For quantum electron waves, the group velocity is twice the phase velocity, for water waves, its half the phase velocity. This article tries to differentiate those differences in detail. A fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1. Jan 20, 20 the resulting wave packet travels with the socalled group velocity which in the above case is faster than the phase velocity. Since many frequencies need not be the same as the group velocity. Group, velocity, phase velocity and dispersion in a wave packet. Phase velocity and group velocity this note is part of the supporting online material for the textbook \solid state physics. 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. The idea of a group velocity distinct from a waves phase velocity was first proposed by w. Lets continue using the superposed wave equation from above. The phase velocity of light is the velocity with which phase fronts propagate in a medium. The group velocity is what has to be considered for the code measurements, i.
Information transmission plane wave does not convey information must be modulated e. The group velocity is directly proportional to phase velocity. Relation between group velocity and phase velocity at byjus. 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. Pdf phase velocity and group velocity in cylindrical and. If we have two waves that are slightly di erent in frequency.
Difference between phase velocity and group velocity velocity. Feb 16, 2019 here is a simple way to think about the difference. Purchase wave propagation and group velocity 1st edition. The group velocity of a wave is the velocity with which the overall envelope shape of the waves. Group velocity and phase velocity university of british. Thus we see direct dependence of group velocity on phase. The qualifier phase is used here, because the speed of a.
This is the velocity at which the phase of any one frequency component of the wave travels. Clearly the velocity of a monochrome light equals its equiphase surface propagation velocity. 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. Particle waves and group velocity university of kansas. The wave packet envelope is shown in magenta color in dashed lines. 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. Pdf closedform expressions are derived for phase velocity and group velocity in cylindrical and spherical sound waves. Negative radiation pressure and negative effective refractive index via dielectric birefringence pdf. 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 phase velocity is the speed at which one member of the phase group travels. Fundamental relation between phase and group velocity.
A fundamental relation between phase and group velocity, and. They arise in quantum mechanics in the time development of the state function for the continuous case, i. In an anomalous medium, the group velocity is greater than the phase velocity. This is defined by chosen here to equal 1, where is the angular velocity and is the wave number. Jun 26, 20 group velocity phase velocity animation case 5.
To begin, lets think about how to solve the wave equation in a dispersive system with initial condition ax,t0fx 9. The phase velocity of a wave is the speed at which a given phase of a wave travels through space, equal to. In nondispersive materials, opposite group and phase velocity corresponds to fields that are mostly oriented in the propagation direction. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g. Pdf fundamental relation between phase and group velocity, and. 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. The group velocity is the speed of the overall shape of a modulated wave called the envelope.
Brillouin, wave propagation and group velocity, academic press, new york, 1960, chapter i. It is a velocity of displacement of the packet as a whole. This notes is meant to be a supplement and not a substitute for. Calculate the group velocity of a relativistic particle. When phase velocity increases, proportionately group velocity will also increase. Difference between phase velocity and group velocity.
Group velocity and phase velocity san jose state university. When the group velocity depends on frequency, this is known as group velocity dispersion, or gvd. Phase control of group velocity of light in an ingangan quantum dot nanostructure article pdf available in quantum electronics 459. 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. This is not true for light propagating in materials otherwise a prism wouldnt work or most other kinds of waves, such waves in water.
Whats the differnce between phase and group velocity. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science and engineering. The ratio between the speed of light c and the phase velocity v p is known as the refractive index, n c v p ck. However, in the case of a superposed wave, we need to carefully define its propagation velocity. 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. Group velocity group delay is the length of a circuit or transmission line, divided by its group velocity. The group velocity is always less than the speed of light, we like to think of that this is because the em wave is pingponging back and forth as it travels down the guide.
Pdf phase control of group velocity of light in an ingan. Phase velocity and group velocity as a function of the total. Phase velocity definition is the velocity of a wave motion as determined by the product of the wavelength and frequency called also wave velocity. In vacuum, the phase velocity is c 299 792 458 ms, independent of the optical frequency, and equals the group velocity.
In order to understand the nature of the group velocity, lets consider a race. Group velocity in a waveguide is speed at which em energy travels in the guide. 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. Phase and group velocity matching for cumulative harmonic generation in lamb waves article pdf available in journal of applied physics 1099. 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. Phase velocity and group velocity in a wave packet physics pages.
Phase and group velocity are two important and related concepts in wave mechanics. 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 phase velocity is given in terms of the wavelength. Pdf we demonstrate that the ratio of group to phase velocity has a simple relationship to the orientation of the electromagnetic field. Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f. Calculate the phase velocity of a relativistic particle.
Light waves in the vacuum have the special property that the velocity is exactly the same for all wavelengths. The group velocity often depends on frequency we have seen that the phase velocity depends on, because n does. 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. To me, it doesnt make much sense what it means for phase velocity to exceed c.
A onedimensional harmonic wave figure 1 is described by the equation. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, ma 029. For such a component, any given phase of the wave for example, the crest will appear to travel at the phase velocity. What is the difference between phase velocity and group velocity. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. Explain why it makes sense that the group velocity is a vector while the phase velocity is not. Pdf phase and group velocity matching for cumulative. As it is explained in the next sections, the troposphere is a non dispersive medium for radio waves at the gnss frequencies.
Although we arent smart enough to bore you with the details of how this is derived order a copy of pozar from our book page, group velocity is slowed by the inverse of the factor that phase velocity is increased. This is the group velocity and is which can be found if you have giving 7 v. The main difference between phase velocity and group velocity is that phase velocity is greater than the group velocity in a normal medium. Rp photonics encyclopedia group velocity, light, envelope. Fundamental relation between phase and group velocity, and. In the top left is a group that moves at the same velocity as the individual wave that makes it. The group velocity is the speed of the constructed wave, created the group of waves. Group and phase velocity wolfram demonstrations project. Assume a particle of mass m and traveling with a speed of v. Group vs phase velocity an analogy that may be useful for understanding the difference comes from velodrome cycling. 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. It is related to the wavenumber and the angular optical frequency.
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