You can nd more information on the distinction between the phase velocity and the group velocity of a wave is a. In an anomalous medium, the group velocity is greater than the phase velocity. Is the velocity of a particle related to the velocity of. To understand the difference between phase and group velocity of waves, consider the following analogy. Phase and group velocity are two important and related concepts in wave mechanics. The carrier wave the blue line travels at the phase velocity and is given by u2. In the case of the wave equation above, the phase velocity and group velocity are both constant, i. Difference between phase velocity and group velocity. We demonstrate that the ratio of group to phase velocity has a simple relationship to the orien tation of the electromagnetic field. Phase velocity definition of phase velocity by merriam. Phase velocity definition is the velocity of a wave motion as determined by the product of the wavelength and frequency called also wave velocity. It is vital to have a solid understanding in both phase velocity and group velocity in order to excel in such fields. Introduction the concept of the group velocity of waves was introduced by hamilton in the.
The group velocity of a wave is the velocity with which the overall envelope shape of the waves amplitudesknown as the modulation or envelope of the wavepropagates through space 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. So how can one talk about a point of constant phase here to obtain the group velocity as it is a product of sin and cos. Here is a simple way to think about the difference. Rp photonics encyclopedia group velocity, light, envelope. The ratio between the speed of light c and the phase velocity v p is known as the refractive index, n cv p ck.
From this you can see that when the cutoff frequency is reached, the phase velocity goes to infinity. Phase velocity and group velocity this note is part of the supporting online material for the textbook \solid state physics. The group velocity can be very different from the phase velocity, only when the phase velocity has a very large derivative. The phase velocity of a guided wave in a wave guide is greater than the speed of light because the wavelength of the guided wave is basically greater then the corresponding wave length in free. Relation between group and particle velocity a material particle in motion is equivalent to a group of waves or wave packet. In the top left is a group that moves at the same velocity as the individual wave that makes it. Thus any wavelike solution to the 1d wave equation travels at a constant speed. In nondispersive materials, opposite group and phase velocity. The group velocity is the velocity of the wave with lower frequency, but the phase velocity is the velocity of the wave with higher frequency. What is the difference between phase velocity and group velocity. Phase velocity the resulting wave may be a perturbation that acts over a short distance, i. General theorems on the equivalence of group velocity and.
For debroglie mass waves, the particle frequency is a linear function of the particle energy e, so it is typical to write the group velocity in the following form. A group of people, say city marathon runners, start. Group velocity and phase velocity are not necessarily the same. A fundamental relation between phase and group velocity, and. This angle is also the angle between the field vectors e and d. Phase velocity vs group velocity group velocity is applicable when there is information or modulation added to a carrier frequency.
Whats the differnce between phase and group velocity. Relation between phase velocity and group velocity learn only in 1. The phase and group velocities are not equal when the wave is dispersive. Group velocity is the speed at which electromagnetic information travels. The phase velocity of a wave is the speed at which a given phase of a wave travels through space, equal to. We demonstrate that the ratio of group to phase velocity has a simple relationship to the orientation of the. 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. Calculate the phase velocity of a relativistic particle. Group velocity and phase velocity of the electromagnetic waves is the same in free space. Group and phase velocity inversions for the general. Difference between phase and group velocity physics youtube.
A fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1. Similarly we can get the group velocity vg by keeping the amplitude a constant. Calculate the group velocity of a relativistic particle. Waveguide group velocity, phase velocity, cut off frequency. Usually, group velocity is not equal to phase velocity, except in empty space. Group and phase velocities in the r theory of matter waves. Nov 18, 2017 relation between phase velocity and group velocity learn only in 1. The momentum carried away slows down the particle, and this result is similar with the previous one, but is obtained from different considerations fig. The envelope of a wave packet see figure on right will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity. Phase velocity is defined as the velocity of a single wave.
The two frequencies are close enough so that they form wave packets, or beats. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science. Why phase velocity is greater than the speed of light. Note that the group velocity is a bit slower than the phase velocity of the individual waves. The phase velocity is given in terms of the wavelength. May 05, 2018 relation between phase velocity and group velocity by education study duration. They play a vital role in fields such as wave mechanics, optics, quantum mechanics and even sound engineering. From this phase velocity, relative dielectric constant of is derived using kobayashis formula. Difference between phase velocity and group velocity pediaa. In schrodinger formulation, zero energy is at zero velocity, and the phase velocity is half the group velocity the group velocity being equal to the particle velocity in all cases that i know of. Is the velocity of a particle related to the velocity of its. The group velocity is simply the derivative of frequency. In a normal medium, the phase velocity is greater than the group velocity.
When one or more waves superimpose together and results in a new wave,the individual velocities adds. Suppose there is a series of ocean waves approaching the shore at an oblique angle that is, they are not coming in perpendicular to the shore line. Group velocity and phase velocity mathematics stack exchange. The envelope the green line is given by u1 and travels at the group velocity. Velocity with which the phase of a wave travels is called phase velocity. This is defined by chosen here to equal 1, where is the angular velocity and is the wave number. A fundamental relation between phase and group velocity. The kinematic definition of group velocity uses the concept of stationary phase, and the velocity of energy pro pagation is defined as the energy ffux divided by the energy density. In this example, the frequency of the second wave is slightly higher than the first.
The product of the phase velocity and the group velocity the product between the phase velocity and the group velocity of any particle massive or massless equals the square of the speed of light in vacuum. They arise in quantum mechanics in the time development of the state function for the continuous case, i. Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f. The group velocity is the speed of the constructed wave, created the group of waves. Relation between group velocity and phase velocity waves can be in the group and such groups are called as wave packets, so the velocity with a wave packet travels is called group velocity. Group and phase velocity wolfram demonstrations project. Group delay is the length of a circuit or transmission line, divided by its group velocity. When a wave is dispersive, a wavepacket will change shape as it propagates.
Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, ma 029. The conditions and sample solutions for superluminal signalling are given. About the propagation of light in dispersive media 1. A onedimensional harmonic wave figure 1 is described by the equation. Clearly the velocity of a monochrome light equals its equiphase surface propagation velocity. The situation is very different in a medium with negative group velocity. Phase velocity definition of phase velocity by merriamwebster. The wavenumber k can be considered as the change in spectral phase per unit length the group velocity is the velocity with which the envelope of a pulse propagates in a medium, assuming a long pulse with narrow bandwidth so that higherorder chromatic dispersion is not relevant and the absence of nonlinear effects i. Relation between group velocity and phase velocity at byjus.
What is the difference between phase velocity and group. Ece theory, r theory, group and phase velocity in r theory, superluminal signalling. Relation between phase velocity and group velocity. Derive an expression for the group velocity of the extraordinary wave in a uniaxial crystal as a function of the polar angle 0 of the propagation vector. Group velocity is less than velocity of electromagnetic waves in free space. For airfilled waveguide, the phase velocity reduces further to. Relation between phase velocity and group velocity by education study duration. However, in the case of a superposed wave, we need to carefully define its propagation velocity.
So vg is the partial derivative of relationship between group velocity vg and phase velocity vp. Understanding heisenberg uncertainty principle duration. 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. Particle waves and group velocity university of kansas. Lets continue using the superposed wave equation from above. Relation between group and particle velocity engineering. The difference between group velocity and phase velocity is most relevant for light that is modulated at high frequencies or for pulses of light.
The group velocity is the speed of the overall shape of a modulated wave called the envelope. The wavenumber k can be considered as the change in spectral phase per unit length the group velocity is the velocity with which the envelope of a pulse propagates in a medium, assuming a long pulse with narrow bandwidth so that higherorder chromatic dispersion is not relevant and the absence of. Fundamental relation between phase and group velocity, and. Since the phase difference between the vibrations is continually changing, the specification of some initial nonzero phase difference is in general not of major. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g.
The phase velocity is the speed at which one member of the phase group travels. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. For, the phase retardation of optical wave at the position is expressed as where if there is the socalled velocity mismatching between the modulation wave and the optical wave, the modulation index becomes a periodical function of with period when a travelingwave eom has a suitable domaininverted halfperiod of, qvm occurs and accordingly a large modulation index is. Derive an expression for the angle a between the phase velocity and the group velocity. The half period of the domain inversion at the modulation frequency of near 16 ghz is derived as mm with the phase velocity of ms and the group velocity of 1. Assume a particle of mass m and traveling with a speed of v. Following the same steps, we get the group velocity of the superposed wave. So, the phase velocity equals to the group velocity. The speed of the wave packets is called the group velocity. As an example, if you send a pulsed carrier thru a network, the group velocity can be derived from the time delay or difference between the input rise time to the output rise time and the distance traveled. The square of the phase velocity c2 is inversely proportional to the mass density r of the rope medium and proportional to the.
Difference between phase velocity and group velocity compare. Well the trick indeed is to recognise the two separate wave components and treat these for now as two separate waves and calculate their phase velocity ie the rate of movement of points of constant phase in each wave. Waves on deep water with surface tension t and density. Difference between phase velocity and group velocity velocity. Find the angle at which a is maximized and obtain an expression for a max. This is typical for normal dispersion, where the speed of waves decreases as frequency increases. Fundamental relation between phase and group velocity. Jun 08, 2007 the phase and group velocities are not equal when the wave is dispersive. Phase velocity of a wave the rate at which the phase of the wave propagates in space group velocity of a wave the velocity with which the overall shape of the waves amplitudes wave packet propagates through space. Phase velocity estimation of a microstrip line in a. This angle is also the angle between the field vectors e.
Group, velocity, phase velocity and dispersion in a wave packet. Phase velocity and group velocity are two very important concepts in physics. Review of basic formulas for waves in spacetime x,t or perspacetime. Relation between phase velocity and group velocity learn. In non dispersive mediums, the phase velocity of waves does not depend on the wavelength. It is the velocity which the electromagnetic waves changes it phase in the waveguide during propagation. Phase velocity and group velocity as a function of the total.
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