Long‐wave anisotropy in stratified media: A numerical test

Geophysics ◽  
1991 ◽  
Vol 56 (2) ◽  
pp. 245-254 ◽  
Author(s):  
J. M. Carcione ◽  
D. Kosloff ◽  
A. Behle
Keyword(s):  
Layer Thickness ◽  
Numerical Test ◽  
Transversely Isotropic ◽  
Seismic Signal ◽  
Layer Spacing ◽  
Wave Approximation ◽  
Numerical Tests ◽  
Long Wave ◽  
Long Wave Approximation ◽  
Minimum Ratio

When a seismic signal propagates in a stratified earth, there is anisotropy if the dominant wavelength is long enough compared to the layer thickness. In this situation, the layered medium can be replaced by an equivalent nondispersive transversely isotropic medium. Theoretical and experimental analyses of the required minimum ratio of seismic wavelength to layer spacing based on kinematic considerations yield different results, with a much higher value in the experimental test. The present work investigates the effects of layering by wave simulation and attempts to establish quantitatively the minimum ratio for which the long‐wave approximation starts to be valid. We consider two‐constituent periodically layered media and analyze the long‐wave approximation for different material compositions and different material proportions in 1-D and 2-D media. The evaluation of the minimum ratio compares snapshots and synthetic seismograms visually and through a measure of coherence. Layering induces scattering with wave dispersion or anisotropy depending upon the wavelength‐to‐layer thickness ratio. The modeling confirms the dispersive characteristics of the wave field, the scattering effects in the form of coda waves at short wavelengths, and the smoothed transversely isotropic behavior at long wavelengths. 1-D numerical tests for different media indicate that the minimum ratio is highest for the midrange of compositions, i.e., equal amount of each material, and for stronger reflection coefficients between the constituents. For epoxy‐glass, the value is around R = 8, while for sandstone‐limestone, it is between R = 5 and R = 6. Recent wave‐propagation experiments done in epoxy‐glass also imply a highest minimum ratio for midrange of composition; however, the 1-D numerical tests confirm the long‐wave approximation at shorter wavelengths than experimentally. The 2-D case shows that the more anisotropic the equivalent medium, the higher the minimum ratio, and that the approximation depends upon the propagation angle with longer wavelengths required in the direction of the layering.

An experimental test of P-wave anisotropy in stratified media

Geophysics ◽  
1984 ◽  
Vol 49 (4) ◽  
pp. 374-378 ◽  
Author(s):  
Patrick J. Melia ◽  
Richard L. Carlson
Keyword(s):  
Layer Thickness ◽  
Field Measurements ◽  
Transversely Isotropic ◽  
Compressional Wave ◽  
P Wave ◽  
Stratified Media ◽  
Layer Spacing ◽  
Long Wave ◽  
Significant Difference ◽  
Isotropic Media

In theory, stratified media in which the layers are elastically homogeneous and isotropic approximate transversely isotropic media with an axis of symmetry perpendicular to layering when the seismic wavelength is sufficiently longer than the layer spacing. The phenomenon has apparently been observed in field measurements, and acoustic anisotropy in deep‐sea sediments, measured in the laboratory, has been attributed to fine‐scale bedding laminations. However, to the best of our knowledge, no rigorous test of the theory has been made. We have made a partial test by making laboratory measurements of compressional‐wave velocities parallel and perpendicular to layering in fabricated samples consisting of glass and epoxy. We found no statistically significant difference between observation and theory in this limited test. Further, having used several frequencies, we found that the velocities progressively change from the long‐wave values toward those predicted by the time‐average relation, as expected. Finally, it has been proposed that the long‐wave approximation holds when the ratio of the seismic wavelength to layer thickness (λ/d) is 10–100. We found that the minimum ratio was highest in the midrange of composition (half glass, half epoxy), even though the samples in that range have the smallest combined layer thickness. This result suggests that the frequency dependence of anisotropy in layered media is a function of the proportions of the materials as well as the thickness of the layers.

Nonlinear dispersion hyperbolic models of continuous media

2007 ◽  
Vol 5 ◽  
pp. 273-278
Author(s):  
V.Yu Liapidevskii
Keyword(s):  
Boussinesq Equations ◽  
Nonlinear Dispersion ◽  
Order Of Accuracy ◽  
Flow Models ◽  
Wave Approximation ◽  
Long Wave ◽  
Primary Model ◽  
Nonequilibrium Flows ◽  
Internal Inertia ◽  
Long Wave Approximation

Nonequilibrium flows of an inhomogeneous liquid in channels and pipes are considered in the long-wave approximation. Nonlinear dispersion hyperbolic flow models are derived allowing taking into account the influence of internal inertia during the relative motion of phases upon the structure of nonlinear wave fronts. The asymptotic derivation of dispersion hyperbolic models is shown on the example of classical Boussinesq equations. It is shown that the hyperbolic approximation of the equations has the same order of accuracy as the primary model.

scholarly journals NUMERICAL MODELS OF HUGE TSUNAMIS OFF THE SANRIKU COAST

1976 ◽  
Vol 1 (15) ◽  
pp. 61
Author(s):  
Toshio Iwasaki
Keyword(s):  
Numerical Models ◽  
Source Model ◽  
Wave Source ◽  
Sanriku Coast ◽  
Wave Approximation ◽  
Long Wave ◽  
Small Wave ◽  
Wave Heights ◽  
Deep Region ◽  
Long Wave Approximation

Although numerical computations of the generation and propagation of tsunamis are successfully achieved in recent years, modeling of their wave sources is still a big problem. Three kinds of, wave source model, that is statistical, oceanographic and fault model, are studied in this paper. It is found that the first model gives reasonable wave heights as shown in the previous paper, the second one presents roughly one half of those for the first model and the last one produces too small wave heights. Based on the analysis of computed results, nature of undulations off from the shore boundary, directivity of wave propagation and the spindle shaped leading part are discussed. Comparing magnitude of various wave parameters for the leading wave along the minor axis of the wave source, it is shown that the long wave approximation modified by the slope effect illustrates the tsunamis in deep region of the sea and the slope effect is most dominant in shallow region.

Nonlinear dynamics of electrostatic Faraday instability in thin films

2018 ◽  
Vol 855 ◽  
Author(s):  
Dipin S. Pillai ◽  
R. Narayanan
Keyword(s):  
Thin Films ◽  
Standing Waves ◽  
Gas Holdup ◽  
Conducting Liquid ◽  
Mode Instability ◽  
Wave Approximation ◽  
Long Wave ◽  
Faraday Instability ◽  
Unstable Configuration ◽  
Long Wave Approximation

The nonlinear evolution of an interface between a perfect conducting liquid and a perfect dielectric gas subject to periodic electrostatic forcing is studied under the long-wave approximation. It is shown that inertial thin films become unstable to finite-wavelength Faraday modes at the onset, prior to the long-wave pillaring instability reported in the lubrication limit. It is further shown that the pillaring-mode instability is subcritical in nature, with the interface approaching either the top or the bottom wall, depending on the liquid–gas holdup. On the other hand, the Faraday modes exhibit subharmonic or harmonic oscillations that nonlinearly saturate to standing waves at low forcing amplitudes. Unlike the pillaring mode, wherein the interface approaches the wall, Faraday modes may exhibit saturated standing waves when the instability is subcritical. At higher forcing amplitudes, the interface may approach either wall, again depending on the liquid–gas holdup. It is also shown that a gravitationally unstable configuration of such thin films, under the long-wave approximation, cannot be stabilized by periodic electrostatic forcing, unlike mechanical Faraday forcing. In this case, it is observed that the interface exhibits oscillatory sliding behaviour, approaching the wall in an ‘earthworm-like’ motion.

Effect of gravity on the dynamics of non-isothermic ultra-thin two-layer films

2010 ◽  
Vol 661 ◽  
pp. 1-31 ◽  
Author(s):  
ALEXANDER NEPOMNYASHCHY ◽  
ILYA SIMANOVSKII
Keyword(s):  
Nonlinear Flow ◽  
Quiescent State ◽  
Horizontal Temperature Gradient ◽  
One Dimensional ◽  
Thermocapillary Flows ◽  
Wave Approximation ◽  
Long Wave ◽  
The Difference ◽  
Thermocapillary Forces ◽  
Long Wave Approximation

The effect of gravity on the dynamics of non-isothermic ultra-thin two-layer films is studied in this paper. The joint action of disjoining pressure and thermocapillary forces is taken into account. The problem is considered in a long-wave approximation. The linear stability of a quiescent state and thermocapillary flows is investigated. It has been found that the influence of the upper fluid density is significantly stronger than that of the difference of fluid densities. Nonlinear flow regimes are studied by means of numerical simulations. The gravity can lead to the formation of stripes or holes instead of droplets. The two-dimensional wavy patterns are replaced by one-dimensional waves with the fronts inclined or transverse to the direction of the horizontal temperature gradient.

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