.
On guitar strings and organ pipes there are other standing waves.
The two waves that produce standing waves may come from the side of the glass.

Evidence for resonance, standing waves, and constructive and destructive interference can be found in a closer look at earthquakes.

A building may be vibrated for several seconds with a driving frequency matching that of the natural frequencies of the building, causing one building to collapse, while neighboring buildings do not.
Buildings of a certain height are often destroyed.

Constructive interference occurs when the earthquake waves travel along the surface of Earth and reflect off denser rocks.

The areas closer to the epicenter are usually undamaged.

The waves are moving in opposite directions.

There are fixed locations in space where the oscillations occur.

Waves from the ends of musical instruments' strings can be seen as standing waves.

The strings are fixed because they can't move there.

The propagation speed of the waves on the strings is related to the standing waves on the strings.
The wavelength is determined by the distance between the points where the string is fixed.

The tension in the string can be adjusted to change the frequencies.

The higher the tension, the higher the frequencies.
Anyone who has ever watched a string instrument being tuned knows this observation.
In later chapters, we will see that standing waves are crucial to many resonance phenomena, such as sounding boxes on string instruments.

The string is at its fundamental frequencies.

The first and second frequencies are shown.

A warbling combination is usually considered unpleasant when two adjacent keys are struck on a piano.

Two waves of the same frequencies are superpositioning.
Jet aircraft, particularly the two-engine variety, are examples of another example.
The sound of the engines goes up and down.
The sound waves have the same frequencies but different loudness.
The jet engine noise and the warbling of the piano are both caused by interference as the waves go in and out of phase.

There are two waves of slightly different frequencies but the same amplitude.

The waves alternate in time between constructive interference and destructive interference, giving the resulting wave a time-varying amplitude.

The average of the two waves is the wave resulting from the superposition.

Adding two waves together can be used to determine the beat Frequency.

The results show that the wave has twice the amplitude and the average Frequency of the two waves, but it also fluctuates at the beat Frequency.

The first cosine term in the expression causes the amplitude to go up and down.

The wave with Frequency is the second term.
The result is valid for all waves.

Beatings are used in the work of piano tuners.

They listen for beats and adjust the string until the beats go away.

Most keys hit multiple strings, and these strings are adjusted until they have nearly the same frequencies and give a slow beat for richness.

The guitars and mandolins are also used for tuning.

Beatings can sometimes be annoying, but they have many applications.

A useful way to compare frequencies is by observing beats.
There are different applications of beats as compared to radar speed traps.

A friend holds one end of a jump rope while you hold the other.

If your friend holds her end still, you can move your end up and down.

The rope could have waves with a certain amount of amplitudes, or it could have waves with no amplitude at all.

The wavelength will result in both constructive and destructive interference.

There are areas of wave interference where there is no motion.

There are areas of wave interference where the motion is at its maximum point.

There is bound to be some wave interference with multiple speakers putting out sounds in the room.

The interference is mostly destructive in the dull areas.
The interference is mostly constructive in the louder areas.

The interference pattern can be created by adding a second source or a pair of slits.