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Wave interference

Wave interference is the phenomenon in which two or more waves superpose in a medium, resulting in a resultant wave whose amplitude is the algebraic sum of the individual wave amplitudes at each point, governed by the principle of superposition.^[1] This interaction occurs when waves from different sources or reflected waves overlap in space and time, leading to regions of enhanced or reduced intensity depending on their relative phases.^[2] The principle of superposition applies to linear waves, such as mechanical waves (sound and water) and electromagnetic waves (light), where the waves pass through each other without altering their individual paths after interaction.^[3] Interference manifests in two primary forms: constructive interference, where waves align in phase—such that crests coincide with crests and troughs with troughs—causing amplitudes to add and produce a wave with greater amplitude and intensity; and destructive interference, where waves are out of phase—crests aligning with troughs—leading to amplitude subtraction and potentially complete cancellation if amplitudes are equal.^[1] For constructive interference to occur, the path difference between waves must be an integer multiple of the wavelength, while destructive interference requires a path difference of an odd multiple of half the wavelength.^[4] These conditions are fundamental to observing stable interference patterns, which require coherent sources—waves with a constant phase relationship.^[2] A landmark demonstration of wave interference is Thomas Young's double-slit experiment conducted in 1801, which provided key evidence for the wave nature of light by producing an alternating pattern of bright and dark fringes on a screen due to the interference of light waves passing through two closely spaced slits.^[5] In this setup, each slit acts as a coherent source, and the interference pattern arises from the superposition of waves with path differences determined by slit separation and distance to the screen.^[6] Similar patterns occur with other waves, such as sound from two speakers creating zones of loud and quiet regions, or water waves from multiple sources forming nodal lines of minimal disturbance.^[1] Wave interference has broad applications across physics and engineering, including anti-reflective coatings on optical lenses that minimize destructive interference to reduce glare, diffraction gratings used in spectroscopy to separate light wavelengths via constructive interference at specific angles, and noise-canceling technology in headphones that employs destructive interference to attenuate unwanted sound waves.^[7]^[8] In acoustics, interference explains standing waves in musical instruments, where fixed ends create nodes and antinodes at resonant frequencies.^[2] These principles extend to modern fields like quantum mechanics, where particle-wave duality leads to interference in electron double-slit experiments, underscoring the universal nature of wave behavior.^[9]

Fundamentals

Definition and Principles

Wave interference occurs when two or more waves overlap in space and time, resulting in a new wave pattern that is the sum of the individual waves.^[2] This phenomenon arises from the interaction of coherent waves, where their displacements at any point combine to produce variations in the overall amplitude.^[3] In constructive interference, waves align in phase, meaning their crests or troughs coincide, leading to an increased amplitude in the resultant wave as the individual amplitudes add together.^[4] Conversely, destructive interference happens when waves are out of phase, such that crests align with troughs, causing the amplitudes to subtract and potentially cancel each other out, reducing or eliminating the resultant wave at certain points.^[2] These effects depend on the relative phases of the waves, which are determined by factors such as path differences or time delays. Understanding wave interference assumes familiarity with basic wave properties: waves are periodic disturbances that propagate through a medium or space, characterized by amplitude (the maximum displacement from equilibrium), wavelength (the distance between successive crests), frequency (the number of cycles per unit time), and phase (the position within the cycle).^[10] These attributes enable the superposition of waves, the principle underlying interference, where the total disturbance is the algebraic sum of individual disturbances.^[3] The phenomenon was first systematically observed and described by Thomas Young in his 1801 Bakerian Lecture to the Royal Society, where he demonstrated interference patterns with light, providing key evidence for its wave nature. The term "interference" derives from the Latin roots inter- ("between") and ferīre ("to strike"), originally connoting mutual striking or clashing, and was applied to wave optics by Young in the early 19th century.^[11]^[12]

Superposition Principle

The superposition principle asserts that, for waves governed by linear wave equations, the net displacement of the medium at any point and time is the algebraic sum of the displacements produced by each individual wave acting alone.^[13] This principle holds because the wave equation itself is linear, meaning that if two functions satisfy the equation, their linear combination also satisfies it.^[14] The one-dimensional wave equation, \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, exemplifies this linearity, where u(x,t) represents the displacement, c is the wave speed, and solutions such as traveling waves superpose to form the general solution.^[15] The principle applies specifically to linear media, where wave amplitudes are sufficiently small to neglect nonlinear effects, such as those leading to shock waves or higher-order interactions; in nonlinear media, superposition breaks down, and waves do not simply add algebraically.^[16] To illustrate, consider two one-dimensional sinusoidal waves of equal amplitude A, wavenumber k, and angular frequency \omega, traveling in the same direction but with a phase difference \phi:
y_1(x,t) = A \sin(kx - \omega t),
y_2(x,t) = A \sin(kx - \omega t + \phi).
The total displacement is then y(x,t) = y_1 + y_2.^[17] Applying the trigonometric identity for the sum of sines yields
y(x,t) = 2A \cos\left(\frac{\phi}{2}\right) \sin\left(kx - \omega t + \frac{\phi}{2}\right),
revealing that the effective amplitude $2A \left|\cos(\phi/2)\right| depends on the phase difference \phi; when \phi = 2n\pi (for integer n), the waves reinforce maximally, while \phi = (2n+1)\pi leads to complete cancellation./14%3A_Waves/14.06%3A_Superposition_of_waves_and_interference) This phase-dependent outcome forms the basis for interference phenomena arising from superposition.^[18]

Mathematical Frameworks

Real-Valued Wave Functions

In wave interference, real-valued wave functions are commonly represented as sinusoidal displacements or fields, such as \psi(x, t) = A \cos(kx - \omega t + \phi), where A is the amplitude, k = 2\pi / \lambda is the wave number, \omega = 2\pi f is the angular frequency, and \phi is the phase offset.^[19] This form captures the oscillatory nature of classical waves like sound or light in one dimension, with the argument kx - \omega t + \phi determining the position and time dependence.^[20] For the interference of two such monochromatic waves with the same frequency, the superposition principle yields the resultant wave function \psi(x, t) = \psi_1(x, t) + \psi_2(x, t), where \psi_1(x, t) = A_1 \cos(kx - \omega t + \phi_1) and \psi_2(x, t) = A_2 \cos(kx - \omega t + \phi_2).^[21] The phase difference \delta = \phi_2 - \phi_1 governs the interference pattern, leading to a resultant amplitude R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos \delta}. Since intensity I is proportional to the square of the amplitude, the instantaneous intensity is I \propto R^2. For time-averaged intensity over many cycles, assuming monochromatic waves, the cross term averages to $2 \sqrt{I_1 I_2} \cos \delta, yielding I = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos \delta, where I_1 \propto A_1^2 and I_2 \propto A_2^2.^[21] This formula shows maxima when \cos \delta = 1 (constructive interference, I = (\sqrt{I_1} + \sqrt{I_2})^2) and minima when \cos \delta = -1 (destructive interference, I = (\sqrt{I_1} - \sqrt{I_2})^2). The phase difference \delta often arises from path length differences in propagation, expressed as \delta = 2\pi d \sin \theta / \lambda in setups like double-slit experiments, where d is the slit separation, \theta is the observation angle, and \lambda is the wavelength.^[22] Constructive interference occurs at \delta = 2\pi m (integer m), and destructive at \delta = (2m+1)\pi, resulting in intensity fringes spaced such that the path difference changes by \lambda/2 between adjacent maximum and minimum.^[4] Observable interference patterns require temporal and spatial coherence, meaning a stable phase relationship over the observation time and across the wave sources, with equal or nearly equal amplitudes A_1 \approx A_2 to produce high-contrast fringes.^[4] While real-valued functions effectively describe classical wave interference and intensity patterns, they become cumbersome for analyzing propagation and phase shifts in complex geometries, as adding multiple cosines with varying phases requires tedious trigonometric expansions.

Complex-Valued Wave Functions

In wave physics, monochromatic waves are often represented using complex-valued functions to facilitate mathematical analysis of interference phenomena. A general form for a plane wave propagating in the positive x-direction is given by the real part of a complex exponential: \psi(x, t) = \Re \left\{ A e^{i(kx - \omega t + \phi)} \right\}, where A is the amplitude, k is the wave number, \omega is the angular frequency, and \phi is the phase constant.^[23] This representation relies on Euler's formula, e^{i\theta} = \cos \theta + i \sin \theta, which decomposes the exponential into real and imaginary components, allowing the physical wave to be extracted as the real part.^[24] The complex form simplifies calculations because differentiation and integration of exponentials preserve the functional form, unlike trigonometric functions.^[25] For interference between multiple waves, the complex notation employs phasors, which are vectors in the complex plane representing the amplitude and phase of each wave. The total field at a point is the vector sum of individual phasors, E = E_1 + E_2 + \cdots + E_n, where each E_j = A_j e^{i\phi_j} is a complex number.^[26] The observable intensity is then proportional to the square of the magnitude of this resultant, I \propto |E|^2 = E \cdot E^*, where E^* denotes the complex conjugate.^[27] Expanding for two waves, |E_1 + E_2|^2 = |E_1|^2 + |E_2|^2 + 2 \Re \{ E_1^* E_2 \}, reveals the interference term $2 \Re \{ E_1^* E_2 \} = 2 A_1 A_2 \cos \delta, where \delta is the phase difference; this cross term captures constructive and destructive interference without explicitly computing time-dependent oscillations.^[28] The primary advantages of complex-valued wave functions lie in their handling of phase relationships and propagation effects. Phase shifts, such as those from path differences or material interactions, are represented multiplicatively as factors of e^{i\Delta\phi}, simplifying algebraic manipulations in interference and diffraction problems compared to real trigonometric forms.^[29] This notation also streamlines derivations for diffraction patterns, where Fourier transforms of complex apertures directly yield intensity distributions.^[30] In three-dimensional wave propagation, complex functions satisfy the Helmholtz equation, (\nabla^2 + k^2) \psi = 0, which is the time-independent form of the wave equation for harmonic fields; solutions like e^{i\mathbf{k} \cdot \mathbf{r}} describe plane waves, enabling efficient modeling of interference in inhomogeneous media.^[31] The equivalence between real and complex representations ensures physical consistency: the complex function is a mathematical tool, and only its real part corresponds to measurable fields like electric or pressure displacements.^[32] This approach transitions naturally from simpler real-valued descriptions by embedding them within the complex framework, where the real part extraction yields identical results for intensity and interference patterns.^[20]

Interference Mechanisms

Between Plane Waves

Plane waves represent a fundamental solution to the wave equation in homogeneous media, characterized by wavefronts that are infinite planes of constant phase perpendicular to the direction of propagation.^[33] Their electric field can be expressed mathematically as \mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}, where \mathbf{k} is the wave vector, \mathbf{r} is the position vector, \omega is the angular frequency, and the phase remains uniform across planes normal to \mathbf{k}.^[33] Interference between two coherent plane waves of equal frequency and amplitude propagating in different directions produces a pattern of straight fringes parallel to the line of intersection of the two wavefronts.^[34] The phase difference arises from the path difference \delta = \mathbf{k} \cdot \Delta \mathbf{r}, where \Delta \mathbf{r} is the displacement vector between points on the two wavefronts, leading to constructive interference where \delta = 2m\pi (for integer m) and destructive interference where \delta = (2m+1)\pi.^[35] For waves intersecting at an angle \theta, the resulting intensity pattern is I = 4I_0 \cos^2(\phi/2), with \phi = (2\pi/\lambda) d \sin(\theta/2), where d is the distance along the observation plane perpendicular to the bisector.^[35] The spacing between adjacent bright fringes, known as the fringe width \Delta x, depends on the wavelength \lambda and the angle \theta between the wave vectors, given by \Delta x = \lambda / (2 \sin(\theta/2)).^[35] This linear pattern arises because the path difference varies linearly across the plane perpendicular to both propagation directions, unlike curved patterns from diverging waves.^[34] Complex-valued representations facilitate analysis by allowing phasor addition to compute the resultant amplitude.^[36] Polarization significantly influences the visibility of interference fringes between plane waves, as full constructive and destructive interference requires the electric fields to oscillate in the same plane.^[37] If the polarizations are parallel, the interference contrast is maximum; however, for orthogonal polarizations, the fields do not add coherently, resulting in no observable fringes and uniform intensity.^[37] In cases of partial alignment at an angle \psi between polarization directions, the fringe visibility reduces by a factor of \cos \psi, with cross-polarization components contributing incoherently to the background intensity.^[37] When multiple plane waves interfere, more complex periodic patterns emerge, such as those underlying diffraction gratings, where equally spaced plane waves at discrete angles produce intensity maxima satisfying the grating equation d (\sin \theta_i + \sin \theta_d) = m \lambda.^[38] For instance, a set of plane waves with wave vectors differing by small angles can form a one-dimensional intensity grating with period determined by the angular spread.^[38] Superposition of two such sets with slightly mismatched periodicities generates moiré patterns, characterized by large-scale beats in intensity that amplify deviations in spacing or orientation between the wave arrays.^[39] These patterns are observable in optics when multiple coherent beams, such as from a laser array, overlap at low angles.^[39]

Between Spherical Waves

Spherical waves emanate radially from point sources, with their mathematical form in the far field approximated as \psi(r, t) \approx \frac{A}{r} e^{i(kr - \omega t)}, where the amplitude A/r decreases inversely with distance r to conserve energy as the wavefront expands over a growing spherical surface. This radial propagation distinguishes spherical waves from plane waves, incorporating both phase progression kr and temporal oscillation \omega t.^[40] Interference between two coherent spherical waves arises prominently in setups like Young's double-slit experiment, where the slits function as secondary point sources emitting identical spherical waves. The path length difference \delta to an observation point is given by \delta \approx d \sin [\theta](/page/Theta) under the far-field approximation, with d as the source separation and \theta the angle from the central axis; constructive interference produces intensity maxima when \delta = m\lambda, where m is an integer and \lambda the wavelength.^[41] The resulting fringe patterns consist of loci where the phase difference is constant, forming hyperbolic curves (or hyperboloids in three dimensions) with the two sources as foci; in the near field close to the sources, these fringes approximate circular arcs centered midway between them. The phase condition for these fringes is \frac{2\pi}{\lambda} (r_1 - r_2) + \phi_1 - \phi_2 = constant, where r_1 and r_2 are distances from each source and \phi_1, \phi_2 are initial phase shifts.^[34] The Huygens-Fresnel principle underpins this interference by positing that every point on an advancing wavefront acts as a source of secondary spherical wavelets, or sphericallets, whose superposition determines the form and propagation of the overall wave. These wavelets interfere constructively in the forward direction to propagate the wavefront while destructively canceling backward components. Due to the $1/r amplitude decay, unequal path lengths r_1 \neq r_2 result in disparate intensities I_1 \propto 1/r_1^2 and I_2 \propto 1/r_2^2 at the observation point, reducing fringe visibility defined as V = \frac{2 \sqrt{I_1 I_2}}{I_1 + I_2}; maximum visibility (V=1) occurs only where paths are equal, such as along the perpendicular bisector between equidistant sources.^[42]

With Multiple Waves

When multiple waves superpose, the principle of superposition extends the two-wave case to an arbitrary number N of coherent waves, resulting in a total electric field \mathbf{E} = \sum_{i=1}^N \mathbf{E}_i. The intensity at a point is then given by I = \left| \sum_{i=1}^N \mathbf{E}_i \right|^2 = \sum_{i=1}^N |\mathbf{E}_i|^2 + 2 \sum_{i<j} \mathrm{Re} (\mathbf{E}_i^* \cdot \mathbf{E}_j ), where the phase differences between waves i and j are included in the complex fields.^[43] This expression shows that the total intensity includes individual wave intensities plus pairwise interference terms, leading to complex patterns with higher-order maxima and minima as N increases.^[44] The nature of these patterns depends critically on phase relationships among the waves. For controlled phases, such as in multi-slit experiments, constructive interference can produce sharp principal maxima with subsidiary peaks between them; for instance, in a triple-slit setup, the interference pattern exhibits a central maximum flanked by higher-order maxima that are about one-ninth as intense as the principal ones due to the vector addition of three equal-amplitude phasors.^[44] In phasor diagrams, equal-amplitude waves are represented as vectors whose resultant length—and thus intensity—varies with relative phases: aligned phasors yield maximum intensity I = N^2 I_0, while evenly spaced phases around a circle result in zero net amplitude.^[45] Random phases, however, average out cross terms, yielding incoherent addition where I \approx \sum I_i, though partial coherence can still produce observable fringes limited by the coherence length.^[46] Examples illustrate these effects vividly. In white-light interference, the short coherence length (on the order of micrometers for broadband sources) restricts multi-beam fringes to regions of near-zero path difference, beyond which the pattern washes out into uniform illumination without distinct interference.^[47] Conversely, with fully coherent sources like lasers, controlled multi-wave setups enable precise pattern formation, but increasing N heightens sensitivity to misalignment. A key limitation arises from the growing complexity: in incoherent or partially coherent scenarios with many waves, random phase variations lead to granular intensity fluctuations known as speckle patterns, where bright and dark spots form due to localized constructive and destructive interference amid overall randomness.^[48] This contrasts with the ordered fringes of few-wave cases and complicates applications requiring uniform illumination.

Interference in Specific Domains

Optical Interference

Optical interference refers to the phenomenon where light waves, as electromagnetic waves in the visible, ultraviolet (UV), and infrared (IR) spectra, superimpose to produce patterns of constructive and destructive interference. This occurs when light from a coherent source interacts, leading to observable fringes or color variations that demonstrate wave nature. Visibility of these patterns demands both temporal coherence, related to the light's monochromaticity, and spatial coherence, ensuring phase consistency across the wavefront.^[49] Temporal coherence requires the light to be nearly monochromatic, with the spectral bandwidth satisfying \Delta \lambda \ll \lambda, where \Delta \lambda is the wavelength spread and \lambda is the central wavelength; this ensures a sufficiently long coherence length for path differences in the setup. Spatial coherence necessitates that the source size be small compared to the path differences between interfering beams, preventing phase randomization and allowing stable fringe visibility. For instance, in Young's double-slit experiment, which divides the wavefront, a single coherent source illuminates two closely spaced slits, producing interference fringes on a screen due to the phase difference from varying path lengths. In contrast, the Michelson interferometer divides the amplitude using a beam splitter, directing light along two perpendicular paths that recombine to form circular fringes, adjustable by mirror displacement.^[50]^[51]^[52] Thin-film interference arises from multiple reflections at the boundaries of a thin transparent layer, such as in soap bubbles, where light undergoes partial reflection and transmission. A key feature is the \pi phase shift (equivalent to a half-wavelength path difference) upon external reflection from a medium of higher refractive index, while internal reflections experience no such shift; this leads to destructive interference for certain thicknesses in reflection, producing iridescent colors as the film thins or varies. For example, in a soap bubble, the top surface reflection gains a \pi phase shift, while the bottom does not, resulting in color bands corresponding to constructive interference for specific wavelengths based on twice the film thickness. Polarization plays a crucial role in optical interference, particularly in birefringent materials like calcite crystals, which split incoming light into two orthogonally polarized rays: the ordinary ray (o-ray) and extraordinary ray (e-ray) with different refractive indices. This double refraction causes the e-ray to deviate, producing double images of objects viewed through the crystal, as the rays follow distinct paths and interfere differently upon recombination. In calcite, the negative birefringence means the e-ray travels faster than the o-ray, enhancing the separation and visibility of polarization-dependent interference effects.^[53]^[54]

Acoustic and Mechanical Interference

Acoustic waves are longitudinal pressure waves consisting of compressions and rarefactions that propagate through elastic media such as fluids and solids.^[55] In air at 20°C, these waves travel at a speed of approximately 343 m/s, determined by the medium's density and compressibility.^[56] In water, the speed is significantly higher, around 1480 m/s at 20°C, due to the denser medium. Interference occurs when multiple acoustic waves superpose, leading to patterns of reinforcement and cancellation. In enclosed structures like organ pipes or resonance tubes, reflected waves interfere with incident waves to form standing waves, characterized by nodes—points of minimal pressure variation—and antinodes—points of maximum pressure variation.^[57] For a pipe closed at one end, the fundamental standing wave mode features a displacement node at the closed end and an antinode at the open end, with the pipe length equal to a quarter wavelength.^[58] In open environments, acoustic interference is demonstrated by setups with two coherent sources, such as speakers emitting identical frequencies. Constructive interference produces louder zones where path differences are integer multiples of the wavelength, while destructive interference creates quieter zones at odd multiples of half-wavelengths, allowing spatial mapping of interference fringes.^[4] Mechanical waves encompass both longitudinal and transverse types in solids, with transverse waves on strings providing a classic example of interference. When waves reflect at the ends of a fixed string, they superpose to form standing waves, with fixed nodes at the boundaries and antinodes midway for the fundamental mode.^[59] Interference in coupled mechanical oscillators, such as connected pendulums or masses on springs, results in normal modes where synchronized motions amplify or cancel, leading to collective oscillations at discrete frequencies.^[60] Dispersion in acoustic propagation arises when wave speed varies with frequency, distorting broadband signals. In water, particularly in oceanic contexts, frequency-dependent attenuation causes higher frequencies to attenuate more rapidly than lower ones, while sound speed slightly increases with frequency due to dispersion, distorting broadband signals, spreading pulses, and altering interference patterns for complex sounds like marine noise.^[61] A practical application of acoustic destructive interference is in noise-cancelling headphones, which use microphones to detect ambient sound and generate inverted waves that superpose with the noise, reducing its amplitude by up to 20-30 dB in low-frequency ranges.^[62]

Electromagnetic and Radio Interference

Electromagnetic waves in the radio frequency range, spanning 3 kHz to 300 MHz, and the microwave range, from 300 MHz to 300 GHz, exhibit interference phenomena that influence signal propagation and reception.^[63] These longer wavelengths compared to optical frequencies allow interference patterns to develop over extended distances in free space, often leading to constructive or destructive superposition that enhances or degrades communication links. In radio systems, interference is both a challenge and a tool, as waves from multiple sources or reflectors can overlap, creating zones of amplification or nulls that affect broadcast coverage.^[64] The wavelength \lambda of radio waves is determined by the fundamental relation \lambda = c / f, where c is the speed of light in vacuum ($3 \times 10^8 m/s) and f is the frequency, resulting in wavelengths from kilometers at megahertz frequencies to centimeters at gigahertz bands.^[65] This wavelength scale governs interference in antenna systems, where phasing exploits wave superposition to steer signals. For instance, in beamforming applications, precise timing of wave emissions from antenna elements creates directed beams by aligning phases for constructive interference in targeted directions, improving efficiency in radar and wireless networks.^[66] Phased array antennas exemplify controlled interference, with the phase difference \delta between adjacent elements separated by distance d at an angle \theta from the array axis given by

\delta = \frac{2\pi}{\lambda} d \cos \theta.

^[67] This relation enables electronic steering without mechanical movement, producing main lobes of high-intensity radiation in desired directions while forming nulls or side lobes elsewhere through destructive interference. Such arrays are critical in modern radio applications like 5G base stations, where beamforming mitigates interference from multipath propagation by focusing energy and suppressing off-axis signals.^[68] Atmospheric effects, particularly ionospheric refraction, significantly alter path differences for long-range radio waves in the high-frequency (HF) band (3-30 MHz). The ionosphere's ionized layers refract these waves, bending them back toward Earth and enabling skywave propagation over thousands of kilometers, but variations in electron density introduce phase shifts that cause multipath interference and signal fading.^[69] Day-night cycles and solar activity further modulate these refractions, impacting path lengths and leading to constructive or destructive interference at receivers. Unwanted electromagnetic noise interference arises from coupling between radio signals and external sources, such as power lines or other transmitters, degrading receiver sensitivity through superimposed oscillations. Mitigation relies on shielding, where conductive enclosures reflect or absorb incident waves, confining fields and preventing ingress of radio-frequency interference (RFI). Techniques like Faraday cages or metallic coatings achieve attenuation levels exceeding 40 dB in the MHz range, ensuring reliable operation in dense electromagnetic environments.^[70]

Quantum Aspects

Quantum Wave Interference

In quantum mechanics, the wave nature of particles was first proposed by Louis de Broglie in 1924, who hypothesized that every particle of momentum p possesses an associated wave with wavelength \lambda = h / p, where h is Planck's constant. This de Broglie relation laid the foundation for describing particles via wave functions, bridging classical wave interference concepts to quantum phenomena, albeit with a probabilistic interpretation rather than classical field intensities. The time-dependent Schrödinger equation governs the evolution of the quantum wave function \psi(\mathbf{r}, t), given by i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, where \hat{H} is the Hamiltonian operator and \hbar = h / 2\pi.^[71] Unlike classical waves, the observable probability density for finding a particle is |\psi|^2, as established by Max Born's interpretation in 1926, which quantifies the likelihood of measurement outcomes rather than energy flux. Interference arises from the superposition principle: for a single particle, the total wave function \psi = \psi_1 + \psi_2 yields a probability density |\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2 \operatorname{Re}(\psi_1^* \psi_2), where the cross term introduces constructive or destructive interference, fundamentally differing from incoherent classical addition. Quantum coherence, essential for observable interference patterns, refers to the phase stability of the wave function superposition; however, interactions with the environment lead to decoherence, rapidly suppressing the interference term through entanglement with environmental degrees of freedom, as formalized in Wojciech Zurek's framework. In the path integral formulation developed by Richard Feynman in 1948, quantum interference emerges from summing complex amplitudes over all possible paths between initial and final states, with phases determined by the action S, yielding \psi \propto \sum_{\text{paths}} e^{i S / \hbar}; constructive interference occurs for paths near the classical trajectory, while others cancel out. This approach underscores how quantum wave interference encodes the probabilistic nature of particle trajectories without classical analogs.

Matter Wave Interference

Matter wave interference refers to the phenomenon where de Broglie waves associated with massive particles exhibit constructive and destructive interference patterns, providing empirical evidence for wave-particle duality. This contrasts with classical particle behavior and has been demonstrated through diffraction and interferometry experiments with electrons, neutrons, atoms, and larger molecules. The underlying quantum wave functions of these particles enable such interference, as predicted by de Broglie's hypothesis.^[72] The first direct observation of matter wave interference came from electron diffraction experiments conducted by Clinton Davisson and Lester Germer in 1927. They directed a beam of electrons onto a nickel crystal target and observed intensity maxima in the scattered electron distribution, corresponding to diffraction peaks that matched the predictions of the Bragg condition using the de Broglie wavelength \lambda = h / p, where h is Planck's constant and p is the electron momentum. This verified the wave nature of electrons for particles with wavelengths on the order of 0.165 nm at 54 eV energy, confirming de Broglie's relation experimentally.^[72] Neutron interferometry provided further confirmation with neutral massive particles. In 1975, Robert Colella, Albert Overhauser, and Samuel Werner utilized a silicon crystal interferometer to split and recombine a beam of thermal neutrons (wavelength ~0.18 nm), observing a phase shift in the interference pattern due to the neutrons' interaction with Earth's gravitational field. The measured phase shift \Delta\phi = \frac{m^2 g A}{\hbar^2 k} agreed with general relativity and quantum mechanics, where m is neutron mass, g is gravitational acceleration, A is the interferometer area, \hbar is reduced Planck's constant, and k is the wave number, demonstrating gravity's effect on matter waves.^[73] Atom interferometry extended these observations to bosonic atoms using Bose-Einstein condensates (BECs), which enhance coherence through quantum degeneracy. A seminal demonstration in 2004 involved splitting a rubidium-87 BEC in an optical double-well potential to create a trapped-atom Mach-Zehnder interferometer, revealing interference fringes with visibilities up to 80% after recombination. This macroscopic quantum interference, involving ~10^4 atoms at temperatures near 100 nK, highlighted collective wave behavior in dilute gases.^[74] A key challenge in matter wave interference experiments is maintaining long coherence lengths, limited by thermal motion that introduces velocity spreads and dephasing. For room-temperature beams, thermal velocities (~300 m/s for atoms) reduce the de Broglie coherence length to micrometers, necessitating laser cooling to microkelvin temperatures to extend it to millimeters or more for observable fringes. Blackbody radiation and environmental scattering further degrade coherence, particularly for larger particles, requiring ultra-high vacuum and cryogenic conditions.^[75]^[76] Recent advances have pushed matter wave interference toward testing quantum limits with increasingly massive objects. In 1999, Markus Arndt and colleagues observed de Broglie interference of C₆₀ fullerene molecules (mass ~720 u) via diffraction from a silicon nitride grating, achieving visibilities of 0.9% with molecular velocities ~220 m/s and wavelengths ~2.5 pm. This experiment with 60-atom clusters verified wave-particle duality for objects approaching nanoscale sizes, probing decoherence mechanisms and the boundary between quantum and classical regimes. More recent experiments have extended this to larger objects; for instance, in 2019, quantum superposition and interference were observed for complex molecules comprising up to 2,000 atoms and masses exceeding 25,000 u.^[77]^[78]

Applications

Beats and Frequency Modulation

Beats occur when two coherent waves of nearly identical frequencies interfere, resulting in a periodic variation in the amplitude of the combined wave while the average frequency remains the average of the individual frequencies. This temporal interference produces an audible or detectable pulsing effect known as the beat frequency, which equals the absolute difference between the two wave frequencies, |f₁ - f₂|. The phenomenon arises from the superposition principle, where the waves add constructively and destructively over time due to their phase difference evolving at the beat rate.^[79] The mathematical description of beats can be derived using trigonometric identities. Consider two cosine waves with angular frequencies ω₁ and ω₂ (where ω = 2πf):

\cos(\omega_1 t) + \cos(\omega_2 t) = 2 \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \cos\left(\frac{\omega_1 - \omega_2}{2} t\right)

Here, the first cosine term represents a high-frequency carrier wave at the average angular frequency (ω₁ + ω₂)/2, while the second term modulates its amplitude at the low-frequency beat angular frequency (ω₁ - ω₂)/2. This envelope modulation visually and audibly manifests as beats when |ω₁ - ω₂| is small compared to the carrier.^[79] In musical applications, beats serve as a practical tool for tuning instruments by ear. Musicians compare a reference tone, such as from a tuning fork, with the instrument's note; the beat frequency indicates the frequency mismatch, and adjustments continue until the beats disappear, achieving perfect consonance at zero beat frequency. For example, piano tuners listen for beats between a struck tuning fork and piano strings to fine-tune octaves and intervals./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/17%3A_Sound/17.07%3A_Beats) In radar systems, Doppler-induced beats enable velocity measurement. Continuous-wave Doppler radar transmits a signal that reflects off a moving target, producing a frequency shift due to the Doppler effect; mixing the returned signal with the transmitted one generates a beat frequency proportional to the target's radial speed, typically f_d = 2v f_0 / c, where v is velocity, f_0 is the transmitted frequency, and c is the speed of light. This beat frequency is detected and processed to determine speed in applications like police radar.^[80] Frequency modulation (FM) extends the beat concept to signal transmission, where a carrier wave's frequency is varied by a modulating signal, effectively creating multiple beat-like interactions that produce sidebands around the carrier. The spectrum consists of the carrier at frequency f_c plus pairs of sidebands at f_c ± n f_m (n = 1, 2, ...), where f_m is the modulating frequency; these arise from the interference of phase-shifted wave components in the modulated waveform, as described by Bessel function expansions. This sideband structure enhances noise resistance compared to amplitude modulation while maintaining bandwidth efficiency for broadcasting.^[79] In nonlinear media, intense beats can drive harmonic generation through wave-wave interactions. The large amplitude variations of the beat envelope induce higher-order nonlinearities, producing harmonics at integer multiples of the carrier and beat frequencies; for instance, in plasma waves, dispersive beats lead to spatial modulation that generates ion-acoustic harmonics, converting a portion of the input energy into these higher frequencies. Such effects are observable in high-intensity acoustic or optical setups where the beat intensity exceeds linear thresholds.^[81]

Interferometry Techniques

Interferometry techniques exploit the interference of waves to achieve precise measurements of length, displacement, and surface characteristics by detecting minute phase differences between wave paths. The fundamental principle relies on the phase difference \delta introduced by a path length variation \Delta L, given by \delta = \frac{2\pi}{\lambda} \Delta L, where \lambda is the wavelength of the wave.^[82] This allows for resolutions as fine as \lambda/1000 or better in stabilized setups, enabling sub-wavelength accuracy in displacement detection.^[83] The Michelson interferometer, a cornerstone of these techniques, consists of a beam splitter that divides an incoming light beam into two paths, each reflecting off a mirror before recombining at the splitter to form interference fringes.^[84] Changes in the path length, such as mirror displacement, shift these fringes, with each fringe corresponding to a path difference of \lambda/2, allowing precise quantification of \Delta L from the number of fringes observed.^[85] This configuration has been instrumental in applications requiring stable, high-precision length measurements, such as calibrating standards of length. In contrast, the Mach-Zehnder interferometer employs two beam splitters to separate and recombine the beam, creating an open path suitable for inserting samples or dynamic elements without the folded geometry of the Michelson design.^[86] The first splitter divides the beam, the paths propagate separately (often through different media), and the second splitter interferes them, producing fringes sensitive to phase shifts from refractive index changes or vibrations.^[87] This setup excels in real-time, dynamic measurements, such as analyzing fluid flows or transient deformations, where the ability to monitor evolving interference patterns provides temporal resolution. A prominent application of laser-based interferometry is in gravitational wave detection, as demonstrated by the Laser Interferometer Gravitational-Wave Observatory (LIGO). LIGO's Michelson-inspired interferometers, with arm lengths of 4 km, use laser light at 1064 nm to detect spacetime distortions as small as $10^{-18} m, corresponding to phase shifts from passing gravitational waves.^[84] The first direct detection occurred on September 14, 2015, from the merger of two black holes 1.3 billion light-years away, confirming predictions from general relativity.^[88] Holographic interferometry extends these principles by recording the interference pattern between an object beam and a reference beam on a photosensitive medium, such as a hologram plate, to capture three-dimensional wavefront information.^[89] Upon reconstruction, a second exposure after object deformation reveals fringe patterns that map surface displacements in all directions, with sensitivity to changes as small as \lambda/10.^[89] This technique, pioneered in the 1960s, is widely used for non-destructive 3D deformation analysis in engineering, such as stress testing of materials under load.^[89]

Diffraction and Pattern Formation

Diffraction phenomena arise fundamentally from the interference of waves emanating from different points on an aperture or obstacle, as described by Huygens' principle, which posits that every point on a wavefront acts as a source of secondary spherical wavelets that propagate forward and interfere with one another.^[90]^[91] This principle explains how waves bend around edges and spread out, producing characteristic patterns rather than geometric shadows. In the context of a slit, each infinitesimal segment serves as a coherent source, leading to constructive and destructive interference that shapes the observed intensity distribution on a screen.^[92] For single-slit diffraction, the pattern consists of a central bright maximum flanked by alternating minima and secondary maxima, forming an envelope that modulates the overall intensity. The intensity distribution is proportional to the square of the sinc function, I(\theta) \propto \left( \frac{\sin \beta}{\beta} \right)^2, where \beta = \frac{\pi a \sin \theta}{\lambda}, with a as the slit width, \lambda the wavelength, and \theta the angle from the center. Minima occur at angles satisfying a \sin \theta = m \lambda for integer m = \pm 1, \pm 2, \dots, where path differences from opposite slit edges lead to destructive interference across the entire aperture.^[93]^[94] This envelope broadens as the slit width decreases relative to the wavelength, highlighting diffraction's role in limiting resolution.^[95] In the double-slit configuration, the interference pattern features equally spaced fringes from the phase difference between waves from the two slits, but this is modulated by the single-slit diffraction envelope due to the finite width of each slit. The overall intensity is the product of the double-slit interference term, I \propto \cos^2 \left( \frac{\pi d \sin \theta}{\lambda} \right), where d is the slit separation, and the single-slit envelope. Fringes are brightest near the center and fade outward within the envelope's boundaries, demonstrating how diffraction imposes an angular spread on the interference.^[96]^[97] This combination produces a pattern where multiple interference maxima fit within the central diffraction lobe, with the number of visible fringes scaling inversely with slit width.^[98] Diffraction gratings, consisting of many closely spaced slits, enhance interference effects to produce sharp principal maxima at angles given by the grating equation, d \sin \theta = m \lambda, where d is the spacing between slits and m = 0, \pm 1, \pm 2, \dots denotes the order. Constructive interference occurs when the path difference between adjacent slits is an integer multiple of the wavelength, resulting in narrow peaks separated by broad minima.^[99]^[100] Higher orders appear at larger angles, with the pattern's resolution improving with the number of slits, as the envelope from individual slit diffraction becomes secondary to the collective interference.^[101] Distinctions between diffraction regimes depend on the observation distance relative to the aperture size and wavelength. Fraunhofer diffraction applies in the far field, where the screen is sufficiently distant (typically z \gg a^2 / \lambda) that incoming and outgoing waves approximate plane waves, simplifying calculations via Fourier transforms of the aperture function.^[102] In contrast, Fresnel diffraction governs the near field, involving curved (spherical) wavefronts and more complex quadratic phase factors in the propagation integral.^[103]^[104] These approximations capture pattern evolution from near the aperture, where curvature effects dominate, to the far field, where angular distributions stabilize.

References

[1]
16.5: Interference of Waves - Maricopa Open Digital Press
To analyze the interference of two or more waves, we use the principle of superposition. For mechanical waves, the principle of superposition states that if ...
[2]
Interference of Waves - Physics
Mar 3, 1999 · Interference occurs when waves meet, either adding together (constructive) or canceling each other out (destructive), based on their peaks and ...Missing: definition | Show results with:definition
[3]
16.5 Interference of Waves – University Physics Volume 1
To analyze the interference of two or more waves, we use the principle of superposition. For mechanical waves, the principle of superposition states that if ...
[4]
Constructive and Destructive Interference - UConn Physics
Constructive interference occurs when waves add to form a larger wave, while destructive interference occurs when waves cancel each other out.
[5]
18.3 Two-Slit Interference - Front Matter
The Double-Slit Experiment was originally conducted by Thomas Young in 1801 to show that light does indeed behave like a wave. Just like water waves ...
[6]
27.3 Young's Double Slit Experiment – College Physics
(b) Pure destructive interference occurs when identical waves are exactly out of phase, or shifted by half a wavelength. When light passes through narrow slits, ...
[7]
[PDF] Wave Interference and Diffraction Part 1: Introduction, Double Slt
→Effects and applications. ◇Double slit. ◇Single slit. ◇Diffraction gratings. ◇Anti-reflective coatings on lenses. ◇Highly reflective coatings for mirrors.
[8]
Young's Double-Slit Experiment - Richard Fitzpatrick
Young reasoned that if light were actually a wave phenomenon, as he suspected, then a similar interference effect should occur for light. This line of reasoning ...
[9]
CHEM 101 - Electromagnetic radiation and waves
May 10, 2024 · All propagating (traveling) waves have four basic characteristics: speed, frequency, wavelength, and amplitude.
[10]
Interference - Etymology, Origin & Meaning
Interfere originates from 1783, meaning to intermeddle. Its meaning evolved in physics (1802), telephony (1887), football (1894), and chess (1913).
[11]
The origins of the concept of interference - Journals
The concept of interference is implicit in Newton's explanation of the anomaly of the tides in the Gulf of Tongkin, but Thomas Young was the first to ...
[12]
Superposition of Waves - Graduate Program in Acoustics
The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time.Missing: linear | Show results with:linear
[13]
[PDF] Chapter 16 Waves I
The principle of superposition y y cy cy c c. + is a direct consequence of the linearity of the wave equation. This principle can be expressed as follows: 1.
[14]
[PDF] d'Alembert's solution of one-dimensional wave equation - Purdue Math
Jan 26, 2021 · To simplify our computation, we can use the Superposition Principle: first find a solution with arbitrary given φ and ψ = 0, then find a ...<|control11|><|separator|>
[15]
Superposition of Waves
The principle of superposition: displacement of the medium at any point in space or time is the sum of individual displacements;; only true if medium is ...
[16]
Waves, Superposition, Dispersion - UMD Physics
May 7, 2024 · This is important to understand: the frequency (and period) are determined by the source of the wave, and the wavelength comes from properties ...
[17]
16.10 Superposition and Interference - College Physics 2e | OpenStax
Jul 13, 2022 · The superposition of most waves produces a combination of constructive and destructive interference and can vary from place to place and time ...
[18]
Wave Functions – University Physics Volume 3 - UCF Pressbooks
The wave function of a light wave is given by E(x,t), and its energy density is given by |E{|}^{2}, where E is the electric field strength.
[19]
[PDF] Wave Optics 1 - Duke Physics
Wave optics covers interference and diffraction of waves, where waves combine and interfere, and diffraction occurs when a wavefront is obstructed.
[20]
[PDF] SIMG-455 Physical Optics SPSP-455 Optical Physics
Mar 10, 2008 · E1 = ^e1E1 cos [k1 r !1t]. E2 = ^e2E2 cos [k2 r !2t]. I = I1 + I2 + 2h^e1E1 cos [k1 r !1t] ^e2E2 cos [k2 r !2t]i. = I1 + I2 + 2E1E2(^e1. ^e2) ...
[21]
[PDF] Chapter 13 - Interference and Diffraction - MIT OpenCourseWare
The advantage of our present approach is that it is a real derivation. ... We will then use (13.19) to calculate the intensity of the wave at various values of z, ...
[22]
2.3: Representation of Waves via Complex Functions
Mar 31, 2025 · Complex numbers are often used to represent wavefunctions. All such representations depend ultimately on a fundamental mathematical identity, known as Euler's ...
[23]
[PDF] Complex Numbers: Euler's ID, Waves, and Phasor Math
Feb 15, 2021 · Euler's ID is incredibly useful. We'll often use it in this form: rei(ωt+φ) = rejφeiωt. (4) where r is the magnitude, eiφ expresses the ...
[24]
[PDF] Notes on the Phasor Diagram Method
Eei(ωt+φ) = E[cos(ωt +φ)+isin(ωt +φ)] is the second vector. The resultant vector is the sum of these two vectors. Q. What does the additon of these vectors, ...
[25]
Complex Numbers & Phasors in Polar and Rectangular Form
Phasor notation is the process of constructing a single complex number that has the amplitude and the phase angle of the given sinusoidal waveform. Then phasor ...
[26]
Waves, Interference, and Diffraction - Stony Brook University
Thomas Young's two slit interference experiment(shown below) involves a plane wave incident on a wall with 2 small slits a distance d apart.
[27]
[PDF] Lecture 18: Antennas and interference
We will try to write this complex representation of the electric field in a form where E0 is real, ... (E1 − E2)2. 2 sin2 ∆φ. 2. (3) where ∆φ = φ1 − φ2. The ...
[28]
Advanced diffraction: waves, interference and complex numbers
Apr 7, 2009 · Advanced diffraction involves understanding waves, their interaction with matter, and how waves add up, using complex numbers to represent ...
[29]
[PDF] Waves II: Interference and Diffraction - Kevin Zhou
where we've switch to complex notation, and the quantities r1 and r2 depend on y. In principle the two terms don't need to have the same magnitude, e.g. if ...
[30]
[PDF] Chapter 2 The Wave Equation
We have replaced the complex amplitude E by ˆE since we're now dealing with an amplitude per unit frequency, i.e. ˆE = lim∆ω→0[E/∆ω]. We have also included ω in.
[31]
[PDF] 3.0 Wave Equation
This yields the basic equation of two-beam interference. Ao. 2 = A1. 2 + A2. 2 + 2 A1 A2 Cos@φ2 − φ1D. If A1 = A2. Ao. 2 = 2 A1. 2 H1 + Cos@φ2 − φ1DL = 4 A1. 2 ...
[32]
[PDF] ME-5304. Wave Optics - WPI
The two canonical solutions of the Helmholtz equation in a homogenous medium are: (1) the plane wave, and (2) the spherical wave. (1) The plane wave. The plane ...
[33]
[PDF] Basic Interference and Classes of Interferometers
• Basic Interference. –Two plane waves. –Two spherical waves. –Plane wave and spherical wave. • Classes of Interferometers. –Division of wavefront. –Division of ...
[34]
Interference of Two Plane Waves Propagating in Different Directions
Jun 11, 2018 · and we express the fringe width: Δx=πksinα. We get the final formula after substituting 2πλ for the mangitude of the wave vector k: Δx=λ2sinα.
[35]
[PDF] Chapter 11 Waves and Imaging
We first consider “interference” of two traveling waves that oscillate with the same frequency and then generalize that to the interference of many such waves, ...
[36]
[PDF] Lecture 14: Polarization
Exponentials just make a lot of the algebra easier. 1. Page 2. 2 Linear polarization. We say a plane wave is linearly polarized if there is no phase difference ...
[37]
[PDF] Diffraction Grating: Wavelengths to Directions & Spectrum Analyzer
Pigure 2.5-4 The interference of two plane waves at an angle results in a sinusoidal intensity pattern of period λ/sin 0. wave and recording the resultant ...
[38]
[PDF] Moiré Method - University of Washington
Moiré effect is the mechanical interference of light by superimposed network of lines. The pattern of broad dark lines that is observed is called a moiré.
[39]
[PDF] SPHERICAL WAVES - UT Physics
Waves like these are called divergent spherical waves because their wave-fronts are spheres spreading out from the center as r = vphaset + const. In these notes ...
[40]
Multiple-Slit Interference – University Physics Volume 3
Interference fringe patterns for two, three and four slits. As the number of slits increases, more secondary maxima appear, but the principal maxima become ...Missing: triple | Show results with:triple
[41]
Coherence
Incidentally, two waves are said to be coherent if their phase difference is constant in time, and incoherent if their phase difference varies significantly in ...Missing: formula | Show results with:formula
[42]
Coherence Scanning Interferometry | Optical Profilometer Training
Coherence between two waves refers to the ability of the waves to interfere with each other in an observable way. White light has a very short coherence length ...
[43]
[PDF] Customizing speckle intensity statistics - Yale University
May 9, 2018 · Speckle formation is a phenomenon inherent to both classical and quantum waves. Characterized by a random granular structure, a speckle pattern ...
[44]
Conditions for Interference - Physics
The sources of the waves must be coherent, which means they emit identical waves with a constant phase difference. · The waves should be monochromatic - they ...
[45]
[PDF] Coherence and Source Requirements
Coherence Length. Distance light travels during coherence time. Coherence length lc = cτc = c. ∆ν=λ. 2. ∆λ. Page 3. Interference and Interferometry - James C.
[46]
Science, Optics and You - Thomas Young's Double Slit Experiment
Jan 11, 2017 · This interactive tutorial explores how coherent light waves interact when passed through two closely spaced slits.
[47]
[PDF] The Michelson Interferometer - UMD Physics
The Michelson interferometer causes interference by splitting a beam of light into two parts. Each part is made to travel a different path and brought back ...
[48]
Principles of Birefringence | Nikon's MicroscopyU
Because calcite is a negatively birefringent crystal, the ordinary wave is the slow wave and the extraordinary wave is the fast wave. Birefringent Crystals in a ...
[49]
10. Birefringence | UCLA Physics & Astronomy - Lab Manuals
Calcite crystals are available to show the double image of birefringence. One is mounted on a slide projector to produce two circular images from a single hole.
[50]
[PDF] Acoustics: the study of sound waves
Sound waves are longitudinal waves of compression and rarefaction in which the air molecules move back and forth parallel to the direction of wave travel ...
[51]
Sound Waves - Physics
Sound waves are longitudinal waves. In air, or any other medium, sound waves ... Speed of Sound (m/s). Air (0 degrees C), 331. Air (20 degrees C), 343.Missing: acoustic | Show results with:acoustic
[52]
Standing Sound Waves (Longitudinal Standing Waves)
May 17, 2012 · Sound waves are longitudinal waves and the particle motion associated with a standing sound wave in a pipe is directed along the length of the pipe.
[53]
17.5 Sound Interference and Resonance: Standing Waves in Air ...
Sound interference in air columns creates standing waves, with the lowest frequency being the fundamental. In a closed tube, the node is at the closed end and ...
[54]
Standing Waves - HyperPhysics
Standing waves are modes of vibration in extended objects, formed by reflection and interference, where the medium appears to vibrate in segments.
[55]
[PDF] Chapter 9 - Normal Modes and Waves
Dec 6, 2000 · We discuss the physics of wave phenomena and the motivation and use of Fourier transforms. 9.1 Coupled Oscillators and Normal Modes. Terms such ...
[56]
The effects of frequency-dependent attenuation and dispersion on ...
A sound speed difference between the object and water shifts all markers backward or forward. Frequency-dependent attenuation and dispersion may alter the ...
[57]
[PDF] Noise-Cancelling Headphones
Noise-canceling headphones use destructive interference by capturing, inverting, and outputting the captured noise to cancel out environmental noise.
[58]
Radiofrequency and Microwave Radiation - Overview | Occupational Safety and Health Administration
### Summary of RF and Microwave Frequencies and Interference
[59]
What is Electromagnetic Interference (EMI)? - RF Page
Apr 25, 2025 · Radio Frequency Interference (RFI) is the most common type of EMI as the electromagnetic radiation emitted from radio waves, microwaves, or ...Types of Electromagnetic... · Sources of Electromagnetic...
[60]
The Ionosphere - Radio Jove
The frequency and wavelength of an electromagnetic wave are related by the equation: c = λ x f where f is the frequency of the wave, c is the speed of light ...
[61]
What is a Phased Array Antenna? - Ansys
Jul 8, 2025 · In a process called beamforming, phased array systems send a signal at the same frequency from each antenna element, but the phase and magnitude ...
[62]
Direction of arrival estimation performance comparison of dual ...
Jan 1, 2014 · is an N × 1 spatial steering vector. Γ(φt) = 2π(d/λ)cos(θ)cos(ϕt) is the inter-element phase shift, which is a function of the target DOA ...1 Introduction · 2 Signal Model · 3.2 Ab-Stap Scheme
[63]
Phased Array Antenna Patterns—Part 1 - Analog Devices
This allows us to compute L, the delta distance of wave propagation, as L = dsin(θ). The time delay to steer our beam is equal to the time it will take for the ...
[64]
[PDF] Ionospheric radio propagation - NIST Technical Series Publications
... Radio Emissions. 43. References. 44. Chapter 2. Theory of Wave Propagation. 2.1,. Purpose. 45. 2.2. Electromagnetic Waves. 45. 2.2.1. Electrostatics and ...
[65]
https://radiojove.gsfc.nasa.gov/education/activities/iono.html
[66]
An Undulatory Theory of the Mechanics of Atoms and Molecules
The paper gives an account of the author's work on a new form of quantum theory. §1. The Hamiltonian analogy between mechanics and optics.
[67]
Diffraction of Electrons by a Crystal of Nickel | Phys. Rev.
Feb 3, 2025 · Quantum Milestones, 1927: Electrons Act Like Waves. Published 3 February, 2025. Davisson and Germer showed that electrons scatter from a ...
[68]
Observation of Gravitationally Induced Quantum Interference
Jun 9, 1975 · We have used a neutron interferometer to observe the quantum-mechanical phase shift of neutrons caused by their interaction with Earth's gravitational field.
[69]
Atom Interferometry with Bose-Einstein Condensates in a Double ...
Feb 6, 2004 · A trapped-atom interferometer was demonstrated using gaseous Bose-Einstein condensates coherently split by deforming an optical single-well potential into a ...Missing: seminal | Show results with:seminal
[70]
Thermal limitation of far-field matter-wave interference | Phys. Rev. A
May 8, 2006 · The present paper focuses on one of the basic effects known to destroy the coherence of matter waves, namely, the heat radiation emitted by the ...Article Text · Iii. Decoherence In... · Iv. Thermal Scaling BehaviorMissing: challenges | Show results with:challenges
[71]
[PDF] Experimental challenges for high-mass matter-wave interference ...
Jan 26, 2023 · We discuss recent advances towards matter-wave interference experiments with free beams of metallic and di- electric nanoparticles. They require ...
[72]
Wave–particle duality of C60 molecules - Nature
Oct 14, 1999 · Here we report the observation of de Broglie wave interference of C 60 molecules by diffraction at a material absorption grating.
[73]
The Feynman Lectures on Physics Vol. I Ch. 48: Beats
We know that the sound wave solution in one dimension is ei(ωt−kx), with ω=kcs, but we also know that in three dimensions a wave would be represented by ei(ωt−k ...
[74]
Police RADAR - HyperPhysics
RADAR speed detectors bounce microwave radiation off of moving vehicles and detect the reflected waves. These waves are shifted in frequency by the Doppler ...
[75]
Nonlinear harmonic generation of ion‐acoustic waves with dispersion
Apr 1, 1976 · The observation of a dispersive effect on the nonlinear generation of ion‐acoustic wave harmonics is reported. This effect is a spatial beat ...
[76]
[PDF] Two-Beam Interference Equation Interferometric optical testing is ...
• φ is the phase of the wave in radians: 0 ≤ φ ≤ 2π. • φ1 − φ2 = φ is the phase difference between the test and reference beams. Page 2. 2. Interferometric ...
[77]
The Michelson Interferometer – University Physics Volume 3
A precision instrument that produces interference fringes by splitting a light beam into two parts and then recombining them after they have traveled different ...<|separator|>
[78]
What is an Interferometer? | LIGO Lab | Caltech
The configuration of a basic Michelson interferometer (seen at right) consists of a light source, a beam-splitter, two mirrors, and a photodetector (the black ...
[79]
[PDF] The Michelson Interferometer - CSUB
Michelson's interferometer has become widely used for measuring the wavelength of light, for measuring extremely small distances, and for investigating optical ...
[80]
[PDF] How does a Mach–Zehnder interferometer work? - cs.Princeton
The Mach–Zehnder interferometer is a particularly simple device for demonstrating interference by division of amplitude. A light beam is first split into.
[81]
Mach-Zehnder Interferometer - an overview | ScienceDirect Topics
The Mach-Zehnder interferometer is widely used for studies of fluid flow, heat transfer, and the temperature distribution in plasmas.
[82]
Observation of Gravitational Waves from a Binary Black Hole Merger
Feb 11, 2016 · This is the first direct detection of gravitational waves and the first observation of a binary black hole merger. Comments: 16 pages, 4 figures.
[83]
The discovery of holographic interferometry, its development and ...
Jun 16, 2022 · This paper recounts the discovery of holographic interferometry, discusses its development, and itemizes some of its major applications.
[84]
6.2: Diffraction - Physics LibreTexts
Jan 14, 2019 · The Huygens-Fresnel principle states that every point on a wavefront is a source of wavelets. These wavelets spread out in the forward direction ...Huygens's Principle · Young's Double Slit Experiment · Single Slit Diffraction
[85]
17.1 Understanding Diffraction and Interference - Physics | OpenStax
Mar 26, 2020 · Huygens's principle states, “Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as ...
[86]
27.2 Huygens's Principle: Diffraction – College Physics
The bending of a wave around the edges of an opening or an obstacle is called diffraction. Diffraction is a wave characteristic and occurs for all types of ...
[87]
4.2: Single-Slit Diffraction - Physics LibreTexts
Mar 26, 2025 · Diffraction can send a wave around the edges of an opening or other obstacle. A single slit produces an interference pattern characterized ...
[88]
Single Slit Diffraction | Physics - Lumen Learning
There is destructive interference for a single slit when D sin θ = mλ, (form = 1,–1,2,–2,3, . . .), where D is the slit width, λ is the light's wavelength, θ ...
[89]
Diffraction by a single slit
That combination of sin(x)/x is called the sinc function. Recall that the variable β in this equation contains sin(θ). it tells us how the intensity changes as ...
[90]
4.4: Double-Slit Diffraction - Physics LibreTexts
Mar 16, 2025 · It is a product of the interference pattern of waves from separate slits and the diffraction of waves from within one slit. Figure shows a graph ...Missing: modulated | Show results with:modulated
[91]
[PDF] 7 Interference and Diffraction - Xie Chen
Now if we take into consideration the fact that the slits have finite width, the interference pattern shown above is going to be modulated by the diffraction ...Missing: fringes | Show results with:fringes
[92]
[PDF] Chapter 14 Interference and Diffraction - MIT
Interference is the combination of two or more waves to form a composite wave, based on such principle. The idea of the superposition principle is illustrated ...<|control11|><|separator|>
[93]
Diffraction and Interference
Huygens' principle tells us that each part of the slit can be thought of as an emitter of waves. All these waves interfere to produce the diffraction pattern.
[94]
[PDF] A Brief Introduction to Diffraction Gratings
Fully-constructive interference fringes thus occur at d sin θ = mλ, as before. • There are now more minima (dark fringes), because there are more ways to get ...
[95]
27.4 Multiple Slit Diffraction – College Physics chapters 1-17
There is constructive interference for a diffraction grating when dsinθ=mλ(form=0,1,–1,2,–2,…)dsinθ=mλ(form=0,1,–1,2,–2,…) size 12{d”sin”θ=mλ,`m=”0,”`”1 ...
[96]
6.6: Fresnel and Fraunhofer Approximations - Physics LibreTexts
Sep 16, 2022 · The Fresnel and Fraunhofer approximation are two approximations of the Rayleigh-Sommerfeld integral (6.13). The approximations are based on the assumption that ...Fresnel Approximation · Fraunhofer Approximation · Examples of Fresnel and...
[97]
[PDF] Lecture 38 – Diffraction - Purdue Physics
Fresnel and Fraunhofer Diffraction. • Fraunhofer diffraction. – Far field: ≫ /. – is the smaller of the distance to the source or to the screen. • Fresnel ...
[98]
[PDF] PhysLab - Diffraction
The Fraunhofer approximation is the limiting case of the Fresnel approximation when the field is observed at a distance far after the aperture (called the far ...<|control11|><|separator|>