Dispersion relation
In wave physics, a dispersion relation is a mathematical equation that relates the angular frequency \omega of a wave to its wave number k, encapsulating the propagation characteristics of waves in a physical medium.[1] This relation is derived from the governing equations of the system, such as the wave equation for mechanical waves or Maxwell's equations for electromagnetic waves, and it determines how different wave components evolve over space and time.[2] For instance, in vacuum, the dispersion relation for electromagnetic waves is linear and given by \omega = c k, where c is the speed of light, implying non-dispersive propagation where all wavelengths travel at the same speed.[1] The dispersion relation distinguishes between phase velocity v_p = \omega / k, which describes the speed of individual wave crests, and group velocity v_g = d\omega / dk, which governs the propagation of wave packets and the transport of energy or information.[3] In dispersive media, where the relation is nonlinear (e.g., \omega^2 = c^2 k^2 + \omega_p^2 for electromagnetic waves in a plasma, with \omega_p as the plasma frequency), different frequencies travel at different speeds, causing wave packets to spread out over time—a phenomenon central to signal distortion in optics and communications.[1] Linear dispersion relations, by contrast, yield constant velocities and no spreading, as seen in ideal sound waves where \omega = v k and v is the speed of sound.[2] Dispersion relations find broad applications across physics, including acoustics (e.g., sound propagation in air or solids), fluid dynamics (e.g., surface gravity waves on water with \omega^2 = g k \tanh(k h), where g is gravity and h is water depth), and solid-state physics (e.g., phonon dispersion in crystals, influencing thermal conductivity and lattice vibrations)./08%3A_Waves/8.02%3A_The_Dispersion_Relation) In quantum mechanics, they underpin de Broglie relations for matter waves, linking particle momentum p = \hbar k to energy E = \hbar \omega.[1] These relations also reveal phenomena like band gaps in periodic structures and instabilities in plasmas, making them essential for analyzing wave phenomena in engineering and natural systems.[2]Basic Concepts
Definition and Terminology
A dispersion relation describes the functional dependence of the angular frequency \omega on the wave number k for waves propagating in a medium, typically expressed as \omega = \omega(k).[4] This relation arises in the study of linear wave equations, where small-amplitude approximations allow the superposition of plane wave solutions.[4] The wave number k is defined as k = 2\pi / \lambda, where \lambda is the wavelength, representing the spatial periodicity of the wave.[5] Similarly, the angular frequency \omega is given by \omega = 2\pi f, with f denoting the ordinary frequency, capturing the temporal oscillation rate in radians per second.[6] The fundamental form of a plane wave solution is e^{i(kx - \omega t)}, where x is the position and t is time; this complex exponential encodes both the oscillatory and propagating nature of the wave.[4] In dispersive media, the dispersion relation \omega(k) is nonlinear in k, leading to a frequency-dependent propagation speed that causes wave packets to spread over time.[7] Conversely, non-dispersive propagation occurs when \omega(k) is linear, such as \omega = c k for some constant c, resulting in all frequency components traveling at the same constant speed without distortion.[8] Key velocities associated with the dispersion relation include the phase velocity v_p = \omega / k, which describes the speed of constant-phase surfaces, and the group velocity v_g = d\omega / dk, which indicates the propagation speed of the wave packet envelope.[4] These concepts are central to understanding wave behavior in contexts like acoustics, optics, and quantum mechanics, always under the framework of linear, small-amplitude waves.[9]Phase and Group Velocities
The phase velocity v_p of a monochromatic plane wave is defined as the velocity at which a surface of constant phase propagates through the medium. For a wave described by the dispersion relation \omega = \omega(k), where \omega is the angular frequency and k is the wavenumber, the phase velocity is derived from the phase factor \phi = kz - \omega t in the wave expression e^{i(kz - \omega t)}. Setting the total phase constant for a moving point, \frac{dz}{dt} = \frac{\omega}{k}, yields v_p = \frac{\omega}{k}. This represents the speed at which individual crests or troughs of the wave advance, though it does not necessarily correspond to the propagation of energy or information.[10] The group velocity v_g, in contrast, describes the velocity of the overall envelope of a wave packet formed by the superposition of waves with wavenumbers centered around some k_0. To derive it, consider the dispersion relation expanded via Taylor series around k_0: \omega(k) \approx \omega(k_0) + \left. \frac{d\omega}{dk} \right|_{k=k_0} (k - k_0) + \frac{1}{2} \left. \frac{d^2\omega}{dk^2} \right|_{k=k_0} (k - k_0)^2 + \cdots The wave packet is then a product of a carrier wave e^{i(k_0 z - \omega(k_0) t)} and an envelope modulated by the spread in k. The envelope propagates at v_g = \left. \frac{d\omega}{dk} \right|_{k=k_0}, as this term determines the shift in the phase synchronization across the packet. Physically, v_g corresponds to the velocity of energy transport in the wave, since the energy density follows the envelope in linear media.[10] In non-dispersive media, where \omega(k) = v k for constant v, the phase and group velocities coincide: v_p = v_g = v, and wave packets maintain their shape without broadening. However, in dispersive media, where \frac{d^2 \omega}{dk^2} \neq 0, v_p \neq v_g, causing the wave packet to spread over time as different frequency components travel at varying speeds; this pulse broadening limits signal integrity in applications like optical communications.[11] In dispersive systems, the signal velocity—the speed at which information or a detectable front propagates—must respect causality and cannot exceed the speed of light in vacuum. Analysis shows this signal velocity equals the group velocity at the point of stationary phase, ensuring no superluminal information transfer despite possible anomalous values of v_p or v_g in certain frequency ranges.Non-Dispersive Waves
Electromagnetic Waves in Vacuum
In vacuum, electromagnetic waves arise as solutions to Maxwell's equations in free space, which in the absence of charges and currents take the form: \nabla \cdot \mathbf{E} = 0, \quad \nabla \cdot \mathbf{B} = 0, \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}, where \mathbf{E} is the electric field, \mathbf{B} is the magnetic field, \mu_0 is the vacuum permeability, and \epsilon_0 is the vacuum permittivity.[12] To derive the wave equation, take the curl of Faraday's law \nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t: \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}). Substitute Ampère's law with Maxwell's correction \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \partial \mathbf{E}/\partial t, yielding \nabla \times (\nabla \times \mathbf{E}) = -\mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}. Using the vector identity \nabla \times (\nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} and \nabla \cdot \mathbf{E} = 0, this simplifies to the wave equation \nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}, or equivalently, \frac{\partial^2 \mathbf{E}}{\partial t^2} = c^2 \nabla^2 \mathbf{E}, where c = 1/\sqrt{\mu_0 \epsilon_0} is the speed of light in vacuum.[12] A similar wave equation holds for \mathbf{B}.[12] Assuming plane-wave solutions of the form \mathbf{E} = \mathbf{E}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}, substitution into the wave equation yields the dispersion relation \omega = c |\mathbf{k}| for transverse electromagnetic waves, where \mathbf{k} is the wave vector and \omega is the angular frequency.[12] This linear relation indicates that the phase velocity v_p = \omega / |\mathbf{k}| = c and the group velocity v_g = d\omega / d|\mathbf{k}| = c are both constant and equal to the speed of light.[13] The constancy of both velocities implies no dispersion in vacuum: all frequencies propagate at the same speed c, so wave packets do not broaden over distance.[14] Electromagnetic waves in vacuum are transverse, as the condition \nabla \cdot \mathbf{E} = 0 for plane waves requires \mathbf{k} \cdot \mathbf{E}_0 = 0, meaning the electric field is perpendicular to the propagation direction.[15] The polarization of the wave is defined by the orientation of \mathbf{E}_0 in the plane perpendicular to \mathbf{k}, which can be linear, circular, or elliptical.[15]Uniform Waves on a String
The transverse displacement y(x, t) of small-amplitude waves propagating along an idealized infinite uniform string under constant tension satisfies the one-dimensional wave equation derived from Newton's second law applied to a small string element.[16] The net transverse force on the element arises from the difference in the vertical components of tension at its ends, leading to \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}, where the wave speed v = \sqrt{T / \mu}, with T the constant tension and \mu the constant linear mass density.[16] This equation assumes small slopes (|\partial y / \partial x| \ll 1) to linearize the tension components, transverse motion only, and no external forces or material damping.[16] For plane wave solutions of the form y(x, t) = A \cos(kx - \omega t + \phi), substitution into the wave equation yields the linear dispersion relation \omega(k) = v |k|, where \omega is the angular frequency and k is the wavenumber.[4] This relation is independent of frequency, characteristic of non-dispersive propagation where all wave components travel at the same speed.[4] The phase velocity v_p = \omega / k = v and the group velocity v_g = d\omega / dk = v are both equal to the wave speed and constant, independent of frequency or wavenumber.[4] Consequently, wave packets maintain their shape without spreading during propagation.[4] In the case of a finite-length string fixed at both ends, boundary conditions y(0, t) = y(L, t) = 0 quantize the allowed wavenumbers to k_n = n \pi / L (for positive integer n), resulting in standing wave modes with frequencies \omega_n = v k_n.[17] However, the focus here remains on traveling waves in the infinite string limit, where the dispersion relation governs free propagation without boundary-induced quantization.[17] The idealized model neglects dispersion arising from bending stiffness, which introduces a higher-order restoring force proportional to the fourth spatial derivative of displacement, causing phase velocity to increase with frequency and leading to inharmonic partials.[18] Similarly, nonlinear effects, such as those from large-amplitude vibrations altering tension, are excluded, as they would frequency-dependently modify propagation.[18]Dispersive Waves in Classical Media
Optical Waves in Dispersive Materials
In dispersive optical materials, such as glasses and crystals, the propagation of electromagnetic waves is characterized by a frequency-dependent refractive index n(\omega), leading to the dispersion relation k = \frac{\omega n(\omega)}{c}, where k is the wave number, \omega is the angular frequency, and c is the speed of light in vacuum. This relation emerges from Maxwell's equations for plane waves in isotropic media with a frequency-dependent dielectric permittivity \epsilon(\omega) = n^2(\omega), reflecting the resonant interaction of light with bound electrons in the material.[19] The refractive index n(\omega) is commonly modeled using the empirical Sellmeier equation, which approximates n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i}, where \lambda = 2\pi c / \omega is the vacuum wavelength, and the coefficients B_i and C_i (related to oscillator strengths and resonance wavelengths) are determined from experimental measurements for specific materials. This form provides accurate predictions over broad wavelength ranges away from strong absorptions and was originally proposed by Wolfgang Sellmeier to explain anomalous color sequences in spectra.[20][21] Materials exhibit normal dispersion in spectral regions distant from electronic absorption bands, where \frac{dn}{d\omega} > 0, so higher frequencies propagate more slowly than lower ones, as seen in visible light through crown glass. In contrast, anomalous dispersion occurs near ultraviolet or infrared resonances, where \frac{dn}{d\omega} < 0, causing the refractive index to decrease with increasing frequency and potentially leading to superluminal phase velocities, though energy transport remains subluminal. These behaviors arise from the Lorentz oscillator model of atomic polarizability.[22] A practical consequence of optical dispersion is chromatic dispersion in waveguides like silica optical fibers, where varying group velocities cause temporal broadening of light pulses, limiting high-speed data transmission over long distances. The group velocity is expressed as v_g = \frac{c}{n + \omega \frac{dn}{d\omega}}, quantifying how pulse envelopes propagate; in normal dispersion regimes typical for telecommunications wavelengths around 1550 nm, v_g < c/n, resulting in positive dispersion parameter D > 0 that spreads pulses.[23][24] The real and imaginary parts of the permittivity \epsilon(\omega) = \epsilon_r(\omega) + i \epsilon_i(\omega), which determine refraction and absorption respectively, are interconnected by the Kramers-Kronig relations, principal-value integrals ensuring the causal response of materials to electromagnetic fields. These relations, derived from the analyticity of the dielectric function in the complex frequency plane, were independently formulated by Hendrik Kramers and Ralph Kronig in the late 1920s.Deep Water Waves
Deep water waves refer to small-amplitude gravity waves propagating on the surface of an inviscid, incompressible fluid occupying a domain of infinite depth, where the water depth h greatly exceeds the wavelength \lambda such that kh \gg 1, with k = 2\pi/\lambda being the wavenumber.[25] These waves are governed by the linearized Euler equations, assuming irrotational flow, which allows the introduction of a velocity potential \phi satisfying Laplace's equation \nabla^2 \phi = 0.[26] The flow is two-dimensional, with waves propagating in the x-direction and vertical coordinate z increasing upward from the mean surface at z=0. To derive the dispersion relation, assume a monochromatic wave solution of the form \phi(x,z,t) = \mathrm{Re} \{ \hat{\phi}(z) e^{i(kx - \omega t)} \}, where \omega is the angular frequency. Substituting into Laplace's equation yields \hat{\phi}(z) = A e^{kz} for the deep water case, as the exponentially decaying solution ensures boundedness as z \to -\infty.[25] The linearized kinematic boundary condition at the free surface z=0 relates the vertical velocity to the surface elevation \eta, giving \partial_t \eta = \partial_z \phi. The dynamic boundary condition, enforcing constant pressure at the surface, linearizes to \partial_t \phi + g \eta = 0 at z=0. Combining these yields the dispersion relation \omega^2 = g k in the deep water limit.[26] The phase velocity is v_p = \omega / k = \sqrt{g / k} = \sqrt{g \lambda / (2\pi)}, which increases with wavelength, meaning longer waves propagate faster than shorter ones.[25] The group velocity, representing the speed of energy transport, is v_g = d\omega / dk = \frac{1}{2} v_p = \frac{1}{2} \sqrt{g / k}.[25] This dispersion causes wave packets to spread out over time, with longer wavelength components advancing ahead of shorter ones, which explains the evolution of wind-generated wave fields into organized swell patterns where dominant longer waves separate from trailing shorter waves.[27] This approximation holds for small-amplitude waves where the surface displacement is much less than the wavelength (a \ll \lambda) and under the deep water condition kh \gg 1, typically valid for ocean waves with periods less than about 20 seconds.[28] Particle orbits are circular near the surface with radius decaying exponentially with depth as a e^{k z}, confining motion to the upper layer.[25]Acoustic Phonons in Solids
In crystalline solids, acoustic phonons arise from the collective vibrations of atoms in the lattice, representing quantized modes of sound waves that propagate through the material. These modes are derived from the harmonic approximation of lattice dynamics, where atoms are treated as point masses connected by springs. Phonons, as the quanta of these vibrations, provide a quantum mechanical description of the classical lattice oscillations.[29] The simplest model for these vibrations is the one-dimensional monatomic chain, consisting of identical atoms of mass m separated by equilibrium distance a, interacting via nearest-neighbor harmonic springs with force constant K. The equation of motion for the displacement u_n of the n-th atom is given bym \ddot{u}_n = K (u_{n+1} + u_{n-1} - 2 u_n).
Assuming plane-wave solutions u_n = A e^{i(k n a - \omega t)}, this yields the dispersion relation
\omega(k) = 2 \sqrt{\frac{K}{m}} \left| \sin\left( \frac{k a}{2} \right) \right|,
where k is the wavevector in the first Brillouin zone [-\pi/a, \pi/a]. This relation was foundational in the development of lattice dynamics.[29][30] The acoustic branch corresponds to this single dispersion curve, where all atoms move in phase at long wavelengths. For small k (long wavelengths, k a \ll 1), the relation linearizes to \omega \approx v_s k, with sound speed v_s = a \sqrt{K/m}, mimicking classical sound waves in a continuum. Near the Brillouin zone edge (k \approx \pi/a), the dispersion becomes nonlinear, with \omega flattening due to the periodic lattice structure, leading to standing-wave-like behavior. The group velocity, v_g = d\omega/dk, equals v_s at the zone center (k = 0) for propagating waves but approaches zero at the zone edges, where modes resemble stationary vibrations.[29] In three dimensions, the model extends to a cubic lattice with atoms having three degrees of freedom, resulting in longitudinal acoustic (LA) modes—where displacements are parallel to the wavevector—and two transverse acoustic (TA) modes—where displacements are perpendicular. The dispersion remains linear at low k, with \omega = v_l k for LA modes (longitudinal sound speed v_l) and \omega = v_t k for TA modes (transverse sound speed v_t), derived from the elastic constants of the material. The Debye model approximates this low-frequency behavior by assuming a linear dispersion \omega = v k up to a cutoff wavevector, treating the solid as a continuum of acoustic modes to simplify calculations of thermodynamic properties; here, v is an effective speed averaging v_l and v_t. This approximation, introduced by Debye, captures the essential physics for low-energy excitations.[29][31] Acoustic phonons play a central role in sound propagation, as their linear dispersion at long wavelengths directly corresponds to the transmission of elastic waves through the solid at speeds v_l and v_t. Additionally, they dominate thermal conductivity in insulators and semiconductors, where heat transport occurs via phonon diffusion; the group velocity determines the rate of energy carry, while scattering from defects or anharmonicity limits the mean free path, as described by the Boltzmann transport equation for phonons.[29][32]