Inhomogeneous Isotropic Medium Research Articles

Three-dimensional crack and contact problems with a general geometric configuration

A new method, based on point set theory and properties of orthogonal functions, is developed for determining exact solutions to three-dimensional crack and contact problems with complicated geometric configurations (e.g. a star-convex domain) in an infinite linear elastic medium. The governing equation is a two-dimensional Fredholm integral equation of the first kind. The central idea of this method is the chain extension of an exact solution from a regular subdomain to an irregular entire domain. Examples are given illustrating how this solution procedure can be used to obtain exact closed form solutions for a general Hertz contact problem and various crack problems in an inhomogeneous isotropic medium with an elastic modulus which is a power function of depth.

Three‐dimensional primary zero‐offset reflections

Zero‐offset reflections resulting from point sources are often computed on a large scale in three‐dimensional (3-D) laterally inhomogeneous isotropic media with the help of ray theory. The geometrical‐spreading factor and the number of caustics that determine the shape of the reflected pulse are then generally obtained by integrating the so‐called dynamic ray‐tracing system down and up to the two‐way normal incidence ray. Assuming that this ray is already known, we show that one integration of the dynamic ray‐tracing system in a downward direction with only the initial condition of a point source at the earth’s surface is in fact sufficient to obtain both results. To establish the Fresnel zone of the zero‐offset reflection upon the reflector requires the same single downward integration. By performing a second downward integration (using the initial conditions of a plane wave at the earth’s surface) the complete Fresnel volume around the two‐way normal ray can be found. This should be known to ascertain the validity of the computed zero‐offset event. A careful analysis of the problem as performed here shows that round‐trip integrations of the dynamic ray‐tracing system following the actually propagating wavefront along the two‐way normal ray need never be considered. In fact some useful quantities related to the two‐way normal ray (e.g., the normal‐moveout velocity) require only one single integration in one specific direction only. Finally, a two‐point ray tracing for normal rays can be derived from one‐way dynamic ray tracing.

A uniqueness theorem in inverse scattering of elastic waves

The scattering of plane elastic waves in an isotropic inhomogeneous medium is considered. The existence and uniqueness of the direct problem is stated, and a reciprocity principle for the far fields of the scattered waves formulated. Finally, it is proved that the knowledge of the far-field patterns for a bounded sequence of different frequencies and certain sets of incoming plane waves uniquely determines the density of the medium

Some reverse mappings in 3-D crack and dislocation problems

A new division of geometric regions for internal and external planar cracks is introduced. This allows some reverse mappings between internal and external problems to be revealed in connection with 3-D planar crack and dislocation problems in an infinite elastic medium. Transform factors are given for a mode I crack with arbitrary shape in an infinite inhomogeneous isotropic medium where the elastic modulus is an exponential function of depth. Arbitrary loading for the inhomogeneous material and coplanar dislocation-crack interaction for a homogeneous isotropic medium are analyzed through easy derivations. As one of the applications, the known solution of an external elliptical crack is used to obtain the exact solution in closed form for the crack with the boundary contour % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdiNaai% ikaiabeI7aXjaacMcacqGH9aqpcaWGIbWaaWbaaSqabeaacaaIYaaa% aOWaaOaaaeaadaWcaaqaaiGacogacaGGVbGaai4CamaaCaaaleqaba% GaaGOmaaaakiabeI7aXbqaaiaadggadaahaaWcbeqaaiaaikdaaaaa% aOGaey4kaSYaaSaaaeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaai% aaikdacqaH4oqCaaaakeaacaWGIbWaaWbaaSqabeaacaaIYaaaaaaa% aeqaaaaa!4F66!\[\rho (\theta ) = b^2 \sqrt {\frac{{\cos ^2 \theta }}{{a^2 }} + \frac{{\sin ^{2\theta } }}{{b^2 }}} \]subjected to a pair of concentrated forces at the crack center.

Stress intensity factors for annular cracks in inhomogeneous isotropic materials

A basic procedure is given tor converting a pair of coupled two-dimensional integral equations relevant to mixed boundary value problems with annular type regions to two non-coupled integral equations. The procedure utilizes fundamental solutions to the corresponding internal and external problems in order to separate the coupled equations and write their solution in series form. The method is first applied to axisymmetric shear loading of an annular crack in a homogeneous transversely isotropic material (where the crack plane is parallel to the plane of isotropy) and to uniform tensile loading of an annular crack in a homogeneous isotropic material. Stress intensity factors are compared with previous analysis to test the accuracy of the solution. New results for stress intensity factors are given for the arbitrary normal loading of an annular crack in an inhomogeneous isotropic medium.

A procedure for computing successive approximations in the asymptotic space-time (ST) ray method of Gaussian beams. II. (An isotropic inhomogeneous elastic medium)

The method of spatial rays is applied in the construction of the formulas for the asymptotic expansions of the Gaussian beam type in an elastic medium. The coefficients for higher-order approximations of the expansions constructed are subject to a recurrent process. The process is considered for theP (orS) component of the displacement vector in an inhomogeneous elastic medium taken as an example.

Modeling of the Subsurface Interface Radar

A finite-difference time-domain (FDTD) method is used to solve the 21 2 D problem of the response of an arbitrary source, in particular, an impulsive point source, in a two-dimensional isotropic inhomogeneous medium. Cosine and sine transforms are used to reduce the three-dimensional problem to two dimensions. The complete solution is obtained by linearly superimposing several transformed field components. This provides great savings in terms of computer storage and run time over the three-dimensional FDTD model. A criterion is given to ensure the stability of this finite difference scheme. Examples of application of this analysis to actual problems such as the subsurface interface radar are illustrated.

Generalized WKB Approximation for Stratified Isotropic Chiral Media

The WKB method and its generalization, well known to be applicable to a certain class of achiral isotropic inhomogeneous media, are extended to stratified chiral media for waves with normal incidence. The method is based on an expansion in wave fields, certain combinations of the electric and magnetic fields, whose properties are also extensively studied in the paper. It is seen that for sufficiently slowly varying media, the reflection dyadic due to the stratification can be expressed in closed integral form and, for normal incidence, the result is independent of the chirality parameter κ applied in this paper. The method can be further extended for chiral media to what is known as the Bremmer series method for achiral media.

Characteristic matrix of a periodically inhomogeneous layer

We propose an analytic representation of the characteristic matrix of a periodically inhomogeneous layer; the characteristic matrix of the homogenized medium is defined and it is shown that the structure of characteristic matrix of transversely isotropic media coincides with the structure of the characteristic matrix of a layered inhomogeneous isotropic medium.

Three-dimensional inverse problem for inhomogeneous transversely isotropic media

The basic formula used in the presented paper gives the relation between the P wave travel-time perturbation and the perturbation of an inhomogeneous transversely isotropic medium, expressed by four perturbations of elastic parameters and by two angles of orientation of the axis of symmetry of transverse isotropy in space. The travel time perturbation is computed along the ray in the unperturbed inhomogeneous isotropic medium. Four elastic parameters and two angles are parametrized in the model under study and a system of equations for many rays is constructed. The equations are linear in the sought elastic parameters and nonlinear in the sought angles, and the iterative Levenberg-Marquardt algorithm is thus used to solve them. The theoretical 3-D inverse problem was solved in the presented numerical example. The data, simulating teleseismic data, were computed in the direct problem and then inverted. The results indicate the applicability and limitation of the presented algorithm in real problems.

An inverse scattering problem of elastic waves

The problem of deducing the elastic parameters of an inhomogeneous isotropic medium with spherically symmetric parameters, from the elastic‐wave scattering information, is considered. It is shown that the problem can be reduced to a quantum mechanical inverse scattering problem at fixed energy, if the scattering amplitude as a function of angle for the transverse wave of polarization perpendicular to the scattering plane is known for two different values of frequency. In this case, existing inverse scattering theories at fixed energy enable us to find two functions which are related to the density and shear modulus of the medium. The problem of constructing the elastic parameters from the inversion result is studied and tested for synthetic data. Results of the inversion obtained from the Regge–Newton method and the discrete method are compared for scattering data given at different frequencies.

Propagation of a geometrical optics field in an isotropic inhomogeneous medium

The cross section of an infinitesimally small ray tube (ray pencil) expands or contracts as rays propagate. The square root of the cross‐section ratio between two points on a ray is defined as the divergence factor DF. In a homogeneous medium, rays are straight lines. Consequently, (DF)2 is simply equal to the ratio of Gaussian curvatures of the wave fronts. In an inhomogeneous medium, rays are curved, the DF becomes more complex. In this paper we derive several integral expressions for calculating DF and the related formulas that govern the field and energy variations along a ray.

Eigenwaves in an inhomogeneous azimuthally bigyrotropic medium

The problem of natural electromagnetic waves in an azimuthally magnetized bigyrotropic medium which is inhomogeneous in the radial direction is solved analytically. A system of generalized wave equations is obtained in the longitudinal components of the electrical and magnetic fields. Particular solutions of the system are represented in the form of generalized power series. The series exponents are determined and recursion formulas are built up for calculating their coefficients. The passage to particular cases of a homogeneous gyrotropic and inhomogeneous isotropic medium is realized, and the results are compared with known data from the literature.

Love waves due to a point source in an isotropic elastic medium under initial stress

The propagation of Love type waves due to a point source in an isotropic inhomogeneous medium under initial stress has been investigated in this paper. Firstly, the propagation of Love waves in a homogeneous crust overlying an inhomogeneous substratum, both under compressive initial stress, due to the presence of a point source in the interface has been considered. Finally, the case in which the upper layer is inhomogeneous and the lower medium is homogeneous has been discussed. The Green's function technique developed by Ghosh (1963) has been used to solve the problem.

To: “Straightforward derivation of Hubral’s wavefront curvature differential equation in inhomogeneous isotropic media,” May 1980, GEOPHYSICS, p. 964–967.

The author of the Short Note, “Straightforward Derivation of Hubral’s Wavefront Curvature Differential Equation in Inhomogeneous Isotropic Media,” Th. Krey, has forwarded the following changes to his paper which appeared in May Geophysics, p. 964–967. In equations (19) and (20), [Formula: see text] should be replaced by [Formula: see text]; and in equation (21), C is replaced by (C/v). In equation (10) the product [Formula: see text] is replaced by [Formula: see text]. The last sentence ahead of the “acknowledgement” paragraph should be replaced by the following: The third term appears only if the velocity is a nonlinear function of x, y, z.

Straightforward derivation of Hubral’s wavefront curvature differential equation in inhomogeneous isotropic media

“Wavefront curvatures in three‐dimensional laterally inhomogeneous media with curved interfaces” (Hubral, 1980, this issue) shows a differential equation [formula (4.1)] which describes the alteration of the wavefront curvature matrix along a raypath in the case of an isotropic velocity v which is an arbitrary function of the locus in the three‐dimensional space. Hubral derives his equation by referring to papers of Popov and Pšenčik (1976, 1978) and Hubral (1979).

On the decoupling of P and S waves in inhomogeneous elastic media

Summary. A representation derived by Richards (1974) for P-SV wave displacements in spherically symmetric elastic media is extended to general displacements in a general inhomogeneous isotropic elastic medium, with suitably differentiable elastic constants. This representation in terms of appropriate potentials gives rise to a partial decoupling of the P and S (weighted) displacements, which satisfy simple second order wave equations with lower order coupling terms. The highest order P and S components satisfy homogeneous wave equations that depend only on the P and S velocities a and respectively and are unaffected by the density other than at the source and observer positions. In the study of small high-frequency particle displacements in homogeneous elastic media the motion may be completely separated into P and S components. Each component has a characteristic wave speed and satisfies a simpler equation than the general elastic wave equation. In inhomogeneous media, this separation of P and S waves may still be observed, for example as distinct phases on seismograms, but there is now lower order coupling between the phases. Also a modified form of the Helmholtz representation in terms of a gradient and curl is required for other than the highest order components. In this paper we examine the partial first order decoupling of the P and S components in a general isotropic inhomogeneous medium with differentiable material constants. We begin by deriving for the harmonic displacement u exp (-- iwt) a representation in terms of potentials P and S, u = V(P/p”2) t g,(P/p”Z) + v x (S/p’/Z) t g, x (S/p”2) + C(u)

Remarks on forces and the energy-momentum tensor in macroscopic electrodynamics

The conservation laws which follow from the field equations and their relation to the energy and momentum conservation laws are discussed. On the basis of the electrodynamics of slowly moving bodies, an expression is derived for the density of the force which acts on an isotropic inhomogeneous medium in an electromagnetic field. Attention is concentrated on elucidating the difference between the energymomentum tensors of Minkowski and Abraham. It is emphasized either of these can be used in practice to consider the exchange of energy and momentum between an emitter and a medium in which the emitter is placed. However, to analyze the processes in the medium itself, Abraham's should be used because it takes into account Abraham's volume force, which acts even on a homogeneous medium (whereas according to Minkowski no force acts at all on a transparent, homogeneous medium with density-independent permittivity in an electromagnetic field).

Numerical calculations of elastic waves in laterally inhomogeneous media

In most laterally inhomogeneous crustal structures, waves propagated away from the source can be constructed to first order without knowledge of any backscattered radiation. If this approximation is used, the second-order boundary-value problem for elastic waves in isotropic inhomogeneous media can be reduced to a first-order initial-value problem in space. Numerical solutions can then be economically obtained. Calculations for plane-layered structures agree quantitatively with exact solutions. Waves in more complicated structures agree qualitatively with intuition, observational experience, and model experiments.

Exact solutions ofSH wave equation in transversely isotropic inhomogeneous elastic media

TheSH wave equation in a transversely isotropic inhomogeneous elastic medium, where the elastic parameters and density are functions of vertical coordinate, is considered. A general procedure is given for finding the inhomogeneities for which the equation can be solved in terms of hypergeometric, Whittaker, Bessel and exponential functions. A few simple inhomogeneities and the corresponding solutions in terms of these transcendental functions are presented.