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In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater intensity (constructive interference) or lower amplitude (destructive interference) if the two waves are in phase or out of phase, respectively. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves as well as in loudspeakers as electrical waves.

Etymology

The word interference is derived from the Latin words inter which means "between" and fere which means "hit or strike", and was used in the context of wave superposition by Thomas Young in 1801.^[1][2][3]

Mechanisms

The principle of superposition of waves states that when two or more propagating waves of the same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves.^[4] If a crest of a wave meets a crest of another wave of the same frequency at the same point, then the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave, then the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference. In ideal mediums (water, air are almost ideal) energy is always conserved, at points of destructive interference energy is stored in the elasticity of the medium. For example when we drop 2 pebbles in a pond we see a pattern but eventually waves continue and only when they reach the shore is the energy absorbed away from the medium.

Constructive interference occurs when the phase difference between the waves is an even multiple of π (180°), whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.

Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement. In other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.

Interference of light is a unique phenomenon in that we can never observe superposition of the EM field directly as we can, for example, in water. Superposition in the EM field is an assumed phenomenon and necessary to explain how two light beams pass through each other and continue on their respective paths. Prime examples of light interference are the famous double-slit experiment, laser speckle, anti-reflective coatings and interferometers.

In addition to classical wave model for understanding optical interference, quantum matter waves also demonstrate interference.

Real-valued wave functions

The above can be demonstrated in one dimension by deriving the formula for the sum of two waves. The equation for the amplitude of a sinusoidal wave traveling to the right along the x-axis is

where ${\displaystyle A}$ is the peak amplitude, ${\displaystyle k=2\pi /\lambda }$ is the wavenumber and ${\displaystyle \omega =2\pi f}$ is the angular frequency of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is also traveling to the right where ${\displaystyle \varphi }$ is the phase difference between the waves in radians. The two waves will superpose and add: the sum of the two waves is Using the trigonometric identity for the sum of two cosines: ${\textstyle \cos a+\cos b=2\cos \left({a-b \over 2}\right)\cos \left({a+b \over 2}\right),}$ this can be written This represents a wave at the original frequency, traveling to the right like its components, whose amplitude is proportional to the cosine of ${\displaystyle \varphi /2}$.

• Constructive interference: If the phase difference is an even multiple of π: ${\displaystyle \varphi =\ldots ,-4\pi ,-2\pi ,0,2\pi ,4\pi ,\ldots }$ then ${\displaystyle \left|\cos(\varphi /2)\right|=1}$, so the sum of the two waves is a wave with twice the amplitude
  $${\displaystyle W_{1}+W_{2}=2A\cos(kx-\omega t)}$$

  
• Destructive interference: If the phase difference is an odd multiple of π: ${\displaystyle \varphi =\ldots ,-3\pi ,\,-\pi ,\,\pi ,\,3\pi ,\,5\pi ,\ldots }$ then ${\displaystyle \cos(\varphi /2)=0\,}$, so the sum of the two waves is zero
  $${\displaystyle W_{1}+W_{2}=0}$$

  

Between two plane waves

A simple form of interference pattern is obtained if two plane waves of the same frequency intersect at an angle. Interference is essentially an energy redistribution process. The energy which is lost at the destructive interference is regained at the constructive interference. One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave. Assuming that the two waves are in phase at the point B, then the relative phase changes along the x-axis. The phase difference at the point A is given by

It can be seen that the two waves are in phase when

and are half a cycle out of phase when

Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase. Thus, an interference fringe pattern is produced, where the separation of the maxima is

and d_f is known as the fringe spacing. The fringe spacing increases with increase in wavelength, and with decreasing angle θ.

The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout.

Between two spherical waves

A point source produces a spherical wave. If the light from two point sources overlaps, the interference pattern maps out the way in which the phase difference between the two waves varies in space. This depends on the wavelength and on the separation of the point sources. The figure to the right shows interference between two spherical waves. The wavelength increases from top to bottom, and the distance between the sources increases from left to right.

When the plane of observation is far enough away, the fringe pattern will be a series of almost straight lines, since the waves will then be almost planar.

Multiple beams

Interference occurs when several waves are added together provided that the phase differences between them remain constant over the observation time.

It is sometimes desirable for several waves of the same frequency and amplitude to sum to zero (that is, interfere destructively, cancel). This is the principle behind, for example, 3-phase power and the diffraction grating. In both of these cases, the result is achieved by uniform spacing of the phases.

It is easy to see that a set of waves will cancel if they have the same amplitude and their phases are spaced equally in angle. Using phasors, each wave can be represented as ${\displaystyle Ae^{i\varphi _{n}}}$ for ${\displaystyle N}$ waves from ${\displaystyle n=0}$ to ${\displaystyle n=N-1}$Zdroj:https://en.wikipedia.org?pojem=Interference_pattern
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