Superposition and Interference

When two waves cross in the same medium, they obey the principle of superposition: the resulting displacement at a point is simply the sum of the displacements that each wave would produce on its own at that location. After they meet, the two waves each continue unchanged, as if they had never crossed. This is a remarkable property, valid as long as the amplitudes remain moderate.

When two waves of the same frequency and the same amplitude superpose, we speak of interference. The result depends on their path difference, that is, the difference between the distances traveled from each source to the point considered. If this path difference is a whole number of wavelengths (0, λ, 2λ...), the two waves arrive in phase: their crests coincide, and the resulting amplitude is maximal. This is constructive interference.

If the path difference is a whole number of wavelengths plus half a wavelength (λ/2, 3λ/2...), the two waves arrive in antiphase: the crest of one coincides with the trough of the other, and they cancel each other almost completely. This is destructive interference. Between these two extreme cases, all intermediate situations occur, which gives rise to the famous fringes or cancellation zones observed on the surface of water with two sources vibrating in phase.

In acoustics, this phenomenon explains why some areas of a concert hall seem quieter or louder depending on position: the sound waves reflected by the walls interfere with the direct sound. It is also the principle used in noise-canceling headphones, which generate a wave in antiphase with the ambient noise to cancel it locally.

Beware of a common misconception: interference does not destroy energy during destructive cancellation. The total energy of the system is simply redistributed elsewhere, toward the zones of constructive interference. Nothing is lost, everything is moved around.

constructive interference (in phase)           destructive interference (antiphase)

wave 1:  /\  /\  /\                             wave 1:  /\  /\  /\
wave 2:  /\  /\  /\                             wave 2:  \/  \/  \/
         --------------                                  --------------
sum   :  doubled amplitude                      sum   :  amplitude ~ 0 (cancellation)
(the crests add up)                             (crest + trough cancel out)