# Phase Velocity

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## Phase velocity

The velocity of propagation of a pure sine wave of infinite extent. In one dimension, for example, the form of the disturbance for such a wave is given by *y*(*x*, *t*) = *A* sin [2π(*x*/λ - *t*/*T*)]. Here *x* is the position at which the disturbance *y*(*x*, *t*) exists at time *t*, λ is the wavelength, *T* is the period which is related to the wave frequency by *T* = 1/*f*, and *A* is the disturbance amplitude. The argument of the sine function is called the phase. The phase velocity is the speed with which a point of constant phase can be said to move. Thus *x*/λ - *f**t* = constant, so the phase velocity *v*_{p} is given by *dx/dt* = *v*_{p} = λ*f*. This is the basic relationship connecting phase velocity, wavelength, and frequency. *See* Phase (periodic phenomena), Sine wave, Wave motion

The phase velocity for waves in a medium is determined in part by intrinsic properties of the medium. For all mechanical waves in elastic media, the square of the phase velocity is proportional to the ratio of the appropriate elastic property of the medium to the appropriate inertia property. The phase velocity of electromagnetic waves depends upon the medium as well. In vacuum, the phase velocity *c* is given by *c*^{2} = l/ε_{0}μ_{0} ≈ 9 × 10^{16} m^{2}/s^{2}, where ε_{0} and μ_{0} are respectively the permittivity and permeability of the vacuum. Phase velocity may also depend upon the mode of wave propagation—in general, upon the frequency of the wave. Waves of different frequencies will travel at different speeds, resulting in a phenomenon called dispersion. *See* Electromagnetic radiation, Light, Wave equation

## Phase Velocity

(phase speed), the velocity of propagation of the phase of a harmonic wave. The phase velocity *c* can be expressed in terms of the frequency *f* and the wavelength λ or in terms of the angular frequency ω = 2π*f* and the wave number *k* = 2π/λ: *c* = *f*λ = ω/*k*.

The concept of phase velocity is valid if the harmonic wave is propagated with constant wave form; this condition is always satisfied in linear media. Velocity dispersion occurs if the phase velocity is dependent on frequency or, stated differently, on wavelength. In the absence of dispersion any wave will propagate, without changing its wave form, at a velocity equal to the phase velocity. If dispersion occurs, nonharmonic waves change their wave form, and the conventional concept of velocity no longer applies. In such cases the concepts of group velocity and wave-front velocity become important. The phase velocity for a given frequency can be found experimentally by determining the wavelength in interference experiments. The ratio of phase velocities in two different media can be determined from the refraction of a plane wave at the interface surface of the media, because the refractive index is equal to the ratio of the phase velocities.

M. A. ISAKOVICH