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. Sending a square pulse as a basic communication tool is the main focus of this lecture. Phase velocity and group velocity for beginners vixra. This is the velocity at which the phase of any one frequency component of the wave travels. We can obtain phase velocity from solving the wave equation. This is meant to serve as a general guide, as rock types are classified by a range of physical properties values which can lead to order of magnitude differences in phase 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. Nov 18, 2019 phase velocity last updated november 18, 2019 frequency dispersion in groups of gravity waves on the surface of deep water. Waves, the wave equation, and phase velocity what is a wave. What i dont understand is why phase velocity does not carry informationenergy where as group velocity does. In a wave packet, the carrier wave travels with the phase velocity.
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. Negative radiation pressure and negative effective refractive index via dielectric birefringence pdf. Then, to obtain the wave equation, we need to consider the maxwells equation. 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. Velocity and phase velocity of matter wave physics forums. Relation between group velocity and phase velocity at byjus. This is the group velocity and is which can be found if you have giving 7 v. The latter again is a function of the permittivity of the medium, we have already derived in chapter 3 the refractive index as a function of medium permittivity, given by n v. Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f. Fundamental relation between phase and group velocity, and. If we have two waves that are slightly di erent in frequency. The velocity of a wave can be defined in many different ways, partly because there many different kinds of waves, and partly because we can focus on different aspects or components of any given wave.
In the presence of dispersion, the second term in the denominator. Simultaneous negative phase and group velocity of light in. The group velocity of a wave is the velocity with which the overall envelope shape of the waves. Relation between phase velocity and group velocity. The group velocity is what has to be considered for the code measurements, i. 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. In an anomalous medium, the group velocity is greater than the phase velocity. 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.
Light waves in the vacuum have the special property that the velocity is exactly the same for all wavelengths. Phase velocity and group velocity are two very important concepts in physics. Calculate the group velocity of a relativistic particle. May 12, 2006 the group velocity v group can be expressed as. Phase velocity and group velocity as a function of the total relativistic energy and the relativistic momentum of a particle 2. 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 idea of a group velocity distinct from a waves phase velocity was first proposed by w. Phase velocity and group velocity superposition of light waves when we analyze the resultant of two or more simple harmonic vibrations monochrome light waves, we usually can treat it as the simple sum of the individual vibrations waves as shown below. In its most general form, the groupvelocity vector can be written as e. Phase and group velocity matching for cumulative harmonic generation in lamb waves article pdf available in journal of applied physics 1099. This is not true for light propagating in materials otherwise a prism wouldnt work or most other kinds of waves, such waves in water. Phase velocity definition of phase velocity by merriam. Sep 12, 2012 v v g itex\frac\partial \omega\partial kitex, that is, group velocity equals the velocity of the particle associated with the wave or wave packet.
When phase velocity increases, proportionately group velocity will also increase. 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 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. Phase velocity and group velocity as a function of the angular frequency and the wave number 4. Unlike phase velocity, which can be obtained directly from the christoffel equation, group velocity depends on the phasevelocity function and, in some representations, on the polarization vector.
Here is a simple way to think about the difference. The red square overtakes two green circles when moving from the left to the right of the figure. Thus we see direct dependence of group velocity on phase velocity and viseversa. 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. You can nd more information on the distinction between the. 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. Simultaneous negative phase and group velocity of light in a. Usually, group velocity is not equal to phase velocity, except in empty space. Oct 26, 2005 what i dont understand is why phase velocity does not carry informationenergy where as group velocity does. The latter again is a function of the permittivity of the medium, we have already derived in chapter 3 the refractive index as a function of medium permittivity, given by n vas the permittivity of the medium is different from that of the vacuum, the waves move with different phase velocity in the. 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. Phase velocity and group velocity this note is part of the supporting online material for the textbook \solid state physics.
To understand the difference between phase and group velocity of waves, consider the following analogy. Calculate the phase velocity of a relativistic particle. Review of basic formulas for waves in spacetime x,t or perspacetime. It is vital to have a solid understanding in both phase velocity and group velocity in order to excel in such fields. For the fictitious case of negligible dispersion, the group velocity is identical to the phase velocity, i. The relation between group velocity and phase velocity are.
It is this modulation that represents the signal content. 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. The group velocity is directly proportional to phase velocity. Difference between phase velocity and group velocity. In this deepwater case, the phase velocity is twice the group velocity. A fundamental relation between phase and group velocity. Since each amplitude envelope contains a group of internal waves, this speed is usually called the group velocity, vg. A fundamental relation between phase and group velocity, and. Nov 18, 2017 relation between phase velocity and group velocity learn only in 1. Phase velocity and group velocity in a wave packet physics pages. That is, to within a phase factor of unit modulus the waveform fx,t is a function of a single variable,x. Signal velocity can be different from group velocity and doesnt necessarily have the invariant. Difference between phase velocity and group velocity velocity. Phase and group velocities article pdf available in computer physics communications 207 june 2016 with 1,273 reads how we measure reads.
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. 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. Pdf phase velocity and group velocity bs physics students. The speed of the monochromatic wave is just the phase velocity of the main wave. The table below shows phase velocity for certain rocks at various frequencies. For such a component, any given phase of the wave for example, the crest will appear to travel at the phase velocity. In its most general form, the group velocity vector can be written as e. They play a vital role in fields such as wave mechanics, optics, quantum mechanics and even sound engineering. Group and phase velocity up to now we have just talked about the speed or velocity v of a wave. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, ma 029.
Group velocity and phase velocity university of british. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g. 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. This velocity is called the group velocity and is usually different from the individual phase velocities of the waves that make up the packet. The phase velocity wave velocity of a wave is the velocity with which any given phase of the wave travels. To me, it doesnt make much sense what it means for phase velocity to exceed c.
I look at it as k wave number approaching 0 so wavelength goes to infinity. They arise in quantum mechanics in the time development of the state function for the continuous case, i. You can nd more information on the distinction between the phase velocity and the group velocity of a wave is a. In the context of gnss signals, the carrier waves are propagating with the phase velocity and thence, this is the velocity associated to the carrier measurements. The red square moves with the phase velocity, and the green circles propagate with the group velocity. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science and engineering.
The phase velocity is given in terms of the wavelength. As a result, the peak of a temporally long gaussian. A group of people, say city marathon runners, start. Or, equivalently, in terms of the waves angular frequency. 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. Unlike phase velocity, which can be obtained directly from the christoffel equation, group velocity depends on the phase velocity function and, in some representations, on the polarization vector. The phase velocity is not the velocity at which stu.
Phase velocity and group velocity in a wave packet 2 x. Signal velocity can be different from group velocity and doesnt necessarily have the invariant relationship group velocity has with phase velocity. Particle waves and group velocity university of kansas. Apr 15, 2016 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 phase velocity, ray velocity, and group velocity. Assume a particle of mass m and traveling with a speed of v. Schuster with noting in 1904 that a negative group velocity, i. Walker department of physics, university of natal, durban, 4001, south africa received 22 june 1976 and in revised form 25 october 1976 the phase velocity surface for. What is the difference between phase velocity and group. Pdf phase and group velocity matching for cumulative. Relation between phase velocity and group velocity learn only in 1. A onedimensional harmonic wave figure 1 is described by the equation. The phase velocity, ray velocity, and group velocity surfaces for a magnetoionic medium by a.
Velocity with which the phase of a wave travels is called phase velocity. The propagation of waves through dispersive media often leads to surprising or counterintuitive behavior. A fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1. The phase and group velocity of the wave is dependent upon the refractive index, n. Lee discusses the phenomenon of dispersion in a realistic medium and the strategy to describe this kind of physics situation.
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