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Phase velocity

Phase velocity is the speed at which a point of constant , such as a or trough, propagates in a wave, defined mathematically as the ratio of the \omega to the wave number k, given by v_p = \frac{\omega}{k}. This velocity equivalently equals the product of the wave's f and \lambda, v_p = f \lambda, and applies to various wave types including mechanical, electromagnetic, and quantum waves. In non-dispersive media, where the wave speed is independent of frequency, phase velocity coincides with the , which represents the speed of the wave's overall or . However, in dispersive media—such as , water waves under , or —phase velocity differs from , with the latter given by v_g = \frac{d\omega}{dk}. For instance, in electromagnetic waves within a waveguide, phase velocity exceeds the c (e.g., v_p > c), but this does not violate since phase velocity alone cannot transmit information or . Phase velocity plays a crucial role in understanding wave propagation in fields like , acoustics, and , where it influences phenomena such as and patterns. In , it relates to the de Broglie wave's phase speed for particles, often superluminal, while the corresponds to the particle's actual velocity. Notable examples include sound waves in air, where v_p \approx 343 m/s at 20 °C and standard , and light in , where v_p = [c](/page/c).

Basic Concepts

Definition

Phase velocity is the speed at which a surface of constant , such as a or trough, propagates through a medium in a . This velocity characterizes the motion of specific points on the where the phase remains unchanged, distinguishing it from other aspects of wave propagation. The concept of phase velocity emerged in the 19th century as part of the study of sinusoidal waves, with early discussions by physicists such as and George Gabriel Stokes highlighting the propagation characteristics of periodic waves. An intuitive example occurs in a on a stretched , where the phase velocity determines how quickly the peaks or crests of the sinusoidal disturbance travel along the length of the ./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/16%3A_Waves/16.01%3A_Traveling_Waves) Importantly, phase velocity differs from the particle velocity of the medium itself; for instance, in a wave on a string, the particles oscillate transversely with small amplitudes perpendicular to the direction of propagation, while the phase points advance longitudinally along the string./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/16%3A_Waves/16.01%3A_Traveling_Waves)

Wave Parameters

In the study of waves, several key parameters describe the periodic nature and propagation characteristics essential for calculating phase velocity. The linear frequency, denoted as f, represents the number of wave cycles per unit time and is measured in hertz (Hz), where 1 Hz equals one cycle per second in the International System of Units (SI). The angular frequency, \omega, is related to the linear frequency by the equation \omega = 2\pi f and has units of radians per second (rad/s), providing a measure of the rate of change of the wave's phase in angular terms. The , \lambda, is the spatial distance over which the wave completes one full cycle, typically expressed in meters (m) in units. Complementing this, the , k, quantifies the of the wave and is defined as k = \frac{2\pi}{\lambda}, with units of inverse meters (m^{-1}). These parameters interconnect through the fundamental relation for phase velocity, v_p = f \lambda, which ties the wave's speed to its temporal and spatial periodicity, or equivalently v_p = \frac{\omega}{k} using measures. In periodic waves, these parameters underpin both sinusoidal representations, such as \psi(x, t) = A \sin(\omega t - k x + \phi), where A is the and \phi the initial , and complex exponential forms like \psi(x, t) = A e^{i(\omega t - k x + \phi)}, which facilitate analytical treatments in physics. Standard notations in physics, such as those in SI units, ensure consistency across electromagnetic, acoustic, and analyses, emphasizing f and \lambda for intuitive cycle-based descriptions and \omega and k for phase-based computations.

Mathematical Formulation

Plane Wave Expression

The standard mathematical representation of a monochromatic propagating along the positive x-direction in one dimension is \psi(x, t) = A \cos(kx - \omega t + \phi), where A is the constant , k is the wave number, \omega is the , and \phi is a constant. The velocity v_p emerges from the condition of constant in this expression. Setting of the cosine equal to a constant, kx - \omega t + \phi = \text{constant}, and differentiating implicitly with respect to time t yields k \, dx/dt - \omega = 0. Solving for the velocity gives v_p = dx/dt = \omega / k. This quantity v_p represents the speed at which a surface of constant —such as a or trough—propagates through , distinct from the motion of individual particles in the medium. An equivalent form of the phase velocity is v_p = f \lambda, where f is the ordinary and \lambda is the . The angular relates to the ordinary frequency by \omega = 2\pi f, while the wave number relates to the by k = 2\pi / \lambda. Substituting these relations confirms the equivalence: v_p = \omega / k = (2\pi f) / (2\pi / \lambda) = f \lambda. This alternative expression highlights the intuitive connection between the wave's temporal rate and its spatial periodicity. The model assumes an infinite extent in the transverse directions, monochromatic (single-frequency) excitation, and propagation in a single dimension, serving as an exact solution to the one-dimensional under these conditions. In practice, this is an idealization, as real waves are typically finite in spatial extent, subject to effects, and composed of a superposition of frequencies, leading to phenomena like and not captured by the simple form.

General Dispersion Relation

The dispersion relation describes the relationship between the \omega and the wave number k for propagating in a medium, typically expressed explicitly as \omega = \omega(k), implicitly as f(\omega, k) = 0, or inversely as k = k(\omega). This relation arises from the underlying partial differential equations governing wave propagation and determines the possible wave solutions. For a monochromatic of the form \psi(x, t) = A e^{i(kx - \omega t)}, the phase velocity is defined as v_p = \frac{\omega}{k}, which directly follows from the when it is expressed as \omega(k). In cases where the is linear, such as \omega = v k with constant v, the phase velocity v_p = v remains independent of k. However, for nonlinear relations \omega(k), v_p varies with k, leading to frequency-dependent propagation speeds. This variation is evident in an \omega-k plot, where the phase velocity at a specific k corresponds to the slope of the straight line connecting the to the point (k, \omega(k)) on the dispersion curve. Such graphical representations highlight how different wave components with varying k travel at distinct phase velocities, contributing to wave . The can take implicit forms in certain contexts, such as in where the phase velocity is given by v_p = \frac{c}{n(\omega)}, with c the in vacuum and n(\omega) the as a of ; this implicitly defines \omega(k) through k = \frac{\omega n(\omega)}{c}. To derive the mathematically, consider a general linear , starting from the non-dispersive case \frac{\partial^2 \psi}{\partial t^2} = v^2 \frac{\partial^2 \psi}{\partial x^2}. Substituting the \psi(x, t) = e^{i(kx - \omega t)} yields -\omega^2 = -v^2 k^2, or \omega = v |k|, giving constant v_p = v. For generalized equations, such as those incorporating terms (e.g., higher spatial or -dependent coefficients), the same substitution produces an algebraic equation relating \omega and k, solved to obtain \omega(k) and thus v_p = \frac{\omega(k)}{k}. This approach extracts the phase speed for arbitrary wave types beyond simple sinusoidal cases.

Relations to Other Velocities

Group Velocity

The group velocity v_g of a wave is defined as the velocity at which the envelope of a wave packet propagates, given by the derivative of the \omega with respect to the k, v_g = \frac{d\omega}{dk}. This concept was first introduced by Lord Rayleigh in his analysis of progressive waves, where he identified the velocity of a group of waves as distinct from the velocity of individual wave crests. Physically, the group velocity represents the speed at which the overall or of the wave packet travels, which corresponds to the propagation of or carried by , in contrast to the phase velocity that tracks the motion of constant-phase points within the wave. In dispersive media, where \omega depends nonlinearly on k, the group velocity determines how the wave packet as a whole moves, while individual phases may advance or recede relative to the . To derive the group velocity, consider the superposition of two waves with nearby wavenumbers k and k + \Delta k, and corresponding frequencies \omega(k) and \omega(k + \Delta k). The resulting wave packet can be expressed using the trigonometric identity for the sum of cosines: \cos(kx - \omega t) + \cos((k + \Delta k)x - (\omega + \Delta \omega)t) = 2 \cos\left( \left( k + \frac{\Delta k}{2} \right) x - \left( \omega + \frac{\Delta \omega}{2} \right) t \right) \cos\left( \frac{\Delta k}{2} x - \frac{\Delta \omega}{2} t \right), where the first cosine term describes the and the second the . The envelope moves such that its \frac{\Delta k}{2} x - \frac{\Delta \omega}{2} t remains constant, implying a velocity v_g = \frac{\Delta \omega}{\Delta k}. For a continuous , expand \omega(k + \Delta k) using a Taylor series: \omega(k + \Delta k) \approx \omega(k) + \frac{d\omega}{dk} \Delta k, so \Delta \omega \approx \frac{d\omega}{dk} \Delta k, yielding v_g = \frac{d\omega}{dk} in the limit \Delta k \to 0. This derivation extends to wave packets formed by many nearby waves, where the envelope speed is governed by the slope of the dispersion relation \omega(k). In multidimensional cases, such as waves in with wavevector \vec{k} = (k_x, k_y, k_z), the group velocity becomes a vector \vec{v_g} = \nabla_{\vec{k}} \omega(\vec{k}), with components given by the partial derivatives v_{g,i} = \frac{\partial \omega}{\partial k_i} for i = x, y, z. This generalization describes the direction and speed of energy propagation in anisotropic or vectorial wave systems, such as electromagnetic or in complex media.

Signal Velocity

Signal velocity refers to the speed at which a signal or disturbance propagates through a medium, carrying or from one point to another. It is defined as the velocity of the of the wave packet's , ensuring that it represents the actual propagation of causal effects rather than mere phase or envelope motion. In linear, non-dispersive media, this coincides with the , but in dispersive environments, it is more precisely tied to the high-frequency components that form the wavefront precursor. According to , the signal velocity v_s must always satisfy v_s \leq c, where c is the in , to preserve and prevent information from traveling . This constraint holds even when phase velocities exceed c, as superluminal phase speeds do not transmit usable signals and thus do not violate relativistic principles. Seminal analyses by Sommerfeld and Brillouin demonstrated that —high-frequency transients at the wave's front—propagate at speeds no greater than c, resolving apparent paradoxes in dispersive propagation. In dispersive media, for signals where the is limited, the approximates the , v_s \approx v_g = \frac{d\omega}{dk}, as derived from the of evolution. This approximation arises because the envelope's motion aligns with the information-carrying components under normal conditions. However, the exact is determined by the medium's response to the initial disturbance, often requiring saddle-point methods in the plane to compute the precursor arrival. In regions of anomalous dispersion, where the refractive index decreases with frequency, the signal velocity can deviate significantly from both phase and group velocities, as the latter may become superluminal or negative. Despite such anomalies, causality ensures the signal front remains subluminal, with the true information speed governed by the medium's absorptive properties and the velocity of the forerunner signals. This distinction, first clarified in early 20th-century treatments, underscores that group velocity alone does not always capture causal propagation in highly dispersive regimes.

Behavior in Media

Non-Dispersive Media

In non-dispersive media, the takes the \omega = v k, where v is a constant speed that does not depend on the wave's or k. Consequently, the phase velocity v_p = \frac{\omega}{k} = v remains constant across all frequencies, preventing the spreading or distortion of wave components that occurs in dispersive environments. This uniformity ensures that individual wave crests propagate at the same speed regardless of their frequency content. A key implication of this constancy is that the phase velocity equals both the group velocity v_g = \frac{d\omega}{dk} = v and the signal velocity, which represents the speed of information or energy transport in the wave. As a result, composite wave packets—formed by superpositions of multiple frequencies—retain their overall shape and structure while traveling, without broadening or changing form over distance. This property is essential for applications where waveform integrity must be preserved. Prominent examples include ideal sound waves in air, treated as longitudinal pressure waves in a medium. At 20°C, the phase velocity for these waves is approximately 343 m/s, arising from the balance between the medium's and in the linearized equations of fluid motion. Another example is non-dispersive electromagnetic waves in , where the phase velocity equals the c \approx 3 \times 10^8 m/s, independent of due to the uniformity of . For a on a taut with T and linear \mu, the phase velocity is given by v_p = \sqrt{\frac{T}{\mu}}. This expression derives from Newton's second law applied to a small , where the net transverse force from gradients provides the acceleration, yielding the one-dimensional with a frequency-independent propagation speed. Similarly, for electromagnetic waves in a non-dispersive with constant \epsilon and permeability \mu, the phase velocity is v_p = \frac{1}{\sqrt{\epsilon \mu}}. In , where \epsilon = \epsilon_0 and \mu = \mu_0, this simplifies to c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}, obtained by substituting into the wave equation for the electric and magnetic fields, confirming propagation at a universal constant speed.

Dispersive Media

In dispersive media, the propagation of exhibits frequency-dependent behavior due to a nonlinear , where the wave number k is a nonlinear function of the \omega, or vice versa, resulting in a phase velocity v_p = \omega / k that varies with . This variation arises because the medium's response, such as its , changes with \omega, altering the relationship between phase and components of the wave. The primary effect of this frequency dependence is the broadening of wave pulses as they propagate, since different frequency components travel at different phase velocities, causing the initial pulse shape to distort and spread over distance. For instance, the second derivative of k(\omega), known as , quantifies this spreading, with the pulse width increasing proportionally to the square root of the propagation distance for Gaussian pulses. Notably, in such media, v_p can exceed the in c, as occurs in waveguides above the where the yields k(\omega) = (\omega / c) \sqrt{1 - (\omega_c / \omega)^2} and thus v_p > c. However, this does not enable superluminal signaling, as the information-carrying remains at or below c. In dielectric media, dispersion is closely tied to the refractive index n(\omega), where the phase velocity is expressed as v_p = c / n(\omega), and n(\omega) > 1 typically slows below c, but the frequency variation of n—often modeled as n^2 = 1 + \frac{n_v e^2}{m \varepsilon_0 (\omega_0^2 - \omega^2)} for contributions—induces the dispersive effects. This \omega-dependence leads to phenomena like normal in the visible range, where dn/d\omega > 0. A representative example of dispersion is deep-water surface gravity waves, where the phase velocity increases with \lambda according to v_p = \sqrt{\frac{g \lambda}{2\pi}}, with g the , demonstrating how longer waves outpace shorter ones and thus spread wave packets.

Applications and Examples

Electromagnetic Waves

In vacuum, electromagnetic waves propagate with a phase velocity equal to the , c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}, where \epsilon_0 and \mu_0 are the and permeability of free space, respectively. This velocity is constant and independent of frequency, resulting in the phase velocity being identical to both the and for such waves. In dielectric media, the phase velocity decreases to v_p = \frac{c}{n}, where n = \sqrt{\frac{\epsilon \mu}{\epsilon_0 \mu_0}} is the , determined by the material's \epsilon_r = \frac{\epsilon}{\epsilon_0} and \mu_r = \frac{\mu}{\mu_0}. In regions of anomalous dispersion, where the refractive index decreases with increasing frequency (\frac{dn}{d\omega} < 0), the phase velocity can become less than the group velocity and exceed c (when n < 1); the group velocity can also appear superluminal or negative in highly absorbing regions. However, the signal velocity—which governs information and energy propagation—remains below c, preserving causality. In waveguides, such as rectangular metallic structures, the phase velocity exceeds c for frequencies above the mode's cutoff frequency f_c, expressed as v_p = \frac{\omega}{\beta} = \frac{c}{\sqrt{1 - (f_c/f)^2}} > c, where \beta is the and f is the operating . This superluminal phase velocity arises because the wave's effective path is longer than the guide's axis due to boundary constraints, but the group velocity v_g = c \sqrt{1 - (f_c/f)^2} < c ensures that energy and propagate subluminally, preserving relativistic . The foundational role of phase velocity in electromagnetism traces to James Clerk Maxwell's equations in the 1860s, which derived the wave equation for electromagnetic fields and predicted propagation at speed c, closely matching experimental measurements of light's velocity and confirming light as an electromagnetic phenomenon.

Acoustic Waves

In , the velocity represents the speed at which a particular of the wave propagates through the medium, which for sound waves in fluids and is closely tied to the material's properties and . In non-dispersive approximations for media, this velocity approximates the , providing a baseline for propagation in ideal conditions. For fluids, the phase velocity of depends on the medium's and . In gases, it is given by v_p = \sqrt{\frac{\gamma [P](/page/P′′)}{\rho}}, where \gamma is the adiabatic index, P is the , and \rho is the ; this expression arises from the isentropic relation in the wave equation. In liquids, where effects are negligible, the phase velocity simplifies to v_p = \sqrt{\frac{B}{\rho}}, with B denoting the bulk modulus, reflecting the medium's resistance to uniform compression. These formulas highlight how phase velocity increases with stiffness (higher B or \gamma P) and decreases with density, enabling sound to travel faster in denser yet more rigid liquids like compared to gases. In solids, acoustic waves support both longitudinal and transverse modes, each with distinct phase velocities determined by the Lamé elastic constants \lambda (related to compression) and \mu (shear modulus), along with density \rho. The longitudinal phase velocity is v_p = \sqrt{\frac{\lambda + 2\mu}{\rho}}, accounting for both compressional and shear contributions in the direction of propagation. For transverse waves, polarized perpendicular to the propagation direction, it reduces to v_p = \sqrt{\frac{\mu}{\rho}}, depending solely on rigidity. These velocities enable applications like seismic , where P-waves (longitudinal) arrive before S-waves (transverse) due to the higher longitudinal speed. Real acoustic media exhibit , where phase velocity varies with frequency, deviating from ideal non-dispersive behavior. In air, slight occurs at high frequencies (above ~10 kHz) due to molecular relaxation processes, such as vibrational modes in oxygen and , leading to a phase velocity increase of up to 0.3% at ultrasonic frequencies as derived from Kramers-Kronig relations applied to data. A practical application arises in ultrasound imaging, where the phase velocity in , approximately 1500 m/s, directly impacts axial since \lambda = v_p / f (with f the ) determines the smallest resolvable feature, typically requiring 2–5 MHz pulses for clinical depths. This value, averaged across tissues like muscle and fat, assumes minimal at diagnostic frequencies, ensuring consistent time-of-flight calculations for image reconstruction.