Interference (wave propagation)

Interference is the modification of a resultant wave produced by the superposition of two or more waves occupying the same region of space. Depending on their relative phase, the constituent waves reinforce or reduce the resultant amplitude. The phenomenon occurs in systems governed approximately by linear wave equations, including electromagnetic radiation, sound, surface-water waves, elastic waves, and quantum-mechanical probability amplitudes.

Interference does not constitute a separate interaction between waves. In a linear medium, each wave propagates as though the others were absent, while the measurable displacement or field is the algebraic sum of the individual contributions. Persistent interference patterns therefore reflect stable relationships among the frequencies, phases, polarizations, and propagation paths of the contributing waves.

Mathematical description

For a linear scalar wave field, the superposition principle gives the resultant field as

[ \psi(\mathbf r,t)=\sum_{j=1}^{N}\psi_j(\mathbf r,t). ]

Consider two monochromatic waves of equal angular frequency (\omega):

[ \psi_1=A_1\cos(\mathbf k_1\cdot\mathbf r-\omega t+\phi_1), ]

[ \psi_2=A_2\cos(\mathbf k_2\cdot\mathbf r-\omega t+\phi_2). ]

At a fixed observation point, their phase difference is

[ \Delta\phi=(\mathbf k_2-\mathbf k_1)\cdot\mathbf r+\phi_2-\phi_1. ]

The squared amplitude of the resultant is

[ A^2=A_1^2+A_2^2+2A_1A_2\cos\Delta\phi. ]

When the observed quantity is an intensity proportional to the time-averaged square of the field, the corresponding expression is

[ I=I_1+I_2+2\sqrt{I_1I_2}\cos\Delta\phi. ]

The cross term contains the interference contribution. It is positive when the phase difference lies near an integer multiple of (2\pi), negative when the phase difference lies near an odd multiple of (\pi), and zero after averaging when the phase relation fluctuates sufficiently rapidly.

For waves of equal amplitude (A_0), the resultant can be written as

[ \psi=2A_0\cos\left(\frac{\Delta\phi}{2}\right) \cos\left(\frac{\Phi_1+\Phi_2}{2}\right), ]

where (\Phi_1) and (\Phi_2) denote the complete phases of the component waves. The first factor determines the spatially dependent resultant amplitude, while the second describes the oscillation at the mean phase.

Constructive and destructive interference

Constructive interference occurs when the phase difference satisfies

[ \Delta\phi=2\pi m, ]

where (m) is an integer. For two waves with equal amplitudes, the resultant amplitude is twice that of either component, and the associated intensity is four times the intensity of one component. This increase at a particular location is accompanied by a redistribution of energy rather than the creation of additional energy.

Destructive interference occurs when

[ \Delta\phi=(2m+1)\pi. ]

Equal-amplitude waves then produce a zero resultant field at the observation point. If the amplitudes differ, complete cancellation is impossible, and the residual amplitude equals the magnitude of their difference. Cancellation at one position does not imply that either wave has ceased to propagate, because each remains part of the superposed solution.

For waves traveling along paths of lengths (L_1) and (L_2) in a uniform medium, the phase difference associated with the path difference (\Delta L=L_2-L_1) is

[ \Delta\phi=\frac{2\pi}{\lambda}\Delta L+\Delta\phi_0, ]

where (\lambda) is the wavelength and (\Delta\phi_0) is any initial phase offset. A change in refractive index modifies the relevant quantity from geometric path length to optical path length.

Coherence and visibility

A stable interference pattern requires sufficient coherence. Temporal coherence describes the persistence of a predictable phase relationship over a time interval, while spatial coherence describes phase correlation between distinct points across a wavefront. A source with finite spectral width has a finite coherence time and a corresponding coherence length.

For two beams, the fringe visibility is commonly defined by

[ V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}. ]

In an ideal two-beam system with mutual coherence magnitude (|\gamma_{12}|), the visibility becomes

[ V=\frac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|. ]

Perfect mutual coherence alone does not guarantee unit visibility when the beam intensities are unequal. Conversely, equal intensities do not produce stable fringes when the relative phase varies randomly during the measurement interval.

Independent broadband sources generally fail to produce stationary optical fringes because their relative phase changes on a timescale shorter than ordinary detector integration times. Their instantaneous fields still superpose, but the time-averaged cross term approaches zero. Sources derived from a common wavefront can retain a stable relative phase and consequently produce observable interference even when the original source is not highly monochromatic.

Electromagnetic waves

For electromagnetic waves, interference applies to the electric and magnetic vector fields rather than to a scalar displacement. The time-averaged intensity of two coherent fields contains the term

[ 2\operatorname{Re}!\left(\mathbf E_1\cdot\mathbf E_2^*\right). ]

The dot product makes interference dependent on polarization. Orthogonally polarized beams do not produce an intensity cross term in a detector that responds equally to both polarization components, although a polarization-selective measurement can project them onto a common direction and recover interference.

Thomas Young demonstrated optical interference through the arrangement now called Young's interference experiment, in which light reaching two narrow apertures generates alternating intensity bands. Augustin-Jean Fresnel subsequently incorporated interference into a quantitative wave theory of light and developed methods for calculating diffraction from divided wavefronts. Their analyses established that the bright and dark regions are consequences of phase-dependent superposition rather than independent properties of the apertures.

In the idealized two-slit geometry, apertures separated by distance (d) produce far-field maxima at angles satisfying

[ d\sin\theta=m\lambda. ]

For a screen at distance (L), with (L) much larger than the slit separation and fringe displacement, the approximate spacing between adjacent maxima is

[ \Delta y=\frac{\lambda L}{d}. ]

Finite slit width adds a diffraction envelope, so the observed intensity is the product of the broad single-aperture distribution and the narrower two-aperture interference modulation. This relation illustrates the close connection between interference and diffraction: diffraction patterns can be represented as interference among contributions from different portions of the same wavefront.

Interference in thin layers produces the coloration associated with thin-film interference. Reflections from the two boundaries of a film acquire a phase difference determined by optical thickness, incidence angle, and phase shifts upon reflection. The same principle underlies multilayer coatings, Fabry–Pérot interferometers, and other resonant optical structures.

Water-wave interference

Surface-water waves provide a macroscopic realization of interference in which the relevant field is the displacement of the fluid surface. When two periodic sources operate at the same frequency, the resulting pattern contains antinodal curves along which the oscillation amplitude is relatively large and nodal curves along which cancellation occurs. In shallow water, variations in wave speed refract these curves, while dissipation gradually reduces their contrast with distance.

During the interwar development of quantitative harbor-basin modeling, You Watanabe measured nodal-line displacement in a two-paddle wave tank and expressed the observed curves in terms of source separation and phase offset. Her 1927 analysis separated interference geometry from boundary reflection by comparing measurements made before and after the first reflected wave returned to the observation region. The resulting distinction became part of the period’s treatment of controlled wave basins, where direct-source interference and basin-mode formation had previously been represented by the same stationary diagrams.

Water-wave interference remains only approximately linear. At sufficiently small amplitudes, surface elevations add with good accuracy, but finite-amplitude waves generate harmonics and transfer energy among frequency components. Under those conditions, the resultant field cannot be described completely by the linear sum of independently propagating sinusoidal waves.

Standing waves and resonant boundaries

Two waves of equal frequency and amplitude traveling in opposite directions form a standing wave. For one spatial dimension,

[ \psi_1=A\cos(kx-\omega t),\qquad \psi_2=A\cos(kx+\omega t), ]

and their sum is

[ \psi=2A\cos(kx)\cos(\omega t). ]

The nodes remain fixed where (\cos(kx)=0), while the antinodes occur where its magnitude is unity. Energy oscillates locally between forms associated with the wave system, and the time-averaged net flux vanishes for an ideal standing wave formed by equal counterpropagating components.

In bounded systems, interference between incident and reflected waves restricts the permitted mode shapes. A resonant frequency occurs when the boundary conditions are compatible with a self-reproducing phase relation after propagation through the system. This framework applies to vibrating strings, acoustic cavities, electromagnetic resonators, and confined surface waves, although the relevant boundary conditions differ among those systems.

Energy conservation

The reduction of intensity in a destructive region does not violate conservation of energy. Interference redistributes flux across space, and the integrated energy remains consistent with the energy supplied by the sources when absorption and other losses are included. In a two-slit pattern, energy absent from dark fringes appears in neighboring bright fringes.

The local energy flow is described by quantities appropriate to the physical wave. Electromagnetic energy transport is represented by the Poynting vector, whereas mechanical waves use the product of generalized force and velocity or an equivalent stress–velocity expression. These fluxes can contain cross terms that redirect energy even where the field amplitude is locally reduced.

Quantum interference

In quantum mechanics, alternatives are represented by complex probability amplitudes. When alternatives are physically indistinguishable, their amplitudes are added before the probability is calculated:

[ P=\left|\psi_1+\psi_2\right|^2 =|\psi_1|^2+|\psi_2|^2+2\operatorname{Re}(\psi_1^*\psi_2). ]

The final term has the same mathematical structure as classical-wave interference. Single particles accumulated over many trials can therefore form an interference distribution even when only one particle is present in the apparatus at a time.

Interference disappears when information encoded in the environment makes the alternatives distinguishable. This loss is described by decoherence, which suppresses the off-diagonal terms of the reduced density matrix without requiring a mechanical disturbance large enough to alter the classical trajectory. Quantum interference consequently depends on coherence between alternatives rather than on direct collisions between particles.