Governing Wave Propagation Research Articles

Effect of Temperature and Pore Pressure on Velocity and Attenuation of Seismic Waves in Rocks: Applications to Crustal Exploration: ABSTRACT

Compressional seismic wave velocities have been used with much success in defining structural features likely to serve as hydrocarbon reservoirs. Recent advances in data collection and processing and, as we will describe, a greater understanding of the factors governing wave propagation in rocks indicate much additional information regarding the in-situ temperature, degree of saturation, and pore pressure may be obtained if data are available on seismic attenuation and shear velocity in addition to compressional velocity. Both velocity and attenuation are strongly temperature dependent. In dry sandstones, velocity increases rapidly with temperature to 100°C then levels off. Vp/Vs increases with tempeature. Shear and extensional attenuations increase rapidly with temperature from a Q of about 90 at room temperature to over 500 at 225°C. Shear attenuation is less than extensional attenuation below 200°C; the opposite is true above 200°C. In rock with a small amount of free water, thermal fracturing of rocks becomes dominant above 170°C decreasing velocities because of the increased porosity and increasing attentuation due to additional surface area in contact with water. In partly saturated rocks at elevated temperatures (200°C) attenuation is high in b lk compression and about half as great in shear. In fully saturated rocks, shear attenuation is high and bulk attenuation is low. Compressional velocity is greater in a fully saturated rock than in a dry or partly saturated rock. Shear velocity is less in a saturated rock than in a dry rock. Velocity and attenuation roughly follow an effective stress law. End_of_Article - Last_Page 1361------------

Formation of approximate domains for nth order differential equations governing wave propagation

In order to clarify the two-region approximation governing the propagation of electromagnetic waves in an anisotropic ionized medium, the analysis is generalized to embracenth order differential equations, thereby enabling the salient features behind the approximation to be isolated and investigated.

Guided solitary waves

Transversely periodic solitary-wave solutions of the Boussinesq equations (which govern wave propagation in a weakly dispersive, weakly nonlinear physical system) are determined. The solutions for negative dispersion (e.g., gravity waves) are singular and therefore physically unacceptable. The solutions for positive dispersion (e.g., capillary waves or magnetosonic waves in a plasma) are physically acceptable except in a limited parametric interval, in which they are complex. The two end points of this interval are associated with (two different) resonant interactions among three basic solitary waves, two of which are two-dimensional complex conjugates and the third of which is one-dimensional and real.

Propagation and decay of waves in porous media

A study is made of the propagation of wave fronts in two types of fluid-filled porous media, anisotropic and isotropic. The characteristic equation governing wave propagation velocities in anisotropic media is obtained by using the notion of surfaces of discontinuity. It is found that the wave velocities in such media vary with the direction of propagation and that for a specified direction there are four different wave velocities with which the disturbances can propagate. When the material is isotropic, its frequency equation shows the existence of three distinct waves, two longitudinal waves and one shear wave with velocities independent of the direction of propagation. The decay equations derived for these waves indicate that there are two factors which influence the decay of waves in isotropic porous media. The first is the geometry of the wave front, and the second is the friction between the solid and fluid phases.

General theorems relating to wave propagation governed by self-adjoint and Hermitian self-adjoint differential operators of order 2r

When plane waves are partly reflected and transmitted from a medium differing from free space, various conditions ensure the existence of simple identities between the reflexion and transmission coefficients. When differential equations of order In govern wave propagation, conditions to be placed both upon the generalized medium and upon the generalized waves are investigated so as to ensure the existence of exact analogues to these elementary identities. The theory is developed in terms of selfadjoint and Hermitian self-adjoint differential operators of order 2 The results fallout from the canonical forms for various skew-Hermitian and skew-symmetric matrices J with complex elements, unitary and orthogonal matrices being found so that J should be unitarily and/or orthogonally similar to its appropriate canonical forms.

Electromagnetic wave propagation in non-linear dielectric media

A modification of Bergman's integral operator method is applied to a hodograph system descriptive of linearly polarised plane electromagnetic pulse propagation in non-linear dielectrics. Series solutions of the system are thereby generated corresponding to general D-E and B-H relations. For certain non-linear ‘constitutive laws’ the governing wave propagation equations are reduced to a hyperbolic canonical form associated with the classical wave equation and closed form integration is thereby achieved. The availability of arbitrary parameters in these laws allows approximation to non-linear dielectric laws derived empirically. Moreover, it is noted how a generalised version of Weinstein's Correspondence Principle may be utilised to generate iteratively more general non-linear constitutive laws for which reduction to canonical form may be obtained.

SPECIFICATION OF TWO MODULI OF A VISCOELASTIC MATERIAL *

AbstractIn order to adequately understand the complicated, anelastic processes governing wave propagation in soils or highly weathered rock it seems desirable first to have available a convenient laboratory material for such propagation studies the frequency dependent rheological parameters of which are known beforehand. A property measurement program has been conducted to determine the complex Young's modulus and the complex shear modulus of a commercially available viscoelastic material over a broad spectral region.

Plasma Waves at Singular Turning Points

Connection formulas at a singular turning point are presented for the WKB solutions to a biquadratic differential equation governing wave propagation in an inhomogeneous plasma.

Effective Elastic Coefficients for Wave Propagation in Strained Elastic Crystals

The concept of effective elastic coefficients for wave propagation in strained crystals is examined critically. It would seem natural to expect that wave propagation in a strained crystal be governed by effective elastic coefficients that obey the same symmetry relations as the thermodynamically defined coefficients for crystals of the same symmetry as the strained crystal. However, it follows from publications of Toupin and Bernstein (1961), Truesdell (1961), and Noll and Truesdell (1964) that this is not true in general. The usual symmetry is lost because the effect of initial stress in the equations of motion governing wave propagation is to replace relations like eljkm=ejlkm by relations that depend on the initial stress. The idea of effective elastic coefficients remains useful, but it is important to distinguish them from the thermodynamically defined coefficients and to recognize that they do not obey the same symmetry relations. Two different sets (based on ρV2 and ρ0W2, respectively) of effective elastic coefficients for quartz under hydrostatic pressure are defined and related to the thermodyqnamically defined coefficients, the pressure, and the deformation. Here, W is the “natural velocity,” defined as the unstressed path length divided by the transit time under pressure.

Non-linear effects in electron plasmas

The effect of non-linear terms in the dynamical equations governing wave propagation in plasmas may be analysed by a perturbation procedure which is acceptable for amplitudes which are not too large. The Hamiltonian describing the complete system is separated into two parts: the quadratic part which yields the linearized equations and the non-linear part. The quadratic part may be eliminated by a normal-mode analysis, the `normal modes' comprising travelling waves. The non-linear part then results in interaction between these waves. Two theorems concerning wave interaction are proved. The first relates energy-transfer between a group of interacting waves to the frequency of these waves. These `action-transfer relations' lead to the Manley-Rowe relations for steady-state or quasi-steady-state configurations. The second theorem relates the frequency-displacements of a group of interacting waves to the energies of these waves. The properties of electron plasmas undergoing longitudinal oscillations are re-examined in the light of the preceding theorems. Interaction terms may be classed as `coherent' and `incoherent': the former do not result in energy transfer but only frequency displacement which may be characterized by a dispersion relation. The second group leads to transfer of energy between waves and hence to spectral decay. The interaction between longitudinal (electrostatic) and transverse (electromagnetic) waves in plasmas is considered and it is shown that in a uniform plasma in the absence of magnetic fields, the dominant interaction couples two longitudinal waves with one transverse wave. Hence one would anticipate that the dominant non-linear mechanism for radiation from excited plasmas leads to emission at twice the plasma frequency.