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Phase velocity

Phase velocity is the speed at which a particular phase of a single-frequency component of a wave propagates; distinct from group velocity and affected by dispersion and the medium.

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Overview

The phase velocity of a wave is the rate at which a given phase point — for example a crest or trough — moves through space for a single frequency component. It characterizes the motion of constant-phase surfaces of a sinusoidal component, not necessarily the transport of energy or information. Phase velocity is meaningful for any wave that can be decomposed into monochromatic components, such as sound, water, electromagnetic waves, or quantum matter waves.

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File:Wave_packet_propagation_(phase_faster_than_group,_nondispersive).gif
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Mathematical definition

For a harmonic wave with angular frequency ω and wavenumber k the phase velocity vp is

  • vp = ω / k
  • equivalently vp = f·λ = λ / T, where f is frequency, λ is wavelength and T is period

These relations hold for each monochromatic component. In a nondispersive medium ω is proportional to k and vp is constant; in dispersive media vp varies with frequency.

Dispersion and important distinctions

When the medium is dispersive, different frequencies travel with different phase velocities. The group velocity, vg = dω/dk, typically governs the propagation of wave packets and the flow of energy or information, and can differ substantially from vp. In many contexts the distinction matters: phase velocity describes the motion of a pattern (e.g., a crest) while group velocity describes signal or energy propagation.

Examples and applications

In vacuum electromagnetic waves have vp = c. In a simple dielectric vp = c/n, with n the refractive index. Water waves and waves on strings often show dispersion so vp depends on wavelength. Practical uses of phase-velocity concepts include phase matching in nonlinear optics, antenna and waveguide design, and analysis of surface and plasma waves. For quantum de Broglie waves the phase velocity can exceed c without transmitting information faster than light.

Historical notes and notable facts

The idea of phase velocity arises naturally from classical wave theory and Fourier analysis. Notable phenomena include the possibility of vp exceeding the speed of light in certain media (which does not violate relativity because information speed is limited by vg) and negative phase velocity in engineered metamaterials. For further background on frequency concepts see frequency and for a simple visual reference see a wave crest. The relation to wavelength is summarized at wavelength.

Key points

  • Phase velocity tracks a single-frequency phase pattern, not signal speed.
  • Different from group velocity; both arise from the dispersion relation ω(k).
  • Important in optics, acoustics, water waves, waveguides, and quantum mechanics.
The red dot is always at the point of the same phase (wave crest) and moves with the phase velocity of the blue, monochromatic wave.
The red dot is always at the point of the same phase (wave crest) and moves with the phase velocity of the blue, monochromatic wave.
A wave packet propagates in a non-dispersive medium (e.g. an electromagnetic wave in a vacuum).
A wave packet propagates in a non-dispersive medium (e.g. an electromagnetic wave in a vacuum).
A wave packet propagates in a dispersive medium.
A wave packet propagates in a dispersive medium.
The green points move at group speed, the red one at phase speed.
The green points move at group speed, the red one at phase speed.

Relationship with group velocity and dispersion

Designation

Symbol

Relationships

Amplitude

{\vec A}_{0}

{\displaystyle {\vec {A}}_{0}\perp {\vec {k}}}

Transverse wave

{\displaystyle {\vec {A}}_{0}\|{\vec {k}}}

Longitudinal wave

Wave vector

{\vec {k}}

Direction of propagation

Circular wave number

k\,

{\displaystyle k=|{\vec {k}}|}

Wavelength

\mathbf {\lambda }

{\displaystyle \lambda ={\frac {2\pi }{k}}}

Circular frequency

\mathbf {\omega }

{\displaystyle \mathbf {\omega } {\big (}{\vec {k}}{\big )}}Dispersion relation

Frequency

f

{\displaystyle f={\frac {\omega }{2\pi }}}

Phase velocity

v_{{\mathrm {p}}}

{\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}=\lambda f}

Group speed

v_{{\mathrm {g}}}

{\displaystyle v_{\mathrm {g} }={\frac {\partial \omega }{\partial k}}}

Phase angle

\varphi

{\displaystyle \varphi ={\vec {k}}\cdot {\vec {r}}-\omega t}

For the mathematical description of a wave in a special medium, one needs its waveform, amplitude, frequency, phase angle and the associated wave equation - if necessary with boundary conditions. Nevertheless, different velocities can be assigned to such a clearly defined wave, which should not be confused with the phase velocity.

The speed at which a wave transmits energy or information is the signal speed. For a lossless medium, this is equal to the group velocity, i.e. the velocity of a wave packet. Such a wave packet is composed of monochromatic waves with different frequencies f Each of these monochromatic waves has its own phase velocity:

v_{{\mathrm {p}}}=v_{{\mathrm {p}}}(f).

The functional relationship between phase velocity and frequency is called dispersion.

For electromagnetic waves, the phase velocity v_{{\mathrm {p}}}and the group velocity v_{{\mathrm {g}}}in vacuum is equal to the speed of light c, i.e., the vacuum is nondispersive. In matter, on the other hand, the phase velocity generally depends on the frequency. Because of the relation for the refractive index }n=c/v_{{\mathrm {p}}} , here the frequency dependence of the refractive index n(f)is called dispersion.

Examples

Structure-borne sound

In solids, sound waves can propagate as structure-borne sound. The phase velocities vary depending on the wave type. For example, the phase velocity of the longitudinal wave at room temperature in stainless steel is about 5980 m/s; the phase velocity of the transverse wave is smaller by a factor of about 1.8: approx. 3300 m/s. In thin plates, other types of waves exist, so-called Lamb waves. In the adjacent picture, each branch corresponds to a Lamb wave type (mode). Vertically, the phase velocity vv_{{\mathrm s}}shown in units of the transverse wave velocity , horizontally, the frequency is shown as the product of angular frequency ω \omega and plate thickness din units of the transverse wave velocity. The higher modes S_{1},A_{1},S_{2},A_{2},\dots exist only above certain minimum frequencies and then propagate at very high phase velocities. The A_{0}mode has a vanishing phase velocity for small frequencies.

Matter wave

According to the wave-particle duality, a particle, e.g. an electron with energy Eand momentum p, a wavelength λ \lambda assigned and thus a phase velocity

v_{{\mathrm p}}=f\lambda ={\frac {\omega }{k}}={\frac {E}{p}}.

With Einstein's formula

E=mc^{2}

or in the formulation with the Lorentz factor γ \gamma

E=m_{0}\gamma c^{2}

and the definition of the relativistic momentum p=m_{0}\gamma vfollows

v_{{\mathrm p}}={\frac {m_{0}\gamma c^{2}}{m_{0}\gamma v_{{\mathrm g}}}}={\frac {c^{2}}{v_{{\mathrm g}}}}.

Here cthe speed of light, the highest speed at which energy or information can propagate. The group velocity v_{{\mathrm g}}={\frac {{\mathrm d}\omega }{{\mathrm d}k}}is the particle velocity, which is always less than c Therefore

So the de Broglie phase velocity is always greater than the speed of light. This so-called superluminal velocity of matter waves does not contradict the theory of relativity, because the signal velocity v_{{\mathrm g}}is

Waveguide

Electromagnetic waves in normal waveguides used for power transmission also travel at phase velocities above the speed of light. In the traveling wave accelerator, the phase velocity must be artificially reduced to values below the speed of light by regularly arranged conductive apertures.

Lambmodes for two different materials with Poisson's ratio
 \sigma =0{,}34(e.g. titanium) and σ \sigma =0{,}27(e.g. steel)

Questions and answers

Q: What is phase velocity?

A: Phase velocity is the speed at which the phase of any one frequency component of a wave travels.

Q: How is phase velocity related to given phase of a wave?

A: Any given phase of the wave appears to travel at the phase velocity.

Q: Can phase velocity be calculated?

A: Yes, phase velocity can be calculated.

Q: In what terms is phase velocity given?

A: Phase velocity is given in terms of wavelength λ (lambda) and wave period T.

Q: What is wavelength?

A: Wavelength is the distance between two consecutive crests or troughs of a wave.

Q: What is wave period?

A: Wave period is the time it takes for a wave to complete one full cycle.

Q: Can phase velocity change with changes in wavelength and wave period?

A: Yes, phase velocity can change with changes in wavelength and wave period.

Related articles

Author

AlegsaOnline.com Phase velocity

URL: https://en.alegsaonline.com/art/76353

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