This notes is meant to be a supplement and not a substitute for. Phase velocity and group velocity for beginners vixra. Simultaneous negative phase and group velocity of light in. Phase velocity definition, the velocity with which a simple harmonic wave is propagated, equal to the wavelength divided by the period of vibration. 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. Using a linearized approximation for the quasicompressional phase velocity, vqp in an orthorhombic anisotropic medium, which is a subset of the related quasicompressional qp wave propagation in a general 21 parameter anisotropic medium, a linearized compressional group velocity may be derived as a function of group angles only. 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. Rayleigh waves can be generated by mechanical sources applied at the boundary or in the interior of a halfspace. A criterion is set up to identify the justification of group velocity definition. If the two components are close in frequency, it makes sense to define the average wavenumber k0 and the. Group velocity, phase velocity, phase velocity and group velocity, relation between phase velocity and group velocity 14 comments. In the visible spectral region, typical transparent crystals and optical glasses have refractive indices between 1. Or is the group velocity a weird result of the frequencywavevector uncertainty and superposition of.
As a result, the peak of a temporally long gaussian. Oskooi,2 mihai ibanescu,3 maksim skorobogatiy,4 and steven g. A onedimensional harmonic wave figure 1 is described by the equation. Phase velocity and group velocity are two very important concepts in physics. Fundamental relation between phase and group velocity, and. Rayleigh and love waves in this period range are most sensitive to the shear velocity structure of the lithosphere and can be used in combination. Signal velocity can be different from group velocity and doesnt necessarily have the invariant relationship group velocity has with phase velocity. I think i understand what group velocity is, its the velocity of the overall wavepacket, in the diagram i can see a gaussian wavepacket moving to the right so im guessing thats the group velocity.
Here x is the position at which the disturbance yx, t exists at time t. In one dimension, for example, the form of the disturbance for such a wave is given by yx, t a sin 2. The group velocity also has relation with the phase velocity. The idea of a group velocity distinct from a waves phase velocity was first proposed by.
Group velocity is the speed at which electromagnetic information travels. A linearized group velocity approach for two point qp ray tracing in a layered orthorhombic medium patrick f. Relationship between group velocity vg and phase velocity vp. A superluminal power flow is constructed to show that the electromagnetic energy. Review of basic formulas for waves in spacetime x,t or perspacetime. Pdf phase velocity and group velocity in cylindrical and.
Phase velocity curves are constructed in the range of 10 to 120 seconds. In vacuum, the phase velocity is c 299 792 458 ms, independent of the optical frequency, and equals the group velocity. Relation between phase velocity and group velocity learn only in 1. They arise in quantum mechanics in the time development of the state function for the continuous case, i. Phase velocity is in opposite direction of group velocity. Pdf closedform expressions are derived for phase velocity and group velocity in cylindrical. They play a vital role in fields such as wave mechanics, optics, quantum mechanics and even sound engineering.
For a spherical wave, we can similarly define a wave number. It is vital to have a solid understanding in both phase velocity and group velocity in. Phase velocity and group velocity as a function of the angular frequency and the wave number 4. Generally, the phase velocity of a wave is faster than the group velocity and depends on the angle of the approach.
The propagation of waves through dispersive media often leads to surprising or counterintuitive behavior. Pdf phase velocity approximations in a transversely. Phase velocity and group velocity as a function of the total relativistic energy and the relativistic momentum of a particle 2. To remove the flaws, a modified definition is proposed. 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. For such a component, any given phase of the wave for example, the crest will appear to travel at the phase velocity. Velocity and phase velocity of matter wave physics forums. What is the difference between phase velocity and group velocity. What is the difference between phase velocity and group. The phase velocity of the resulting waveform is the apparent or effective rayleighwave phase velocity. When the stone hits the surface of the water, a circular pattern of waves appears. Johnson1 1department of mathematics, massachusetts institute of technology, cambridge, massachusetts 029, usa 2center for materials science and engineering. Group velocity definition of group velocity by merriam. When phase velocity increases, proportionately group velocity will also increase.
Twostation phase velocity determination for structure in. You can nd more information on the distinction between the phase velocity and the group velocity. Phase velocity is defined as the velocity of a single wave. Phase velocity article about phase velocity by the free. The phase wavefront normal velocity is the quantity selected for this study, as it is perhaps that which is most often chosen as a candidate for approximation, from which other relevant. Phase velocity and group velocity this note is part of the supporting online material for the textbook \solid state physics. The group velocity of a wave is the velocity with which the overall envelope shape of the waves. 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. The velocity of carrier wave is called the phase velocity. When one or more waves superimpose together and results in a new wave,the individual velocities adds. The group velocity of a wave is the velocity with which the overall shape of the waves amplitudes known as the modulation or envelope of the wave propagates through space for example, imagine what happens if a stone is thrown into the middle of a very still pond. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. 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.
Lecture 7 wave function, phase velocity, group velocity. Generally, the phase velocity of a wave is faster than. Usually, group velocity is not equal to phase velocity, except in empty space. The speed at which that crest travels up the shoreline is the phase velocity meaning that phase of the wave is traveling northward at some speed which is greater than the speed of the wave itself. The variations of phase velocity, group velocity, ray direction with phase direction in 3d space, are plotted for a numerical model of general anisotropic poroelastic solid. Group velocity definition is the velocity with which some definite peculiarity of a composite wave train as an interference maximum advances through a medium. Phase and group velocity are two important and related concepts in wave mechanics. Phase velocity definition of phase velocity by merriam. The velocity of propagation of a pure sine wave of infinite extent. Phase velocity vector article about phase velocity. Difference between phase velocity and group velocity. The concept of group velocity is very important when a particle is represented by waves.
In a normal medium, the phase velocity is greater than the group velocity. Here, the pure wave factor is responsible for the ripples in the wavepacket, so the function f must be what defines the overall shape of the wavepacket. Group velocity synonyms, group velocity pronunciation, group velocity translation, english dictionary definition of group velocity. Modified definition of group velocity and electromagnetic. The phase and group velocity of the wave is dependent upon the refractive index, n. The phase velocity is given in terms of the wavelength. This is the quantity determined when one measures the distance which the radiation travels. Does a single photon particle move in the direction of the group velocity or the wave vector. The difference curve is then converted to a phase velocity dispersion curve. The group velocity is what has to be considered for the code measurements, i. For airfilled waveguide, the phase velocity reduces further to.
Thus we see direct dependence of group velocity on phase velocity and viseversa. Information transmission plane wave does not convey information must be modulated e. It is vital to have a solid understanding in both phase velocity and group velocity in order to excel in such fields. Group velocity definition of group velocity by the free. Dispersion pl 2pl 3pl 4pl 2w0 w k slope wk phase velocity figure 3 there are various things to note about this. A linearized group velocity approach for two point qp ray. Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backwardwave structures poru loh,1, ardavan f. The group velocity is directly proportional to phase velocity. In a medium, the phase velocity is typically smaller by a factor n, called the refractive index, which is frequencydependent chromatic dispersion. If these sources are harmonic in time, the different modes of propagation are superimposed foti 2000. Daley abstract using a linearized approximation for the quasicompressional phase velocity, vqp in an orthorhombic anisotropic medium, which is a subset of the related quasi. The table below shows phase velocity for certain rocks at various frequencies.
The same numerical model is used to plot the deviations of polarizations from phase direction and ray direction. Group delay is the length of a circuit or transmission line, divided by its group velocity. From this you can see that when the cutoff frequency is reached, the phase velocity goes to infinity. This is the velocity at which the phase of any one frequency component of the wave travels. Before we continue i shall define the compton wavelength of a particle of rest.
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