Localization estimates for a random discrete wave equation at high frequency

1987 ◽  
Vol 46 (3-4) ◽  
pp. 477-491 ◽  
Author(s):  
William G. Faris
Keyword(s):  
Wave Equation ◽  
High Frequency

Ray equations in retarded Snell midpoint coordinates

Geophysics ◽  
1984 ◽  
Vol 49 (12) ◽  
pp. 2100-2108
Author(s):  
Alfonso González‐Serrano ◽  
Mathew J. Yedlin
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Group Velocity ◽  
High Frequency ◽  
Frame Of Reference ◽  
Velocity Analysis ◽  
Acoustic Wave Equation ◽  
Arbitrary Angle ◽  
Frequency Limit ◽  
Velocity Function

Group velocity (ray) equations describe the dynamic behavior of wave‐equation extrapolators in the high‐frequency limit. They are found in general from the dispersion relation of an arbitrary acoustic wave equation. Wave‐equation operators require a background extrapolation velocity. As an application of the group velocity equations, a sensitivity analysis to the background‐operator velocity illustrates the trade‐off between uncertainty in velocity and precision in imaging. Exact wave extrapolators are most useful when the exact velocity function is known. Wave‐equation imaging for velocity analysis in Snell midpoint coordinates requires velocity‐insensitive extrapolation operators. In this frame of reference, approximations of the exact acoustic wave equation are referenced to an arbitrary angle of propagation. Group velocity equations show that in Snell midpoint coordinates, using wide‐reference propagation angles, the fifteen‐degree wave equation gives satisfactory velocity‐independent images. The forty‐five degree wave equation does not appreciably improve the image.

HIGH FREQUENCY ANALYSIS OF FAMILIES OF SOLUTIONS TO THE EQUATION OF VISCOELASTICITY OF KELVIN–VOIGT

2004 ◽  
Vol 01 (04) ◽  
pp. 789-812 ◽  
Author(s):  
AMEL ATALLAH-BARAKET ◽  
CLOTILDE FERMANIAN KAMMERER
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Initial Data ◽  
Lower Bounds ◽  
Frequency Analysis ◽  
High Frequency ◽  
Weak Limit ◽  
Negative Viscosity

In this paper, we study the evolution of the energy density of a sequence of solutions to the Kelvin–Voigt viscoelasticity equation. We do not suppose lower bounds on the non-negative viscosity matrix. We prove that, in the zone where the viscosity matrix is invertible, this term prevents propagation of concentation and oscillation effects contrary to what happens in the wave equation. We calculate precisely the weak limit of the energy density in terms of microlocal defect measures associated with the initial data under the assumption that the oscillations of the data are not microlocally localized on directions which are in the kernel of the viscosity matrix.

On cylindrical waves in stratified media: high frequency refraction and diffraction at a plane interface

1970 ◽  
Vol 67 (1) ◽  
pp. 133-161 ◽  
Author(s):  
I. Roebuck
Keyword(s):  
Exact Solution ◽  
Wave Equation ◽  
High Frequency ◽  
Line Source ◽  
Dielectric Medium ◽  
Plane Interface ◽  
Stratified Media ◽  
Cylindrical Waves ◽  
Cylindrical Obstacle ◽  
High Frequency Waves

Introduction. The problem of the scattering of high-frequency waves, which emanate from a line source in a homogeneous isotropic dielectric medium and impinge upon a cylindrical obstacle, has been attacked in a variety of ways. In certain cases, where both the shape of the obstacle and the conditions to be satisfied on its boundary are particularly convenient, an exact solution may be found by separation of the wave equation (see, for example, Marcuvitz (l)), but in general some form of approximation is necessary to obtain an explicit answer.

scholarly journals High-frequency homogenization for travelling waves in periodic media

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160066 ◽  
Author(s):  
Davit Harutyunyan ◽  
Graeme W. Milton ◽  
Richard V. Craster
Keyword(s):  
Wave Equation ◽  
High Frequency ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Coupling Constants ◽  
Convergence Theory ◽  
Periodicity Cell ◽  
Periodic Media ◽  
Carrier Wave ◽  
Convergence Type

We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector k and frequency ω 1 plus a modulated Bloch carrier wave having crystal wavevector m and frequency ω 2 . We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω 1 = ω 2 and ( k − m ) ⊙ Λ ∈ 2 π Z d , where Λ =(λ 1 λ 2 …λ d ) is the periodicity cell of the medium and for any two vectors a = ( a 1 , a 2 , … , a d ) , b = ( b 1 , b 2 , … , b d ) ∈ R d , the product a ⊙ b is defined to be the vector ( a 1 b 1 , a 2 b 2 ,…, a d b d ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.

Asymptotic theory of body waves in a radially heterogeneous Earth

1969 ◽  
Vol 59 (5) ◽  
pp. 2039-2059
Author(s):  
Sarva Jit Singh ◽  
Ari Ben-Menahem
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
High Frequency ◽  
Asymptotic Theory ◽  
Heterogeneous Medium ◽  
Body Waves ◽  
Spherically Symmetric ◽  
Vector Wave ◽  
Vector Wave Equation ◽  
Earth Models

abstract Various aspects of elastic wave propagation in a spherically symmetric, non-gravitating, isotropic, inhomogeneous medium are considered. It is shown through a simple example that the high frequency decoupling conditions of the vector wave equation may be approximately satisfied by real Earth models. An asymptotic theory is developed for the decoupled potential equations. This theory is applied to the case of a shear dislocation and to that of a center of compression in a radially heterogeneous medium. Explicit expressions are obtained for the ray-theoretical displacements.

Higher modes of continental Rayleigh waves

1957 ◽  
Vol 47 (3) ◽  
pp. 187-204 ◽  
Author(s):  
Jack Oliver ◽  
Maurice Ewing
Keyword(s):  
Wave Equation ◽  
Particle Motion ◽  
High Frequency ◽  
Rayleigh Waves ◽  
Velocity Curve ◽  
Theoretical Studies ◽  
Higher Modes ◽  
Short Period ◽  
First Time ◽  
Rayleigh Mode

ABSTRACT A long dispersive train of waves corresponding to higher modes of the Rayleigh-wave equation (including Sezawa's M2 wave) for the continental crust-mantle system is positively identified, apparently for the first time. Observed particle motion is elliptical and retrograde, in agreement with theory. Although several theoretical studies have been published in which progressive elliptical particle motion was found, all of these involved values of the elastic constants unsuitable for the present problem. The beginnings of the short-period branches of the higher modes can account for the high-frequency longitudinal and vertical components of the continental surface-wave phase Lg. The large amplitudes and the peculiar appearance of Rg appear to depend on the broad flat minimum of the group velocity curve of the lowest or Rayleigh mode.

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