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. Modified definition of group velocity and electromagnetic. Volume 119, issue 3, 22 february 2008, pages 122126. The classical definition of group velocity has two flaws. Line sizing and pressure drop calculation for two phase fluid flowing through a horizontal pipe. To me, it doesnt make much sense what it means for phase velocity to exceed c. I am getting confused between the concept of group velocity vs. A group of people, say city marathon runners, start. Phase velocity group velocity, difference between group velocity and phase velocity phase velocity the velocity of component waves of a wave packet is called the phase velocity of those waves. To understand the difference between phase and group velocity of waves, consider the following analogy. Kishk university of mississippi usa abstractwe study theoretically the propagation of electromagnetic waves in an in. The propagation of waves through dispersive media often leads to surprising or counterintuitive behavior. Particle waves and group velocity university of kansas.
Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science and engineering. Introduction the concept of the group velocity of waves was introduced by hamilton in the. Group and phase v elo cities if w e ha v et w ow a es that are sligh tly dieren t in frequency and phase constan t a linear sup erp osition of them is still solution the w v e equation e x cos t z if then e x cos t z cos using the fact that cos a b cos cos cos w e ha v e x cos t z cos or e x z t cos t z a t w e ha v e x cos z cos whic h is sk. Ece theory, r theory, group and phase velocity in r theory, superluminal signalling. Calculate the phase velocity of a relativistic particle. Phase velocity definition of phase velocity by merriamwebster. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g. Waves on deep water with surface tension t and density. A fundamental relation between phase and group velocity. Assume a particle of mass m and traveling with a speed of v. 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. Calculate the group velocity of a relativistic particle. In an anomalous medium, the group velocity is greater than the phase velocity.
This is the velocity at which the phase of any one frequency component of the wave travels. A onedimensional harmonic wave figure 1 is described by the equation. Phase and group angles and velocities in anisotropic wave propagation the phase angles and velocities are different from the group angles and velocities. This is typical for normal dispersion, where the speed of waves decreases as frequency increases. In the case of the wave equation above, the phase velocity and group velocity are both constant, i. 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 and phase velocity mathematics stack exchange. Pdf determination of group velocity of propagation of. You can nd more information on the distinction between the phase velocity and the group velocity of a wave is a. Thus any wavelike solution to the 1d wave equation travels at a constant speed.
Pdf phase and group velocity matching for cumulative. Phase matching limitation of highefficiency secondharmonic generation in both phase and groupvelocitymatched structures. The envelope the green line is given by u1 and travels at the group velocity. Difference between phase velocity and group velocity velocity.
The group velocity is the speed of the overall shape of a modulated wave called the envelope. The group velocity is positive, while the phase velocity is negative. The principles of singlephase flow fluid dynamics and heat transfer are relatively well understood, however twophase flow thermofluid dynamics is an order of magnitude more complicated subject than that of the singlephase flow due to the existence of moving and deformable interface and its interactions with the two phases. Pdf we demonstrate that the ratio of group to phase velocity has a simple. The conditions and sample solutions for superluminal signalling are given. Group and phase velocities in the r theory of matter waves. The phase velocity, v, is the velocity of the wave normal to the wavefront and the. Explain why it makes sense that the group velocity is a vector while the phase velocity is not. By combining two sines with slightly different frequencies and wavelengths. Phase velocity and group velocity this note is part of the supporting online material for the textbook \solid state physics.
A fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1. Since in the presence of anisotropy the wavefront is not spherical, the group and phase velocity vectors generally are different. The concepts of phase and group velocity relates to the propagation of. 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. Pdf the concepts of phase, group, and front velocities introduced by. So you get the same answer in the relativistic case as in the classical case for group velocity, even if you use total energy instead of just kinetic. What i dont understand is why phase velocity does not carry informationenergy where as group velocity does.
Oct 26, 2005 what i dont understand is why phase velocity does not carry informationenergy where as group velocity does. Here is a simple way to think about the difference. Difference between phase velocity and group 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. Group and phase velocity wolfram demonstrations project. Fundamental relation between phase and group velocity, and. Pdf fundamental relation between phase and group velocity, and. Note, first, that the issue you describe only arises for phase 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. We shall identify e with the energy density per unit distance and give a proof that in all cases considered the group velocity is given by vg ge, 2. Simultaneous negative phase and group velocity of light in. For the case of positiverefractiveindex media in electrodynamics, where the phase velocity v phase is positive in the forward direction, the group velocity v group can become negative in the regime of anomalous dispersion. What is the difference between phase velocity and group velocity. You can also directly use the definition of group velocity, i. 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. Wavepacket and dispersion, andreas wacker, lund university, september 18, 2017 4 to onedimensional signals, where only the zdirection matters, so that we have a scalar k. Group, velocity, phase velocity and dispersion in a wave packet. Phase and group velocity matching for cumulative harmonic generation in lamb waves article pdf available in journal of applied physics 1099. You can nd more information on the distinction between the. The phase velocity of a wave is the rate at which the phase of the wave propagates in space.
For group velocity, since you are taking the derivative of energy, any constant term, such as the rest energy, drops out. However this result is not physical because all component waves with different. 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. 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. Usually, group velocity is not equal to phase velocity, except in empty space. Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f. To remove the flaws, a modified definition is proposed. This shows a wave with the group velocity and phase velocity going in different directions. Feb 16, 2019 here is a simple way to think about the difference.
Phase and group velocity are two important and related concepts in wave mechanics. What is the difference between phase velocity and group. The group traveltime is defined simply as the traveltime that will yield a group velocity when divided into the distance between source and receiver. How come group and phase velocity can be different in a nondispersive media. The group velocity was switched from subluminal to superluminal through a continuous change of the phase. For such a component, any given phase of the wave for example, the crest will appear to travel at the phase velocity. Relationship between group velocity and group delay. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, ma 029. Phase velocity definition is the velocity of a wave motion as determined by the product of the wavelength and frequency called also wave velocity. Phase matching limitation of highefficiency second.
Lets assume that some channel has same group velocity and phase velocity then does this mean that group delay and phase delay are same for that channel. Aug 11, 2012 phase velocity group velocity, difference between group velocity and phase velocity phase velocity the velocity of component waves of a wave packet is called the phase velocity of those waves. The phase velocity of a wave is the speed at which a given phase of a wave travels through space, equal to. 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. For quantum electron waves, the group velocity is twice the phase velocity, for water waves, its half the phase 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. Since each amplitude envelope contains a group of internal waves, this speed is usually called the group velocity, vg. About the propagation of light in dispersive media 1. The phase velocity is given in terms of the wavelength. This is defined by chosen here to equal 1, where is the angular velocity and is the wave number. 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. Explain why it makes sense that the group velocity is a. 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.
As a result, the peak of a temporally long gaussian pulse. The measurement of lamb waves group velocities was performed on a plate with dimensions 800. Phase velocity is the propagation velocity of an equiphase wave surface of a single sinusoidal wave along a transmission line, and group velocity is the propagation velocity of the envelope of two sinusoidal waves with a slight difference in frequency superimposed. Note that the group velocity is a bit slower than the phase velocity of the individual waves. In this example, the frequency of the second wave is slightly higher than the first. These velocities can be described with secondary constants of the transmission. Phase velocity definition of phase velocity by merriam. A fundamental relation between phase and group velocity, and.
For hyperbolic systems, such as the wave equation, it must be true that the group velocity is bounded from above i. 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. It is this modulation that represents the signal content. They arise in quantum mechanics in the time development of the state function for the continuous case, i. General theorems on the equivalence of group velocity and. Group and phase velocity inversions for the general. 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. Pdf control of group velocity by phasechanging collisions. This is the group velocity and is which can be found if you have giving 7 v. The two frequencies are close enough so that they form wave packets, or beats. I look at it as k wave number approaching 0 so wavelength goes to infinity.
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