In the presence of dispersion, the second term in the denominator. Calculate the phase velocity of a relativistic particle. Phase velocity and group velocity, relation between phase. Dec 06, 2016 the group velocity and the phase velocity are two different ways to characterize how fast a wave is moving through space. When we look at waves breaking on a shore, it is obvious that the waves bring with them a lot of energy. Pdf fundamental relation between phase and group velocity, and.
When the group velocity depends on frequency, this is known as group velocity dispersion, or gvd. The group velocity often depends on frequency we have seen that the phase velocity depends on, because n does. These are plotted and compared for orders 0, 1 and 2, but the expressions are. The groupvelocity vector determines the direction and speed of energy. 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. You can nd more information on the distinction between the phase velocity and the group velocity. Without quantum mechanics a particle is not considered a wave so there is no group velocity or phase velocity, only the velocity of the particle. This is defined by chosen here to equal 1, where is the angular velocity and is the wave number. This is the velocity at which the phase of any one frequency component of the wave travels. Group and phase velocities in the r theory of matter waves.
Group and phase velocity wolfram demonstrations project. 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. About the nonrelativistic limit of the phase velocity of. The relation between group velocity and phase velocity are proportionate. You can nd more information on the distinction between the phase velocity and the group velocity of a wave is a.
Effect of group velocity mismatch 1277 in order to justify this approach, let us consider the follow ing numerical example. The group velocity is the speed of the overall shape of a modulated wave called the envelope. Phase velocity vs group velocity forum for electronics. Phase and group velocity matching for cumulative harmonic generation in lamb waves article pdf available in journal of applied physics 1099. What is the difference between phase velocity and group. Phase velocity and group velocity this note is part of the supporting online material for the textbook \solid state physics. Opti 500 c, spring 2011, lecture 22, chromatic dispersion and pmd 3 phase and group velocity.
The group velocity can be very different from the phase velocity, only when the phase velocity has a very large derivative. Phase velocity and group velocity for beginners in the first section of this paper i derive the formulas for the phase velocity and group velocity as a function. The group velocity and the phase velocity are two different ways to characterize how fast a wave is moving through space. Phase velocity and group velocity clearly the velocity of a monochrome light equals its equiphase surface propagation velocity. It can be seen that this implies that group velocity is less than the speed of light, while phase velocity is greater than the speed of light. Rp photonics encyclopedia group velocity, light, envelope. What is the difference between phase velocity and group velocity. The group velocity differs from the phase velocity. Assume a particle of mass m and traveling with a speed of v.
The inequality of group velocity, speed of light in vacuum, and phase velocity is then. 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. Dispersive medium, phase velocity, group velocity mit opencourseware. Closedform expressions are derived for phase velocity and group velocity in cylindrical and spherical sound waves. Simultaneous negative phase and group velocity of light in. Nov 01, 2005 phase and group angles and velocities in anisotropic wave propagation the phase angles and velocities are different from the group angles and velocities. Group velocity and phase velocity math help boards. For the love of physics walter lewin may 16, 2011 duration. Pdf we demonstrate that the ratio of group to phase velocity has a simple. Recently, zhou and wu 52 proposed to combine the pml with. Rayleigh wave phase dashed lines velocity and group solid lines velocity sensitivity. For such a component, any given phase of the wave for example, the crest will appear to travel at the phase velocity.
Sending a square pulse as a basic communication tool is the main focus of this lecture. However, in the case of a superposed wave, we need to carefully define its propagation velocity. Pdf phase and group velocity matching for cumulative. Jun 08, 2007 the phase and group velocities are not equal when the wave is dispersive. The velocity of carrier wave is called the phase velocity. Fundamental relation between phase and group velocity, and. Why phase velocity is greater than the speed of light inside. Relation between group velocity and phase velocity at byjus. This term genrally comes into picture when there is a single wave, suppose in optical fiber when there is only one mode i. In this paper, first, i derive the formulas for the phase velocity and group velocity as a function of the total relativistic energy and the momentum of a particle.
I started with the relationship between energy and impulse. Jan 20, 20 here, we demonstrate the group and phase velocity phenomena as observed when two signals with different temporal and spectral frequencies wavelength are added to each other. Here, we demonstrate the group and phase velocity phenomena as observed when two signals with different temporal and spectral frequencies wavelength are added to each other. Group velocity when the various frequency components of a waveform have different phase velocities, the phase velocity of the waveform is an average of these velocities the phase velocity of the carrier wave, but the waveform itself moves at a different speed than the underlying carrier wave called the group velocity. Relation between phase velocity and group velocity by. Group velocity of relativistic wave packet physics forums. In order to understand the nature of the group velocity, lets consider a race. Group velocity and phase velocity university of british. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. In crystal17, phase and group elastic wave velocities, as well as the related power.
The threshold condition is that the velocity of the particle should by equal to the phase velocity of the radiation. Here is a simple way to think about the difference. Group and phase velocity of love waves propagating in elastic functionally graded materials piotr kielczynski, marek szalewski, andrzej balcerzak, krzysztof wieja institute of fundamental technological research polish academy of sciences pawinskiego 5b, 02106 warszawa, poland. Write each answer as a multiple of the speed of light c.
Pdf phase velocity and group velocity in cylindrical and. A wave is essentially some kind of oscillation which propagates through space. It is vital to have a solid understanding in both phase velocity and group velocity in order to excel in such fields. Velocity with which the phase of a wave travels is called phase velocity. Opti 500 c, spring 2011, lecture 22, chromatic dispersion and pmd 2 phase and group velocity. This site animates the differences between phase and group velocities. 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. 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. A onedimensional harmonic wave figure 1 is described by the equation. Phase velocity and group velocity are two very important concepts in physics. The group velocity is the speed at which energy is transported. Pdf on mar 2, 2015, daniel munozsantiburcio and others published. How does the phase velocity can be greater than the speed of light inside the waveguide.
The nonrelativistic group velocity and the relativistic group velocity yield exactly the same result. Wave velocity, when not otherwise specified, usually means the phase velocity. For hyperbolic systems, such as the wave equation, it must be true that the group velocity is bounded from above i. The discovery of cerenkov radiation has a long history which started with the. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g.
Nov 18, 2017 relation between phase velocity and group velocity by education study. When a wave is dispersive, a wavepacket will change shape as it propagates. Feb 16, 2019 here is a simple way to think about the difference. Little notice was taken of these abstracts until the concept was inferred independently by stokes and by rayleigh. Is it possible for a group velocity of any wave disturbance to be greater than its component phase. Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f. Group and phase velocity inversions for the general.
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. The group velocity is equal to the real velocity of the particle in both cases. Group and phase velocity of love waves propagating in elastic. In contrast, the nonrelativistic phase velocity and the relativistic phase velocity are not. 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. You can also directly use the definition of group velocity, i. The angle insider the cos term in a certain phase is.
The main difference between phase velocity and group velocity is that phase velocity is greater than the group velocity in a normal medium. Information transmission plane wave does not convey. 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. Particle waves and group velocity university of kansas. 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.
The first experiment, designed for the acquisition of phase velocities, involved the. They arise in quantum mechanics in the time development of the state function for the continuous case, i. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science and engineering. The energy content in an infinite plane wave is obviously infinite, so we are more interested in finding the energy per unit area of the surface. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, ma 029. Difference between phase velocity and group velocity. In order to understand the nature of the group velocity. Why phase velocity is greater than the speed of light inside the waveguide. The envelope the green line is given by u1 and travels at the group velocity. Review of basic formulas for waves in spacetime x,t or perspacetime. Going back to the case of airfilled waveguide, the group velocity is. My groupvelocity inversion overcomes the difficulty of calculating group. A code for the calculation of phase and group velocities.
Phase velocity and group velocity in a wave packet 2 x. They play a vital role in fields such as wave mechanics, optics, quantum mechanics and even sound engineering. The concept of the group velocity of waves was introduced by hamilton in the late eighteen thirties and published in abstracts in 1841. Lee discusses the phenomenon of dispersion in a realistic medium and the strategy to describe this kind of physics situation. Calculate the group velocity of a relativistic particle. You can say that the velocity of the particle cant be higher than the speed of light without quantum mechanics, but then there is no group velocity. Phase velocity and group velocity for beginners in the first section of this paper i derive the formulas for the phase velocity and group velocity as a function of the total relativistic energy and the momentum of a particle. The group velocity also has relation with the phase velocity.
If you have any queries please contact me on my whatsapp number 9416939577. This is not true for light propagating in materials otherwise a prism wouldnt work or most other kinds of waves, such waves in water. Opti 500 c, spring 2011, lecture 22, chromatic dispersion and pmd 4. This article tries to differentiate those differences in detail. Phase and group velocity are two important and related concepts in wave mechanics. Relation between phase velocity and group velocity by education study. Group, velocity, phase velocity and dispersion in a wave packet. For the fictitious case of negligible dispersion, the group velocity is identical to the phase velocity, i. The phase velocity of a wave is the speed at which a given phase of a wave travels through space, equal to. A fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1. Hi, im trying to figure out what the group and phase velocities of a wave packet describing a particle are in the relativistic case.
1004 1065 91 76 193 787 292 1470 222 1031 737 300 880 387 1481 1425 256 518 73 295 1505 1245 821 260 1333 1361 726 763 659 14 597