To understand the difference between phase and group velocity of waves, consider the following analogy. Usually, group velocity is not equal to phase velocity, except in empty space. The tangent red line is the phase velocity and the green dashed line is the group velocity. T radians per unit distance radians per unit time phase velocity vp. Relation between group velocity and phase velocity at byjus.
What is the difference between phase velocity and group. In our application, we have velocity code which generates csv and html reports. A full set of computer generated plots of such surfaces are presented. 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. For the love of physics walter lewin may 16, 2011 duration. The phase velocity is given in terms of the wavelength. The idea of a group velocity distinct from a waves phase velocity was first proposed by w. For airfilled waveguide, the phase velocity reduces further to. For quantum electron waves, the group velocity is twice the phase velocity, for water waves, its half the phase velocity.
In a medium, the phase velocity is typically smaller by a factor n, called the refractive index, which is frequencydependent chromatic dispersion. Studies on phase and group velocities from acoustic logging article pdf available in applied geophysics 91. Difference between phase velocity and group velocity velocity. This is not true for light propagating in materials otherwise a prism wouldnt work or most other kinds of waves, such waves in water. Group velocity and phase velocity san jose state university. Group velocity and phase velocity university of british.
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 defined to be the velocity at which the envelope of a wave travels. Velocity with which the phase of a wave travels is called phase velocity. Phase and group angles and velocities in anisotropic wave propagation the phase angles and velocities are different from the group angles and velocities. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g. Particle waves and group velocity university of kansas. Here is a simple way to think about the difference. If we have two waves that are slightly di erent in frequency.
This is the velocity at which the phase of any one frequency component of the wave travels. The relation between group velocity and phase velocity are proportionate. Pdf phase and group velocity matching for cumulative. Quantities have been made dimensionless using the gravitational acceleration g and period t. 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. Group velocity is less than velocity of electromagnetic waves in free space. It is the velocity which the electromagnetic waves changes it phase in the waveguide during propagation. Or is the group velocity a weird result of the frequencywavevector uncertainty and superposition of. Phase and group velocity matching for cumulative harmonic generation in lamb waves article pdf available in journal of applied physics 1099. Group, velocity, phase velocity and dispersion in a wave packet. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science and engineering. On the other hand, the function f is translating to the right with velocity vgroup d.
Pdf studies on phase and group velocities from acoustic. The phase velocity c p blue line and group velocity c g red line as a function of water depth h, for surface gravity waves of constant period according to airy wave theory. This is the velocity of the whole wavepacket, known as the group velocity. Group and phase velocity wolfram demonstrations project. Phase velocity and group velocity in a wave packet physics pages. It is explained and illustrated in this document, using little animations in the pdf le. A summary is presented of the correct interpretations of the phase velocity, ray velocity, and group velocity surfaces. The group velocity is the speed of the constructed wave, created the group of waves. A fundamental relation between phase and group velocity. Group and phase velocity in the spacetime x,t domain. 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.
This is the group velocity and is which can be found if you have giving 7 v. A group of people, say city marathon runners, start. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is defined by chosen here to equal 1, where is the angular velocity and is the wave number. Phase velocity is in opposite direction of group velocity. 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. The phase velocity is not the velocity at which stu. Phase velocity and group velocity for beginners vixra. The distinction between the phase velocity and the group velocity of a wave is a concept of general signi cance for many di erent waves in physics. The phase velocity, ray velocity, and group velocity.
Its an approximation in the sense that it was derived involving approximating the phase in. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, ma 029. 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. The group velocity of a wave is the velocity with which the overall envelope shape of the waves. The group velocity is the speed of the overall shape of a modulated wave called the envelope. 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. The group velocity direction is the direction in which energy travels. 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. Group velocity the phase velocity of a wave is and comes from the change in the position of the wavefronts as a function of time the waveform moves at a rate that depends on the relative position of the component wavefronts as a function of time.
Group velocity and phase velocity of the electromagnetic waves is the same in free space. 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. Difference between phase velocity and group velocity. In the visible spectral region, typical transparent crystals and optical glasses have refractive indices between 1.
From this you can see that when the cutoff frequency is reached, the phase velocity goes to infinity. Does a single photon particle move in the direction of the group velocity or the wave vector. We can obtain phase velocity from solving the wave equation. Rp photonics encyclopedia group velocity, light, envelope. Group velocity is the speed at which electromagnetic information travels. Group delay is the length of a circuit or transmission line, divided by its group velocity. Fundamental relation between phase and group velocity, and. In vacuum, the phase velocity is c 299 792 458 ms, independent of the optical frequency, and equals the group velocity.
Waves, the wave equation, and phase velocity what is a wave. In the top left is a group that moves at the same velocity as the individual wave that makes it. Vary the group velocity away from unity, and carefully follow one of the packets as it moves across the screen, then carefully follow one of the individual crests. 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 group velocity is usually the more important physical velocity.
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. In an anomalous medium, the group velocity is greater than the phase velocity. Group velocity and phase velocity light speed is really. 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 phase velocity is the speed at which one member of the phase group travels.
Group velocity and phase velocity light speed is really the phase velocity v. Pdf is binary format, velocity generates plain text. Generating pdf using itext and an apache velocity template file. The phase velocity surface for waves propagating in a uniform cold plasma is sometimes misinterpreted as having the shape of a wavefront. The phase velocity of a wave is the speed at which a given phase of a wave travels through space, equal to. Any modulation will propagate at the group velocity, which, in dispersive media, is always less than c. 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. An example of an actual physical application in which we must be careful to distinguish between the phase and the group velocity is the case of electromagnetic waves propagating through a hollow magnetic conductor, often called a waveguide. Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f.
Because the stability of an explicit numerical method such as those that we will examine in the rest of this thesis which solves a system of hyperbolic equations is dependent on propagation velocities, it is worthwhile to spend a few moments here to define phase and group velocities 35,101 for a system such as. 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. 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. This velocity is called the group velocity and is usually different from the individual phase velocities of the waves that make up the packet.