Limited vs. full frequency wave‐equation imaging

2002 ◽  
Author(s):  
J. Bee Bednar ◽  
G. H. Neale
Keyword(s):  
Wave Equation ◽  
Frequency Wave

scholarly journals Simulation of laser propagation in a plasma with a frequency wave equation

2008 ◽  
Vol 227 (4) ◽  
pp. 2610-2625 ◽  
Author(s):  
S. Desroziers ◽  
F. Nataf ◽  
R. Sentis
Keyword(s):  
Wave Equation ◽  
Frequency Wave ◽  
Laser Propagation

The Fresnel volume and transmitted waves

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 653-663 ◽  
Author(s):  
Jesper Spetzler ◽  
Roel Snieder
Keyword(s):  
Wave Equation ◽  
Fresnel Zone ◽  
Wave Theory ◽  
Three Dimensions ◽  
Ray Theory ◽  
Finite Frequency ◽  
Frequency Wave ◽  
Reflected Waves ◽  
Band Limited ◽  
Propagation Of Waves

In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high‐frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite‐frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite‐frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band‐limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite‐frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite‐frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat's theorem in the context of geometric ray theory is a special case of the finite‐frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

A causal and fractional all-frequency wave equation for lossy media

2011 ◽  
Vol 130 (4) ◽  
pp. 2195-2202 ◽  
Author(s):  
Sverre Holm ◽  
Sven Peter Näsholm
Keyword(s):  
Wave Equation ◽  
Frequency Wave ◽  
Lossy Media

scholarly journals Homogenization of the wave equation with non-uniformly oscillating coefficients

2022 ◽  
pp. 108128652110650
Author(s):  
Danial P. Shahraki ◽  
Bojan B. Guzina
Keyword(s):  
Wave Equation ◽  
Wave Motion ◽  
Shear Waves ◽  
Low Frequency ◽  
Wave Dispersion ◽  
Second Order ◽  
Frequency Wave ◽  
Cell Functions ◽  
Varying Coefficients ◽  
Periodic Microstructure

The focus of our work is a dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g., functionally graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in [Formula: see text], [Formula: see text], within the framework of multiple scales expansion. When either [Formula: see text] or [Formula: see text], this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in one dimension by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.

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