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. In order to understand the nature of the group velocity, lets consider a race. Rayleigh wave phase dashed lines velocity and group solid lines velocity sensitivity. The first experiment, designed for the acquisition of phase velocities, involved the. Group velocity and phase velocity math help boards.
A onedimensional harmonic wave figure 1 is described by the equation. The group velocity is the speed at which energy is transported. The nonrelativistic group velocity and the relativistic group velocity yield exactly the same result. Phase velocity and group velocity in a wave packet 2 x. Why phase velocity is greater than the speed of light inside. Jun 08, 2007 the phase and group velocities are not equal when the wave is dispersive. Pdf phase velocity and group velocity in cylindrical and. Pdf fundamental relation between phase and group velocity, and. This is not true for light propagating in materials otherwise a prism wouldnt work or most other kinds of waves, such waves in water.
Pdf we demonstrate that the ratio of group to phase velocity has a simple. Opti 500 c, spring 2011, lecture 22, chromatic dispersion and pmd 3 phase and group velocity. Nov 18, 2017 relation between phase velocity and group velocity by education study. The group velocity differs from the phase velocity. The concept of the group velocity of waves was introduced by hamilton in the late eighteen thirties and published in abstracts in 1841. The relation between group velocity and phase velocity are proportionate. 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 main difference between phase velocity and group velocity is that phase velocity is greater than the group velocity in a normal medium. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science and engineering. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. You can nd more information on the distinction between the phase velocity and the group velocity of a wave is a. Write each answer as a multiple of the speed of light c. They play a vital role in fields such as wave mechanics, optics, quantum mechanics and even sound engineering.
Group velocity and phase velocity university of british. For such a component, any given phase of the wave for example, the crest will appear to travel at the phase velocity. This term genrally comes into picture when there is a single wave, suppose in optical fiber when there is only one mode i. When we look at waves breaking on a shore, it is obvious that the waves bring with them a lot of energy.
For the love of physics walter lewin may 16, 2011 duration. 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. Assume a particle of mass m and traveling with a speed of v. Relation between group velocity and phase velocity at byjus. Opti 500 c, spring 2011, lecture 22, chromatic dispersion and pmd 4. For the fictitious case of negligible dispersion, the group velocity is identical to the phase velocity, i. The threshold condition is that the velocity of the particle should by equal to the phase velocity of the radiation. You can nd more information on the distinction between the phase velocity and the group velocity. 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.
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. What is the difference between phase velocity and group velocity. This is the velocity at which the phase of any one frequency component of the wave travels. If you have any queries please contact me on my whatsapp number 9416939577. The discovery of cerenkov radiation has a long history which started with the. Is it possible for a group velocity of any wave disturbance to be greater than its component phase. In order to understand the nature of the group velocity. Phase velocity and group velocity this note is part of the supporting online material for the textbook \solid state physics. Phase and group velocity are two important and related concepts in wave mechanics. How does the phase velocity can be greater than the speed of light inside the waveguide. 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.
Simultaneous negative phase and group velocity of light in. 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. What is the difference between phase velocity and group. Group, velocity, phase velocity and dispersion in a wave packet. Little notice was taken of these abstracts until the concept was inferred independently by stokes and by rayleigh. 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. 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. 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. 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. 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 fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1. The group velocity is the speed of the overall shape of a modulated wave called the envelope. Why phase velocity is greater than the speed of light inside the waveguide.
Relation between phase velocity and group velocity by. 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. Wave velocity, when not otherwise specified, usually means the phase velocity. Phase velocity vs group velocity forum for electronics. Group velocity and phase velocity light waves in the vacuum have the special property that the velocity is exactly the same for all wavelengths. The velocity of carrier wave is called the phase velocity. Fundamental relation between phase and group velocity, and.
It is vital to have a solid understanding in both phase velocity and group velocity in order to excel in such fields. The groupvelocity vector determines the direction and speed of energy. This article tries to differentiate those differences in detail. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g. 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. Calculate the phase velocity of a relativistic particle. Pdf phase and group velocity matching for cumulative. Data acquisition for the retrieving group velocities via frequencytime analyses. Opti 500 c, spring 2011, lecture 22, chromatic dispersion and pmd 2 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. Relation between phase velocity and group velocity by education study. Information transmission plane wave does not convey.
You can also directly use the definition of group velocity, i. Phase velocity and group velocity clearly the velocity of a monochrome light equals its equiphase surface propagation velocity. Group and phase velocities in the r theory of matter waves. 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. The inequality of group velocity, speed of light in vacuum, and phase velocity is then. A fundamental relation between phase and group velocity, and. The group velocity and the phase velocity are two different ways to characterize how fast a wave is moving through space. Recently, zhou and wu 52 proposed to combine the pml with.
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. 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. Hi, im trying to figure out what the group and phase velocities of a wave packet describing a particle are in the relativistic case. Dec 06, 2016 the group velocity and the phase velocity are two different ways to characterize how fast a wave is moving through space. Group and phase velocity wolfram demonstrations project. 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. These are plotted and compared for orders 0, 1 and 2, but the expressions are.
Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f. Pdf on mar 2, 2015, daniel munozsantiburcio and others published. In crystal17, phase and group elastic wave velocities, as well as the related power. Phase and group velocity matching for cumulative harmonic generation in lamb waves article pdf available in journal of applied physics 1099. Particle waves and group velocity university of kansas. Review of basic formulas for waves in spacetime x,t or perspacetime. Phase velocity and group velocity, relation between phase.
This site animates the differences between phase and group velocities. They arise in quantum mechanics in the time development of the state function for the continuous case, i. Lee discusses the phenomenon of dispersion in a realistic medium and the strategy to describe this kind of physics situation. Closedform expressions are derived for phase velocity and group velocity in cylindrical and spherical sound waves. When the group velocity depends on frequency, this is known as group velocity dispersion, or gvd. However, in the case of a superposed wave, we need to carefully define its propagation velocity.
For hyperbolic systems, such as the wave equation, it must be true that the group velocity is bounded from above i. The group velocity is equal to the real velocity of the particle in both cases. 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. The group velocity also has relation with the phase velocity. Effect of group velocity mismatch 1277 in order to justify this approach, let us consider the follow ing numerical example. 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. About the nonrelativistic limit of the phase velocity of. The group velocity often depends on frequency we have seen that the phase velocity depends on, because n does. The phase velocity of a wave is the speed at which a given phase of a wave travels through space, equal to. In contrast, the nonrelativistic phase velocity and the relativistic phase velocity are not.
Difference between phase velocity and group velocity. Here is a simple way to think about the difference. I started with the relationship between energy and impulse. When a wave is dispersive, a wavepacket will change shape as it propagates. Going back to the case of airfilled waveguide, the group velocity is. Phase velocity and group velocity are two very important concepts in physics. Velocity with which the phase of a wave travels is called phase velocity. Calculate the group velocity of a relativistic particle. A code for the calculation of phase and group velocities. 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. In the presence of dispersion, the second term in the denominator. The angle insider the cos term in a certain phase is.
Group velocity of relativistic wave packet physics forums. The envelope the green line is given by u1 and travels at the group velocity. Sending a square pulse as a basic communication tool is the main focus of this lecture. A wave is essentially some kind of oscillation which propagates through space. Dispersive medium, phase velocity, group velocity mit opencourseware. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, ma 029. Feb 16, 2019 here is a simple way to think about the difference. My groupvelocity inversion overcomes the difficulty of calculating group. Group and phase velocity inversions for the general.
1180 429 1307 647 1406 547 440 1040 745 1255 1308 1030 868 473 1038 904 474 894 25 1157 1197 773 1169 303 804 911 1264 694 418 772 1310 1020 680 1398 1234 1317 950 214