# Phase velocity

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The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

In physics and mathematics, the phase of a periodic function of some real variable is the relative value of that variable within the span of each full period.

Wave propagation is any of the ways in which waves travel.

## Contents

${\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}$

Equivalently, in terms of the wave's angular frequency ω, which specifies angular change per unit of time, and wavenumber (or angular wave number) k, which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) νp,

In physics, angular frequencyω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

${\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}.}$

To understand where this equation comes from, consider a basic sine wave, A cos (kxωt). After time t, the source has produced ωt/2π = ft oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance x to fit the same number of oscillations, kx = ωt.

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

Thus the propagation velocity v is v = x/t = ω/k. The wave propagates faster when higher frequency oscillations are distributed less densely in space. [2] Formally, Φ = kxωt is the phase. Since ω = −dΦ/dt and k = +dΦ/dx, the wave velocity is v = dx/dt = ω/k.

## Relation to group velocity, refractive index and transmission speed

Since a pure sine wave cannot convey any information, some change in amplitude or frequency, known as modulation, is required. By combining two sines with slightly different frequencies and wavelengths,

In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the carrier signal, with a modulating signal that typically contains information to be transmitted. Most radio systems in the 20th century used frequency modulation (FM) or amplitude modulation (AM) for radio broadcast.

${\displaystyle \cos[(k-\Delta k)x-(\omega -\Delta \omega )t]\;+\;\cos[(k+\Delta k)x-(\omega +\Delta \omega )t]=2\;\cos(\Delta kx-\Delta \omega t)\;\cos(kx-\omega t),}$

the amplitude becomes a sinusoid with phase speed Δωk. It is this modulation that represents the signal content. Since each amplitude envelope contains a group of internal waves, this speed is usually called the group velocity, vg. [2]

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

In a given medium, the frequency is some function ω(k) of the wave number, so in general, the phase velocity vp = ω/k and the group velocity vg = dω/dk depend on the frequency and on the medium. The ratio between the speed of light c and the phase velocity vp is known as the refractive index, n = c/vp = ck/ω.

Taking the derivative of ω = ck/n with respect to k, yields the group velocity,

${\displaystyle {\frac {{\text{d}}\omega }{{\text{d}}k}}={\frac {c}{n}}-{\frac {ck}{n^{2}}}\cdot {\frac {{\text{d}}n}{{\text{d}}k}}~.}$

Noting that c/n = vp, indicates that the group speed is equal to the phase speed only when the refractive index is a constant dn/dk = 0, and in this case the phase speed and group speed are independent of frequency, ω/k=dω/dk=c/n. [2]

Otherwise, both the phase velocity and the group velocity vary with frequency, and the medium is called dispersive; the relation ω=ω(k) is known as the dispersion relation of the medium.

The phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin. See dispersion for a full discussion of wave velocities.

## Related Research Articles

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.

Particle velocity is the velocity of a particle in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string.

Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. The SI unit of particle displacement is the metre (m). In most cases this is a longitudinal wave of pressure, but it can also be a transverse wave, such as the vibration of a taut string. In the case of a sound wave travelling through air, the particle displacement is evident in the oscillations of air molecules with, and against, the direction in which the sound wave is travelling.

In optics, group velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium will affect the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the inverse of group velocity of light in a material with respect to angular frequency,

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important. Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

Acoustic waves are a type of longitudinal waves that propagate by means of adiabatic compression and decompression. Longitudinal waves are waves that have the same direction of vibration as their direction of travel. Important quantities for describing acoustic waves are sound pressure, particle velocity, particle displacement and sound intensity. Acoustic waves travel with the speed of sound which depends on the medium they're passing through.

Zero sound is the name given by Lev Landau to the unique quantum vibrations in quantum Fermi liquids.

Equatorial Rossby waves, often called planetary waves, are very long, low frequency waves found near the equator and are derived using the equatorial beta plane approximation.

Precursors are characteristic wave patterns caused by dispersion of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of electromagnetic radiation as well as in fluid surface waves and seismic waves.

A Sverdrup wave is a wave in the ocean, which is affected by gravity and Earth's rotation.

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

In physics, sinusoidalplane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane.

## References

### Footnotes

1. Nemirovsky, Jonathan; Rechtsman, Mikael C; Segev, Mordechai (9 April 2012). "Negative radiation pressure and negative effective refractive index via dielectric birefringence" (PDF). Optics Express. 20 (8): 8907–8914. Bibcode:2012OExpr..20.8907N. doi:10.1364/OE.20.008907. PMID   22513601. Archived from the original (PDF) on 16 October 2013.
2. "Phase, Group, and Signal Velocity". Mathpages.com. Retrieved 2011-07-24.