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Sound fields
Published in Carl Hopkins, Sound Insulation, 2020
To calculate n(ω) we now need to find dk/dω which is equal to the reciprocal of the group velocity, cg. The group velocity is the velocity at which wave energy propagates across the space. For sound waves in air, the group velocity is the same as the phase velocity, c0. Hence the general equation to convert n(ω) to n(f) is () n(f)=2πn(ω)=2πcgdN(k)dk
Three-Dimensional Molecular Electronics and Integrated Circuits for Signal and Information Processing Platforms
Published in Sergey Edward Lyshevski, Nano and Molecular Electronics Handbook, 2018
The group velocity represents the velocity of motion of the group of propagating waves that compose the wave packet. The phase velocity is the velocity of propagation of the phase of a single mth harmonic wave eikm(x−vpht). The wave packet travels with the group velocity. Taking note of E = ℏω and p = ℏk, one obtains vgb=dE(p)/dpandvph=E(p)/p.
Nonlinear Optics and Physics of Supercontinuum Generation in Optical Fibre
Published in Shyamal Bhadra, Ajoy Ghatak, Guided Wave Optics and Photonic Devices, 2017
The dispersion can affect both the phase velocity and the group velocity of a signal. For high efficiency, phase matching is essential in many nonlinear interactions such as parametric generation and the nonlinear contribution to this must also be considered in most circumstances. In addition, group-velocity dispersion matching is of importance, for example, in the situation of soliton–dispersive wave interactions that straddle the zero-dispersion wavelength of the interaction fibre. This process, as will be seen later, gives rise to the important short-wavelength extension of supercontinua. In conventional silica fibre structures, the minimum zero-dispersion wavelength achievable with the use of pure silica is 1270 nm. At wavelengths above this, the dispersion is anomalous. In practical units, the group-delay dispersion is defined in units of picoseconds per nanometre per kilometre (ps/nm km) defined as
High buffering capability of silicon-polymer photonic-crystal coupled cavity waveguide
Published in Waves in Random and Complex Media, 2023
Israa Abood, Sayed Elshahat, Zhengbiao Ouyang
To investigate and heighten the slow light properties, some required relations and definitions are provided and explained. The dispersion relation of the waveguide modes determines the group velocity : where is the speed of light in vacuum and is the group index of the guided mode. From Eq. (1), the can be calculated through simulation by plane-wave expansion (PWE) method. In our study, and k are normalized as the normalized frequency and the normalized wavevector K=, where is the operating wavelength and a is the lattice constant of PhC. Then the can be normalized as:
Love-type surface wave propagation due to interior impulsive point source in a homogeneous-coated anisotropic poroelastic layer over a non-homogeneous extended substance
Published in Waves in Random and Complex Media, 2022
Santanu Manna, Dipendu Pramanik, Saad Althobaiti
In dispersive media, different frequency components of a signal form a group and if such groups propagate at different speeds then it is called group velocity. Therefore, both the phase velocity and group velocity can be derived from the dispersion equation. For the case of non-dispersive media, both the phase and group velocity would be equal. It is also known that the group velocity is referred to as the velocity of energy transport, which is the rate of energy travel as signal. In this paper, it is observed from the dispersion relation of Love-type wave (cf. Equation (48)) that the angular frequency ω is an implicit function of wavenumber k. And the group velocity of the Love-type wave signal for a particular mode is given by
Influence of attached inertia and resonator on the free wave propagation in 2D square frame grid lattice metamaterial
Published in Waves in Random and Complex Media, 2021
Gandharv Mahajan, Avisek Mukherjee, Arnab Banerjee
The point O of the band-structure plot, as shown in Figures 4,5,7,9 and 12, corresponds to the very low frequency or long-wavelength. In this range, the square frame grid lattice can be modelled as an equivalent continuum. The effective properties are valid only within the long wavelength approximation. The scattering of wave increases with the decreasing wavelength. Two branches of the dispersion curve emerge from the origin O: these are the longitudinal waves also known as dilatational waves and transverse waves also known as distortional, shear, or equivoluminal waves. The phase velocity obtains from the secant slope of the line connecting the origin O to the point of interest on the dispersion curve. On the other hand, the tangent to the dispersion curve at any point gives the group velocity. In the low frequency regime, wave propagation in is illustrated in Figure 15.