# Phase Velocity And Group Velocity Pdf

For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. If the waves are propagating through an anisotropic i. Simply, the velocity of each individual runner within the group is different. In this deep-water case, the phase velocity is twice the group velocity.

But, they run within a group. This dual nature of particles and waves allow us to represent a particle in terms of waves.

For example, the phase velocity is equal to the group velocity when sound waves travel through the air as air is a non- dispersive medium for sound waves. This is also a consequence of the linear dispersion that makes all partial waves travel with the same velocity. Remember me on this computer. As a result, the envelope of the wave packet not only moves, but also distorts, in a manner that can be described by the material's group velocity dispersion. The peaks of wavepackets were also seen to move faster than c. The group velocity is, of course, not the speed of light but it is given by the degree of dispersion and the lattice constant.

## BS Physics Students

From Wikipedia, the free encyclopedia. For this, we need a linear dispersion. Then, the imaginary part of the wavevector is arbitrarily discarded and the usual formula for group velocity is applied to the real part of wavevector, i. The longer waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group. In a wave packet, the carrier wave travels with the phase velocity.

## Difference Between Phase Velocity and Group Velocity

This property of particles and waves is known as the wave- particle duality. In order to convey information, superposition of many waves which gives a wave packet is needed. The shorter waves travel faster than the group as a whole, but their amplitudes diminish as they approach the leading edge of the group.

Waves Berkeley Physics Course, Vol. It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket. According to quantum physics, particles sometimes behave like waves. In quantum physics, the idea of group velocity comes with the idea of wave packets or pulses. The envelope moves at the group velocity. In other words, the velocity of the particle is equal to the group of the waves with which the wave packet is made of. The velocity of the group or wave packet is called the group velocity. Then we form a package of such waves centered around a k-point of interest.

The above generalization of group velocity for complex media can behave strangely, and the example of anomalous dispersion serves as a good illustration. The resultant wave packet represents the particle, and it is a group of many waves. The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. However, we must add many waves, having different angular frequencies and amplitudes in particular ways, in order to get a wave packet which represents a particle. ## Difference Between Phase Velocity and Group Velocity

For a linear dispersion, e. Copyright C by Philip Hofmann. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon. Such a situation is especially important when the linear dispersion is situated at the Fermi energy.

## PDF) Phase velocity and group velocity You have to click on the images to start the animations. The most important example for this appears in the electronic band structure of graphene, a single layer of carbon atoms.

As the band disperses only over a limited energy range, the curvature has to change from positive to negative and thus it becomes nearly zero for a small range of k values. So, the position of a particle can be localized within a specific region by using such a wave packet which represents a particle. The only important difference is that the waves do not propagate with the speed of light but with the speed of sound.

So in this model, the velocity of the particle is equal to the velocity of the wave packet. So, the phase velocity equals to the group velocity. Now the partial wave do not all move with the same velocity because of the quadratic dispersion. In non- dispersive mediums, the phase velocity of waves does not depend on the wavelength. Nevertheless, this wave packet is a combination of many waves. Within this concept, decompress pdf file there are two types of velocities namely phase velocity and group velocity. The relationship between the phase velocity and the group velocity depends on the material properties of the medium. The main difference between Phase Velocity and Group Velocity is that phase velocity is greater than the group velocity in a normal medium.

At any given instant, the velocity of a particular runner is different relative to others. For the first, we can choose a wave package around a small k value where the dispersion is linear. The envelope of the wave packet. Introduction to Quantum Mechanics. Based on this concept, waves are used to represent particles as it is an essential step when we deal with particles in quantum physics.

In other projects Wikimedia Commons. The Quantum Theory of Light. 