differential equations - Group Velocity and Phase Velocity - Mathematics Stack Exchange
term u∂u/∂x next to the relative acceleration term ∂u/∂t, under the assumption our attention to wave motions in which the velocity of the fluid is much smaller . imposes a relationship between the frequency ω and the wavenumber k: shorter waves, and an initial group of waves with a mix of different wavelengths. PDF | We demonstrate that the ratio of group to phase velocity has a simple relationship to the orientation of the electromagnetic field. If the oscillation amplitude in the wave is small in relation to the where p is the pressure, ρ the density, g the acceleration due to gravity, φ the velocity potential, the z on k is linear, the group velocity coincides with phase velocity U = ω/k. The group velocity of a short gravity wave in liquid depends on the.
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Although the literature concerning ZGV Lamb modes is rather extensive, to the authors' best knowledge, only a few studies consider the topic of continuously varying material through the plate thickness.
To better understand the limitations and robustness of this type of application and to support further development, there is a need for improved knowledge of ZGV Lamb modes in plates with continuous material variations through the thickness.
Dispersion (water waves) - Wikipedia
Improved understanding also contributes to the overall aim of increased knowledge within the general field of ZGV Lamb modes. This study investigates the behavior of ZGV modes in two synthetic cases defined by two plates with inhomogeneous and nonsymmetric continuous variation of the acoustic bulk wave velocities.
Results are also compared with a corresponding isotropic case. Since only a few prior studies used higher ZGV modes in the testing of plate-like concrete structures, 40,41 the present work is limited to a study of the lowest ZGV mode.
The study is organized in sections as follows: II defines the material variation cases, Sec. IV further explores the cases in a simulated nondestructive testing application. Although the study originates from a perspective of thick concrete structures, the results hold for other inhomogeneous cases as well under the condition of guided wave propagation in plate-like structures through the general assumption of linear elastic wave propagation.
These two cases plates are also compared with a homogeneous, isotropic reference case defined by an isotropic and infinite plate with constant material properties throughout its thickness. Although the unbounded infinite domain of the cases represent a theoretical and idealized condition, results for the lowest ZGV Lamb mode derived under this assumption can be generalized and extended to real plates of finite dimensions at distances greater than one thickness from the plate-edge.
A more detailed description of both cases and their relation to the isotropic reference case is provided below. In total, two scaling functions are used in the study with one function assigned to each case.
Dispersion (water waves)
The scaling functions for case 1 and case 2 are shown in Figs. The two functions are defined similarly, with one term given by a single cycle of a sinusoidal function crest to crest as well as a second constant term of unity. For both cases, the wave cycle has the same frequency and spatial location but it differs in magnitude and sign. The corresponding variations in the longitudinal and transversal wave velocities as function of the thickness coordinate for cases 1 and 2 are shown in Figs.
To highlight the continuous variation of the material, the darkness of the background colors of Figs. Internal gravity waves may also arise in a stratified liquid.
Waves in fluids
If two liquid layers move slipping one on the other, their interface is a tangential discontinuity because the liquid velocity tangential to the surface changes abruptly. The disturbance of the interface may bring about an interfacial instability Kelvin-Helmholtz Instability. An incompressible fluid uniformly rotating as a whole may give rise to internal waves due to Coriolis forces inertia!
Pressure Waves Pressure waves appear in a compressible liquid. Pressure perturbation in a compressible fluid involves perturbation of density, velocity, and other parameters. A disturbance of the finite amplitude propagates in compressible fluid as a simple, or Rieman, wave.
This circumstance results in the profile deformation as the wave propagates, the rarefaction points lagging behind the compression points, and an ambiguity arising, i. Physically the ambiguity gives rise to a discontinuity, known as a shock wave, approaching which from the left and from the right the density is single-valued. The discontinuity displaces in space and attenuates if the velocity of gas flow behind it is not kept constant by an appropriate boundary condition.
Propagating through the surface of a shock wave the gas parameters change sharply see Compressible flows and Shock tubes. However, the substantial temperature elevation behind the shock wave is not attained because in an imperfect gas, as the translational temperature grows, part of energy is expended on excitation of molecule vibrations and dissociation.
In this case, heat capacity depends on temperature, and enthalpy is calculated by the physics methods of statistics.
Propagation of a finite amplitude pressure distrubance. Temperature behind a shock wave with various assumptions. This suggests that in most media rarefaction shocks do not exist and in the rarefaction wave entropy is constant.
The head of the stationary rarefaction wave propagates with the velocity of sound, while the shock wave front may propagate with a velocity tens of times higher than the velocity of sound depending on the boundary conditions.
The velocity and intensity of a blast wave fall with the distance from the point of energy release. The most rapid drop of the shock wave velocity and the pressure behind it is observed in a spherical blast wave. A shock wave propagating in a medium capable of exothermic reaction is called a detonation wave.