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Slow Light in a Silicon-Based Waveguide
Published in Erich Kasper, Jinzhong Yu, Silicon-Based Photonics, 2020
The phase velocity describes the speed at which the phase of the wave propagates. It can be defined asvp =ω/k =c/n, whereω is the angular frequency of the light andk is the wave number. And the group velocity gives the velocity with which a pulse of light propagates through a material or structure system. It can be defined asvg =∂ω/∂k. Slow light indicates that the group velocityvg is much less than the velocityc of light in vacuum. Correspondingly, there is also a concept of fast light (superluminal), whenvg >c. The speed of a signal cannot be higher than the phase velocity. A negative group velocity corresponds to the case when the group velocity opposes the phase velocity [15].
Superposition of Waves
Published in Myeongkyu Lee, Optics for Materials Scientists, 2019
The concept of group velocity plays an important role in optics and other physics fields, because any real harmonic wave, either electromagnetic or elastic, is actually a group of waves with close frequencies. Even so-called “monochromatic light” contains a very narrow range of wavelengths (frequencies). We showed in Chapter 1 that the dielectric constant and refractive index of a material are frequency-dependent. Thus, in such a dispersive medium, the phase velocity of a wave depends on its frequency. If the medium is nondispersive (e.g., vacuum), all harmonic waves present in the group propagate at the same phase velocity. The envelope of the group also propagates at the same velocity. That is, the group velocity is identical to the phase velocity. As a consequence, the envelope of the group maintains its original shape upon propagation. In dispersive media such as typical dielectrics, the harmonic waves forming the group propagate at different phase velocities and the relations among the phases of various harmonic waves continuously change upon propagation. This distorts the form of the group envelope. If the phase velocities of a wave group are very close to one another (i.e., if the signal has a very narrow spectrum), the shape of the propagating envelope is little distorted. Nevertheless, the effect of dispersion makes the group velocity of the waves differ from the phase velocity.
Solving the time-independent Schrödinger equation
Published in Nils O. Petersen, Foundations for Nanoscience and Nanotechnology, 2017
The phase velocity is defined as the distance a particular phase of the wave, such as the peak, moves as a function of time. It can be seen to be the wavelength (distance) moved in one period (time) and as a consequence depends only on their ratio, i.e., vp=λT=ωk $ v_{p} = \frac{\lambda }{T} = \frac{\omega }{k} $ . The phase velocity is therefore independent of k and ω individually. This is evident when comparing the propagating waves in Figures 4.1 and 4.2 where the peak moves the same distance along the position axis for a given time along the time axis. This is best visualized by following the peaks marked by the * symbol and indicated by the black line.
Waves propagation for magneto-thermoelastic medium during the two-temperature theory with the gravitational field
Published in Waves in Random and Complex Media, 2022
Substituting from Equation (39), we get Therefore, Equations (54), (55), and (57)–(60) determine the wave velocity equation for Stoneley waves in a solid elastic media under the influence of the magnetic field, initial stress, and gravity. From Equations (54), (55), and (57)–(60) it follows that the wave velocity of the Stoneley waves depends on the parameters for two-temperature parameters of the material medium, gravity, and the densities of both media. Since the wave velocity Equations (54), (55), and (57)–(60) for Stoneley waves under the present circumstances depend on the particular value of ω and create a dispersion of a general waveform . Furthermore, Equations (54) and (55), are in complete agreement with the corresponding classical results, when the effect of magnetic field, gravity, initial stress, and parameter for two temperatures is ignored. Phase velocity defines the speed at which waves oscillating at a particular frequency propagate and it depends on the real component of the wave number.
Guided Wave Studies for Enhanced Acoustic Emission Inspection
Published in Research in Nondestructive Evaluation, 2021
Waves are dispersive when the phase velocity is a function of frequency. As a result, there appears to be pulse spreading as the wave propagates along the waveguide. Energy can be conserved still showing pulse spreading and loss of magnitude. Group velocity is associated with the superposition of a group of waves of similar frequency leading to a single group velocity. Individual Fourier components travel with slightly different phase velocities as can be imagined in Figure 2. At an instant later in time the pulse shape will change, but the group velocity of the peak of the waveform will be the same. See [11] for more detail and formulas to calculate group velocity from the phase velocity components. The group velocity can often be referred to as the energy transport velocity but has a different value in certain situations. The Poynting Vector is associated with the energy velocity and can be calculated from the particle velocity and stress fields. See Rose [11] or Auld [16].
Propagation of Love wave in multilayered viscoelastic orthotropic medium with initial stress
Published in Waves in Random and Complex Media, 2022
Sourav Kumar Panja, S. C. Mandal
Now a days earthquake is a common natural phenomenon. Seismic waves produced by earthquake draw attention for many researchers to study the wave propagation aspects. Surface waves travel slowly but having larger amplitude can be the most destructive type of seismic wave. Love waves are horizontally polarized shear waves exist in the presence of semiinfinite medium. Generally, Earth's interior and the near-surface geological structures majorly constituted of multiple layers of parallel structures of different materials like rocks, crystalline materials and anisotropic material, underground water, oil and gases also. The study of the Earth's interior structure is not possible explicitly. The results collected from the study of wave propagation through different composite medium are one of the valuable informative source to understand the nature of internal structure of the earth. Some materials exhibit viscoelastic properties like wood carbon-epoxy in orthotropic medium. Ground shaking, earthquake loads, stiffness of soil can be visualized through these works. The earth may be assumed as elastic solid layered medium under high initial stress. It is very much important to study the influences of initial stress on the propagation of Love wave. Phase velocity is the velocity at which the phase of any one frequency component of the wave travels and group velocity is the velocity at which the envelope of the wave travels through space. Passive damping is one of the most effective and commonly used method for noise and vibration control. Damping ratio is useful variable to reduce the amplitude of dynamic instabilities in a structure. Attenuation coefficient is directly proportional to damping ratio. We have designed this problem keeping all these things in mind.