Superposition

The principle of superposition tells us how two or more waves may interact in the same medium. The displacement of the medium at any given point is just the sum of the displacements of each wave.

This simple principle has powerful applications in many domains of science. Complex waveforms observed in nature are often the superposition of simpler waves, and those simpler component waves often provide insight into the physical process that generated the complex waveform.

Credit: NOAA

Pulses can pass through each other.

In the laboratory, two water waves generated on opposite sides of a ripple tank can illustrate how the water waves pass "through" each other. You can see the details in this diagram — a series of "snapshots" of two pulses traveling in opposite directions:

In the diagram, both pulses had displacements above their undisplaced levels. The superposition of the pulses generated an extra large displacement. The pulses are said to interfere constructively.

Two pulses traveling in opposite directions pass through each other. At any given point, the medium is displaced by an amount equal to the sum of the displacement from each pulse.

If one pulse has a displacement above the mean level and one below, the superposition of the two may cancel out or partially cancel. The pulses are said to interfere destructively, as shown in this illustration.

The superposition of two pulses whose displacements are above and below the mean, respectively. The amplitudes partially cancel out. If the pulses had exactly the same amplitude and symmetrical shapes, the pulses would cancel out entirely at some point.

Animations are even better for showing the constructive and destructive interference of waves:

The first animation on this page shows a small pulse from the right and a larger pulse from the left moving toward each other and passing through each other:
This site shows constructive and destructive interference separately: for constructive interference and for destructive interference.
This site has several illustrations and an animation of sound waves interacting.

A standing wave is a wave that does not travel.

A standing wave can also be described as the superposition of two waves traveling in opposite directions, in such a way that the medium as a whole has its own wave motion.

Transverse waves on a rope or Slinky are the easiest way demonstrate visible standing waves. Longitudinal sound waves may also make standing waves, particularly in musical instruments.

Play with an animation applet of standing waves.

Amplitude

One distinction between the wave motion of traveling waves and standing waves is in the amplitude of the wave.

In the traveling wave, each point on the rope or Slinky has the same amplitude of vibration, although each point crests at a different time.
In the standing wave, the amplitude varies according to the location on the string or Slinky. There may be no motion at some points in the standing wave, while other points always have the maximum amplitude.

See an animation comparing standing and traveling waves.

Energy

What about the energy carried by the two traveling waves going in opposite directions?

In a standing wave, the energy stays “standing” as the medium oscillates. The energy of a point on the rope or Slinky alternates between kinetic energy and potential energy.

Generating a standing wave

To generate a standing wave in the classroom, you need a long, flexible rope (or a rubber tube). One end is fixed. The experimenter waves the other end back and forth. When the experimenter hits a particular frequency, the wave traveling down toward the fixed end interacts with the wave reflected from the fixed end to create a standing wave.

The standing wave in this demonstration can take many forms. It may have no motion at the two fixed ends and a maximum displacement in the middle, as shown in this diagram. The red line shows the outline or envelope of where the rope may be as it oscillates back and forth.

Standing wave on a string that is fixed at both ends. An antinode has formed in the middle. Points on the string at the antinode have the maximum amplitude of vibration.

Nodes and antinodes

The fixed points at the ends of the rope are called nodes. The point in the middle, where the rope has maximum amplitude of displacement, is called an antinode.

If the experimenter waves the end of the rope at a different particular frequency, a standing wave may emerge with a node in the middle of the rope as well as at the ends, and antinodes in between.

Standing wave on a string with nodes at both ends and one node in the middle.

In general, standing waves may be set up with more nodes and antinodes. The ends of the rope, if fixed, will always be nodes.

It is possible to set up a standing wave with an antinode at one end, or even two ends. This is the case for the example with sound waves in wind instruments.

Consider an organ pipe that is open at both ends. Vibrations very near an open end will experience reflection like that of a wave on a string with a free end: the reflected wave will be in phase with the incoming wave. The standing wave will have an antinode at the open ends of a pipe. If the organ pipe is closed at one end and open at the other, the closed end will always be a node and the open end an antinode.

Longitudinal pressure waves in an organ pipe that is open at both ends. A standing wave can be set up with antinodes at the open ends, and one, two or more nodes inside. The red line shows the envelope or outline of the superposition of waves inside the pipe.

Beats are interference in time.

The interference we have discussed so far has been interference in space: that is, extra large or small displacements from constructive or destructive interference have occurred at physical points along a medium.

The interference may also happen in time. The superposition of two sound waves, for example, may result in a wave that is extra loud or extra quiet at different points in time.

For example, consider playing two adjacent notes on a piano at the same time. In the same interval of time, the pressure wave from one of the notes might cycle through three wavelengths, while the second note cycles through two wavelengths. These two waves will interfere — sometimes constructively, and sometimes destructively.

Plot of the pressure versus time for two sound waves that are of the same amplitude but slightly different frequency. Note that every other time there is a crest in the Frequency 2 wave, there is also a crest in the Frequency 1 wave. But for the intervening crests in Frequency 2, there is a trough in Frequency 1. So sometimes the amplitudes add to make a greater positive amplitude, and sometimes they cancel out.

Watch this interactive animation showing two different waves, one in white and one in green, and the superposition in yellow below.

When the two waves interfere destructively, no sound is heard. These are the nodes.
When the two waves interfere constructively, the sound is extra loud. These are antinodes, as in the case of standing waves on a string.

In general, when two tones of two different frequencies sound at the same time, the result is a pattern of constructive and destructive interference that varies on a time period equal to the difference in frequencies of the two tones.

For example, if you play the A below low C on the piano (frequency 110 Hz) at the same time as the adjacent A sharp (frequency 116.5 Hz), the difference between them is 116.5 Hz - 110 Hz or 6.5 Hz. This (6.5 Hz) is the frequency at which one hears the "beating" of the sound — the periodic increase and decrease in loudness.

Experiment with beats at this site:
You'll notice that the "beating" is more noticeable for the combination or interference of 330 and 331 Hz, because the beat frequency is 1 Hz. For the interference of 330 and 340 Hz, the beat frequency is a fast 10 Hz. More than 10 or so beats per minute are difficult to detect by ear.

What does it mean to be “in tune”?

A piano tuner makes use of beats to tune a piano. He or she will hit a tuning fork and simultaneously play a piano note that is supposed to vibrate at the same frequency. If the piano tuner hears beats, he or she will adjust the tension on the string (or one of the strings) hit by the hammer on the piano key, slightly changing the piano note’s pitch. When no beats can be heard, the piano note and tuning fork are “in tune.”