Phase velocity

The phase velocity is the velocity of propagation of the same phase of a monochromatic wave. In the case of dispersive media spread monochromatic waves of different frequencies with different phase velocities. Considering wave packets ( waves, which is a sum of components, ie, monochromatic waves ) in dispersive media, then the phase differences of individual components change with time and the shape of the wave changes Pact ( it " melts "). In the top illustration, the red point moves to the phase velocity in the second image, a wave packet is shown, the group velocity is equal to the phase velocities of the individual components. In the third image, the phase velocities of the individual components are different.

The phase velocity is calculated from the wavelength (the distance that is covered ) and the period (the time which is required for this) to

Due to the definitions of frequency, angular frequency and wavenumber circle results in the equivalent representation

The speed of light in vacuum is the maximum transfer rate of energy and information. However, there are numerous cases in which phase velocities occur above the speed of light. Examples are matter waves and waves in waveguides.

Connection with group velocity dispersion and

For the mathematical description of a wave in a special medium is required its waveform, amplitude, frequency, phase angle and the associated wave equation - possibly with boundary conditions. One shaft so clearly defined can still be assigned, which should not be confused with the phase velocity different speeds.

The rate at which to transmit a wave energy or information to the signal speed. This is for a lossless medium is equal to the group velocity, ie the velocity of a wave packet. Such a wave packet is composed of monochromatic waves with different frequencies. Each of these monochromatic waves has its own phase velocity:

The functional relationship between phase velocity and frequency, is referred to as dispersion.

Electromagnetic wave, the phase velocity and the group velocity in vacuum is equal to the speed of light, that is, the vacuum is not dispersive. In matter, the phase velocity is, however, generally depends on the frequency. Due to the relationship for the refractive index here the frequency dependence of the refractive index is referred to as dispersion.

For structure-borne noise

In solids, sound waves can propagate (acoustic emission ), depending on the wave type, the phase velocities are different. For example, the phase velocity of the longitudinal wave at room temperature is made ​​of stainless steel such as 5980 m / s, however, the phase velocity of the transverse wave is smaller by a factor of about 1.8: about 3300 m / s Considering thin plates of thickness, so there are other, more complex types of waves, so-called Lamb waves. The picture to each branch corresponds to a Lambwellentyp (fashion ). In the vertical direction of the phase velocity is shown in units of Transversalwellengeschwindigkeit and horizontal frequency in the form of the product of angular frequency, and the plate thickness in units of Transversalwellengeschwindigkeit. The higher modes exist only above certain minimum frequencies and then propagate with very high phase velocities. Fashion has a vanishing phase velocity for small frequencies.

Example: matter wave

According to de Broglie and the wave -particle duality can be assigned to a particle such as an electron with the energy and the pulse has a wavelength and thus a phase velocity

According to Einstein's formula or in the formulation of the Lorentz factor

Here is the speed of light and is the uppermost limit with which energy or information can spread. is the particle velocity, which is always smaller. It thus follows

That is, the de Broglie phase velocity is always greater than the speed of light. This superluminal velocity of matter waves but does not violate the theory of relativity, since the signal velocity. Also in waveguides waves are transmitted greater the speed of light with phase velocities.