Sometimes as a wave travels through a medium, it encounters another wave, also travelling through the same medium. What happens when these waves collide? It turns out that the waves combine in a relatively intuitive, easy-to-calculate way. Not only that, but there are also plenty of useful applications of ​wave interference​ both in the lab and in every day life.

To know what the combination of waves will do to a given point in the medium at a given point in time, you simply add what they would be doing independently. This is called the ​principle of superposition​.

For example, if you were to plot the two waves on the same graph, you would simply add their individual amplitudes at each point to determine the resultant wave. Sometimes the resultant amplitude will have a larger combined magnitude at that point, and sometimes the effects of the waves will partially or completely cancel each other.

Imagine if we had wave A traveling to the right and wave B traveling to the left. If we look at a certain point in space where wave A had an upward displacement of 2 units, while wave B had a downward displacement of 1 unit, the resultant wave would have an upward displacement of 1 unit: 2 - 1 = 1.

In ​constructive interference​, the displacement of the medium must be in the same direction for both waves. They combine together to make a single wave with a greater amplitude than either wave individually. For perfect constructive interference, the waves must be in phase – meaning their peaks and valleys line up perfectly – and have the same period.

For ​destructive interference​, the displacement of the medium for one wave is in the opposite direction to that of the other wave. The amplitude of the resultant wave will be less than that of the wave with the larger amplitude.

For perfect destructive interference, where the waves cancel each other out to create zero amplitude, the waves must be exactly out of phase – meaning the peak of one lines up perfectly with the valley of the other – and have the same period ​and​ amplitude. (If the amplitudes are not the same, the waves will not cancel out to exactly zero.)

Note that destructive interference doesn't stop the wave; it just brings its amplitude in that particular place to zero. Interference is what happens when waves pass through one another – once the waves are no longer interacting, they go back to their original amplitudes.

Waves can reflect off of surfaces and fixed points wherever the medium they're traveling through changes to a different medium.

If a string is fixed on one side, any wave traveling along the string that hits that fixed point will reflect off of it "upside down," or as a reverse version of the original wave. If a string is free on one side, any wave traveling along the string that hits the end will reflect off of it right-side up. If a string is tied to another string of a different density, when a wave hits that connection part of it will reflect (as if the end of the string were fixed) and part of it will continue.

When a wave in water or air hits a surface, it will reflect off of that surface at the same angle it struck. This is called the incident angle.

Reflected waves can often interfere with themselves which can, in special circumstances, create a special kind of wave known as a standing wave.

Imagine a string with one or both ends fixed. A wave traveling on this string that hits a fixed end will reflect off of that end, traveling in the opposite direction, and interfere with the original wave that created it.

This interference is not necessarily perfectly constructive or destructive unless the string's length is a multiple of half of the wave's wavelength.

[image of fundamental/harmonic standing frequencies]

This creates a standing wave pattern: outgoing original waves interfering with reflected waves as they move in opposite directions. The waves going in opposite directions interfere with each other in such a way that they no longer look like they are moving; instead, it appears as if sections of the string are simply moving up and down in place. This occurs, for example, in guitar strings when they are plucked.

The points on the string that appear fixed are called ​nodes​. Midway between each pair of nodes is a point on the string that reaches maximum amplitude; these points are called ​antinodes​.

The ​fundamental frequency​, or ​first harmonic​, of a string occurs when the length of the string is half of the wavelength of the wave. The standing wave then looks like a single wave peak vibrating up and down; it has one antinode, and one node on each end of the string.

The standing wave with string length equal to the wavelength of the wave is called the second harmonic; it has two antinodes and three nodes, where two nodes are at the ends and one node is in the center. Harmonics are very important to how musical instruments create music.

Noise-cancelling headphones work on the principle of destructive interference of sound waves. A microphone on the headphones detects any low-level noise around you, and then the headphones emit sound waves into your ears that destructively interfere with the ambient noise. This cancels the ambient noise out completely, allowing you to hear your music and podcasts much more clearly in a noisy environment.

Mufflers on cars work similarly, although in a more mechanical fashion. The size of the chambers in a muffler are precisely designed such that once the engine noise enters the muffler, it destructively interferes with its own reflected noise, making the car quieter.

Microwave light emitted by your microwave oven also experiences interference. There are locations inside your microwave where light waves emitted into the interior of the oven constructively and destructively interfere, either heating up your food more or less. This is why most microwave ovens have a rotating plate inside: to keep your food from being completely frozen in some spots and boiling in others. (Not a perfect solution, but it's better than the food staying still!)

Wave interference is a very important consideration when designing concert halls and auditoriums. These rooms can have "dead spots," where the sound from the stage, reflected off of the surfaces in the room, destructively interferes at a certain place in the audience. This can be prevented through careful placement of sound-absorbing and sound-reflecting materials in the walls and ceiling. Some concert halls will have speakers aimed at these spots to enable the audience members sitting there to still hear properly.

Just like with other waves, light waves can interfere with each other and can diffract, or bend, around a barrier or opening. A wave diffracts more when the opening is closer in size to the wavelength of the wave. This diffraction causes an interference pattern – regions where the waves add together and regions where the waves cancel each other out.

Let's take the example of light going through a single horizontal slit. If you imagine a straight line from the center of the slit to the wall, where that line hits the wall should be a bright spot of constructive interference.

We can model the light passing through the slit as a line of multiple point sources that all radiate outwards. Light from sources at the left and right of the slit will have traveled the same distance to get to this particular spot on the wall, and so will be in phase and constructively interfere. The next point in on the left and the next point in on the right will also constructively interfere, and so on, creating a bright maximum in the center.

The first spot where destructive interference will occur can be determined as follows: Imagine the light coming from the point at the left end of the slit (point A) and a point coming from the middle (point B). If the path difference from each of those sources to the wall differs by 1/2λ, 3/2λ and so on, then they will destructively interfere.

If we take the next point in on the left and the next point to the right of the middle, the path length difference between these two source points and the first two would be approximately the same, and so they would also destructively interfere.

This pattern repeats for all remaining pairs of points, meaning that if light coming from point A and point B interferes at a given spot on the wall, then all the light coming through the slit experiences interference at that same spot.

A slightly different diffraction pattern can also be obtained by passing light through two small slits separated by distance a in a double-slit experiment. Here we see constructive interference (bright spots) on the wall anytime the path length difference between light coming from the two slits is a multiple of the wavelength λ.

Scientists use wave interference every day to make exciting discoveries, using interferometers. An interferometer is a scientific instrument that uses the interference of light waves to make measurements and perform experiments.

A basic interferometer takes a laser beam and splits it into two beams. One beam will do very different things or have different things done to it, depending on the question scientists are trying to answer. The beams will then be recombined, but the different experiences they had will have changed them. Scientists can look at the interference of the two now-different laser beams to investigate scientific questions, like the nature of gravitational waves.

The Laser Interferometer Gravitational-wave Observatory (LIGO) is a giant interferometer that sends its split laser beams 2.5 miles (4 km) away and back.

The split beams are at a right angle, so if a gravitational wave passes through the interferometer, it will affect each beam differently. This means that they will interfere with each other when they are recombined, and the interference pattern tells physicists about what caused the gravitational waves. That is how LIGO detected gravitational waves from black holes crashing together, a discovery that won the Nobel Prize in 2017.