What interference means

Each of us has experienced the phenomenon of interference, without being aware of it. The list of phenomena that physicists explain using the concept of interference can be very long:

• when two sea waves generated by "rude" boats reach our little boat in phase, we have experienced that the amplitude of the wave hitting us increases;
• when listening to a concert we may have noticed that, moving from one place to another, the perceived sound can change appreciably in intensity;
• looking at a puddle "polluted" by a thin film of lubricant, we have observed that next to areas of splendid iridescence (also due to interference, which reinforces different colours depending on the thickness of the film) there are very dark areas in which there is almost no reflected light;
• how many times, holding our mobile phone very close to our computer, have we experienced, just before a call arrives, a sort of "crackle" coming out of the speakers, or a "flicker" of the images on the screen? In this last case, however, the word "interference" has a somewhat different meaning from the others: the phone's signal is picked up by the device's circuits and disturbs their operation, and it is not the superposition of two waves that this page is about.

The explanation of these phenomena is linked to the superposition principle: at every point in space where two waves of the same nature arrive simultaneously, the overall oscillation is given by the algebraic sum of the oscillations of the two incident waves taken separately.

constructive superposition	destructive superposition

As the images show, the superposition of waves can be "constructive", in the sense that the overall amplitude of the resulting wave is greater than the individual amplitudes of the waves that add up to form it. In such cases we speak of constructive interference. If instead, as in the second image, the superposition of the waves produces a wave of amplitude smaller than the sum of the two component amplitudes, we speak of destructive interference.

Interference between periodic waves

If, instead of occurring between single pulses, interference occurs between periodic waves of arbitrary frequency, a single point in space can be, at different instants, the site of constructive or destructive interference, which makes the study of interference extremely complex. Moreover, even within constructive (or destructive) interference alone, there can be different "degrees" of interference. In the following animation, for example, we see in grey two periodic waves, which give rise to a wave resulting from their superposition (in black).

NB: in the following examples all the coloured points are at a fixed position on the horizontal axis, while they oscillate freely along the vertical axis. In the second and third animations these points actually seem to move forward. This is an optical illusion, as you can check by placing a vertical ruler on the screen!

The red point undergoes maximum constructive interference, obtained by making the two maxima of the interfering waves meet exactly at that point. At the green point this "meeting condition" is not satisfied, and the superposition still takes place, giving rise to an oscillation of amplitude smaller than the maximum possible. The black point, instead, is the site of maximum destructive interference: it always remains at rest (we also say that the point is a node). The condition is obtained by making sure that the maximum of one of the two interfering waves meets, at the position of the black point, the minimum of the other wave. The black wave does not look like a progressive wave propagating through space; it rather seems to "stand" between two nodes: for this reason it is called a standing wave.

In the second example the two waves have equal amplitude and frequency, but there is a time shift between them: the second interfering wave lags behind the first by a constant time interval. This situation could correspond to that of points in space whose difference in distance from the two sources is constant (this is precisely the definition of a hyperboloid). This condition can also be obtained at points in space aligned with the sources of the interfering waves (a geometer would say that the hyperboloid degenerates into a straight line). The situation could also be that of a single source emitting two waves at successive times with a constant time shift. But if the source is the same, one might as well say that it emits the black wave directly! This wave looks like a "progressive" wave advancing with a speed equal to that of the two waves that compose it. This motion is not transmitted to the coloured points: they oscillate vertically, reaching the maximum amplitude of oscillation at successive times.

In this case a further complication appears, to which a separate page is devoted (dispersion). The two interfering waves have different propagation speeds: the one with the longer wavelength moves twice as fast as the other. As in the previous example, no fixed nodes or antinodes can be identified in space any more but, in addition, in this case the shape of the resulting black wave does not remain unchanged during propagation. In other words, while in the previous case the propagation speed of the maximum of the wave was equal to the propagation speed of the wave, in this case the position of each maximum moves at a speed different from the speeds of the two component waves. See the page on dispersion for more details, and to discover the importance of this phenomenon for many practical applications as well.

The previous analysis shows how complex the study of interference can be in general situations: for this reason one speaks in general of superposition, while the term interference is reserved for the case in which the superposition takes place in the following ways:

• periodic waves of equal frequency (and possibly amplitude) are superposed;
• the sources of the waves oscillate in phase, that is, perfectly synchronously or in any case with a known "phase shift";
• when we speak of constructive (or destructive) interference we always mean totally constructive (or destructive) interference;

Under these assumptions the interference phenomenon becomes "regular", in the sense that a given point in space is always the site of constructive, or always of destructive, interference, unchanging in time (as in the first animation above). Ultimately, the central problem of interference is to establish, given the positions of the sources, the frequencies of oscillation and any phase shifts, whether a given point in space is the site of constructive or destructive interference.

The answer depends on the difference between the distances of the point from the two sources (the path difference). For two sources oscillating in phase:

• if the path difference is a whole number of wavelengths (0, λ, 2λ, ...) the two waves arrive in phase and the interference is constructive;
• if it is an odd number of half wavelengths (λ/2, 3λ/2, ...) the two waves arrive in antiphase and the interference is destructive.

For a more complete treatment see mathematical aspects of interference.

If, instead, even one of these assumptions is not met, interference becomes more complicated and the waves superpose without producing regularities that are easy to analyse quantitatively.

Where does the energy go?

If two waves cancel by destructive interference, where does their energy go? Energy cannot disappear. The answer is that complete cancellation everywhere and at every instant never happens: the energy is redistributed in space or in time, or it is not emitted at all, or it goes back.

• Redistribution in space. Two sources emitting waves of the same frequency produce an interference pattern: at some points the interference is destructive (silence), at others constructive. At the points of constructive interference the amplitudes add up and, since the intensity is proportional to the square of the amplitude, the intensity there is four times that of a single wave, not two. Averaged over all space, the intensity is exactly the sum of those of the two waves: the energy missing at the points of silence is found, in excess, at the points of reinforcement.
• Redistribution in time. Two equal and opposite pulses travelling in opposite directions along a string cancel each other at the instant they overlap: the string is perfectly straight. But it is not at rest: at that instant the parts of the string involved move with twice the velocity, and all the energy has become kinetic energy. Immediately afterwards the two pulses reappear and continue on their way (see superposition principle). Something similar happens in standing waves: at the nodes the string never moves, but the energy has not disappeared; it is at the antinodes, where it continuously changes from potential to kinetic.
• Energy not emitted. If two sources very close to each other (compared with the wavelength) vibrate in phase opposition, their waves cancel almost everywhere. In this case each source, when it compresses the air, finds next to it the rarefaction produced by the other: the air offers it very little resistance, and the two sources give the medium much less energy than they would on their own. The energy therefore stays in the generator (for example in the amplifier driving two loudspeakers connected in antiphase). This is also why a loudspeaker without a cabinet sounds weak at low frequencies: the waves emitted by the two faces of the cone are in phase opposition and cancel each other.
• Energy reflected. In the Quincke tube the sound splits into two branches that then rejoin: when the two paths differ by half a wavelength, no sound reaches the output. The energy does not disappear: it goes back towards the source as a reflected wave.

Check your understanding

Virtual laboratory: interference experiments

If you are interested in visualising interference phenomena, you can use our 2D Waves virtual laboratory. Below we propose a series of guided experiments, but you are free to try new configurations and new experiments. The app is very intuitive; to understand correctly what you observe, just keep in mind that the coloured areas represent the amplitude of the oscillation. The brighter the colour, the greater the amplitude of the oscillations (green for crests, red for troughs). Obviously the dark areas represent the absence of oscillation caused by destructive interference.

Suggested experiments:

• Interference between two or more point sources of the same frequency
• Interference between two point sources of different frequencies
• Interference between waves of different frequencies generated by a single source (beats)
• Reflection from a parabolic mirror

Further reading and links

• Those interested in studying the conditions for interference and beats more deeply from a mathematical point of view can visit the page mathematical aspects of interference.

• The phenomenon of beats, because of its importance in acoustics, deserves a separate in-depth treatment, which you will find in the section Beats

• A particularly important case of interference is that of harmonic waves. In this case the general conditions for interference take a much simpler mathematical form. On the page interference of harmonic waves the question is dealt with in full generality.

• Standing waves are so important for understanding how a musical instrument works that we recommend visiting the page devoted to them.

• Anyone who simply wants to observe other spectacular effects of interference should not miss the page Antennas and interference.
• Related questions and answers:
  • In ripple tank images with two sources, does constructive interference correspond to the bright zones or to the dark ones?
  • I know that two pure sounds with opposite phases cancel each other out; but is it really possible to cancel a sound by adding one with the opposite phase to it?