Stratified Half Research Articles

Effect of air layer location and depth on redistribution of convective heat transfer induced by energy source in a sandwiched porous-air-porous enclosure free from temperature difference at external boundaries

The study deals with nonlinear convection in a three-layered porous-air-porous enclosure with zero temperature difference at the external impermeable thermally conductive boundaries. The upper and lower porous matrices are internally heated with a uniform energy source. To describe redistribution of heat energy between the top and bottom surfaces of the enclosure, we introduce a relative heat transfer coefficient qr that has the conduction and convection parts. The former part tends to unity in the fully filled porous domain. The symmetry of heat transfer from the inner area towards the outer surfaces breaks down if one adds an intermediate air layer with low thermal conductivity. Two sandwiched systems which have equal depth ratio d but distinct air layer location are considered. In system 1, the air layer is located in the upper unstably stratified half of the enclosure. In system 2, this layer is in the lower stably stratified half. The total value of qr always increases due to penetrative convection. At d = 0.1, local convection in system 1 is easily initiated in contrast to the hard-to-generate large-scale convection in system 2 due to a strong destabilization. The latter most effectively enhances heat transport though the top surface with increasing the supercriticality.

Rigid Foundations Resting on an Elastic Layered Soil

A rigid foundation of arbitrary shape resting on a stratified half space (soil), with specific geometrical and elastic properties for its horizontal layers, is examined under the action of a vertical load placed anywhere on the foundation area. On the basis of a purely analytical treatment of the deformability of the soil surface, i.e. without using a finite element discretization of the layered soil mass, a general method is developed in order to determine the contact soil pressures and the resulting settlements through an easily implemented numerical procedure. The possible presence of “inactive” foundation parts not being in contact with the soil surface is taken into account. Parametric studies are performed showing the influence of the variability of the soil layers, as well as of the vertical load location, on the contact pressures developed. In this context, for the cases of a rectangular and a circular foundation the area borders for the location of a vertical load are determined, either for the development of compressive soil pressures all over the footing (core), or for allowing the existence of some inactive part without contact with the soil (zero contact pressure), not exceeding the half of the total foundation surface (limit core).

Random wave propagation in a viscoelastic layered half space

An extremely efficient and accurate solution method is presented for the propagation of stationary random waves in a viscoelastic, transversely isotropic and stratified half space. The efficiency and accuracy are obtained by using the pseudo excitation method (PEM) with the precise integration method (PIM). The solid is multi-layered and located above a semi-infinite space. The excitation sources form a random field which is stationary in the time domain. PEM is used to transform the random wave equation into deterministic equations. In the frequency-wavenumber domain, these equations are ordinary differential equations which can be solved precisely by using PIM. The power spectral densities (PSDs) and the variances of the ground responses can then be computed. The paper presents the full theory and gives results for instructive examples. The comparison between the analytical solutions and the numerical results confirms that the algorithm presented in this paper has exceptionally high precision. In addition, the numerical results presented show that: surface waves are very important for the wave propagation problem discussed; the ground displacement PSDs and variances are significant over bigger regions in the spatial domain when surface waves exist; and as the depth of the source increases the ground displacement PSDs decrease and the regions over which they have significant effect become progressively more restricted to low frequencies while becoming more widely distributed in the spatial domain.

A precise numerical method for Rayleigh waves in a stratified half space

AbstractThis paper presents theory and results for Rayleigh waves propagating in a transversely isotropic stratified solid resting on an elastic semi‐infinite space. It uses the precise integration method (PIM) and the extended Wittrick–Williams (W–W) algorithm. This problem can be reduced into the eigenvalue problem of ordinary differential equations (ODEs) in the frequency–wavenumber domain. The PIM used here is a precise method for solving the ODEs with two‐point boundary conditions, and the eigenvalue counting method of the extended W–W algorithm is used to ensure that all eigenvalues are found without the possibility of any being missed. Hence the theory presented in this paper is exact, in the sense that its precision is limited only by the precision of the computer used. Copyright © 2006 John Wiley & Sons, Ltd.

Inversion of Rayleigh Wave in a Stratified Half Space

Inversion of Rayleigh waves is studied for the first time by using thegenetic algorithm based on the intensities of the fundamental and/orhigher modes. The relations of all modes to the medium structure areinvestigated. It is emphasized that the higher modes and mode jumpingshould be considered in the inversion when the low-velocity layers arecontained in the stratified half space. Shear wave velocity andthickness of each layer are inverted by the genetic algorithm. Comparedwith the least square method, the genetic algorithm has severaladvantages. It usually does not depend on the initial models and theglobal optimal solutions can be obtained easily. The influences of someparameters on the inversion are also analysed and some useful resultsare presented.

The effect of initial stresses on harmonic stress fields within the stratified half plane

The effect of initial stresses on dynamic (harmonic) stress fields within an elastic stratified half plane is investigated. It is assumed that the point-located harmonic force acting on the free plane of the layer by which the half plane is stratified causes this stress field. By employing displacement potentials and the exponential Fourier transform the governing system of partial differential equations of motion is solved. The necessary inverse transformations including rigorous mathematical complexity is performed numerically. The analysis of the numerical results, which shows the influence of the homogeneous initial stresses on the distribution of the stresses on the inter-medium plane, is made. These analyses are examined for various problem parameters and it is assumed that the material of both the layer and the half plane is homogeneous, isotropic, compressible and linearly elastic. It has been observed that the initial stresses may change significantly the values of the superimposed harmonic stresses.

Surface waves in a stratified half space with enhanced continuum properties. Part 2: Analysis of the wave characteristics in regard to high-speed railway tracks

This paper discusses the propagation of surface waves in a stratified half space, which consists of an elastic granular upper layer that rests on a half-infinite elastic substratum. For the upper layer the effect of particle translation and particle rotation are incorporated via a second-gradient continuum formulation and a Cosserat continuum formulation, respectively. Consequently, additional surface wave modes emerge, which cannot be described by a classic elastic formulation. After a general discussion of the surface wave dispersion curves, the critical behaviour of the stratified medium under a moving vibrating load will be examined. Here, the adjective `critical' designates that the wave energy that is generated by the moving load has the same velocity as the load itself, which results in resonance phenomena. The model parameters in this study are chosen to be representative for conditions in a high-speed railway track. The critical load velocities will be determined for several cases, where the effect by the ballast particle size, the ballast layer thickness, the load velocity and the sleeper distance will be analysed.

Surface waves in a stratified half space with enhanced continuum properties. Part 1: Formulation of the boundary value problem

This paper discusses the propagation of surface waves in an elastic stratified half space, which consists of a granular elastic upper layer resting on a classic elastic substratum. We will consider both the case of an upper layer with second-gradient continuum properties, and an upper layer with Cosserat continuum properties. These enhanced continuum formulations are able to describe micro-scale effects as governed by the individual particle displacements and particle rotations. The micro-scale effects become important when the length of the surface waves is of the same order of magnitude as the particle size, which may occur in a high-speed railway track that consists of an upper layer of ballast particles. In an accompanying paper we will therefore analyse the critical structural response of the stratified half space when it is subjected to a moving, harmonically vibrating load. This model is representative for a high-speed railway system.

A transverse aperture‐integral equation solution for edge diffraction by multiple layers of homogeneous material

The transverse aperture‐integral equation method provides a means of computation for diffraction coefficients at blunt edges of a broad class of stratified layers, including sheet‐anisotropy models for conducting composites. In this paper we concentrate on the application of the method when the material profile comprises layers of homogeneous, potentially lossy material. The method proceeds from defining an artificial aperture perpendicular to a semi‐infinite, planar, stratified region and passing through the terminal edge of the region. An integral equation is formulated over this infinite‐extent aperture, and the solution to the integral equation represents the influence of the edge. The kernel in the integral equation is a weighted sum of the Green's functions for the respective half spaces lying on either side of the aperture plane. The vector wave equation is separable in each of these half spaces, resulting in Green's functions that are expressible analytically. The Green's function for the stratified half space is stated in terms of a Sommerfeld‐type integral.

Convergence of difference methods for one-dimensional inverse problems

We consider the one-dimensional inverse problem of reflection seismology. An impulsive plane wave in pressure is applied to toe surface of a stratified elastic half space and the resulting particle velocity is measured at the surface. The characteristic impedance of the medium is to be recovered as a function of travel time. We study two methods for the numerical solution of the inverse problem. One method la a discretization of the Gopinath-Sondhi integral equation. The other is a downward continuation method. Both methods are exact for so-called Goupilllaud-tayered media. Both use second-order difference schemes to reconstruct a solution of the system of partial differential equations governing motion in the medium. A formally second-order accurate equation based on the theory of propagation of singularities is used to recover the impedance. We prove that these two methods are second-order convergent for smoothly stratified media by analyzing the discretization of the Gopinath-Sondhi integral equation.

Electromagnetic scattering from subterranean obstacles in a stratified ground

In this paper we give a general three‐dimensional treatment of electromagnetic scattering from a finite obstacle in a stratified half space. The two generic cases of an obstacle either below all layers or inside one of the layers are treated. The results are an extension of earlier works on scattering in related geometries, treated by means of the T matrix method. Numerical results are presented for subterranean inhomogeneities of various shapes in a lossy one‐layered half space with plane interfaces.

Seismic waves in a stratified half space?III. Piecewise smooth models

Summary. A recursive procedure is introduced for the calculation of the properties of an earth model consisting of regions with smoothly varying properties separated by discontinuities in seismic parameters or parameter gradients. Within each region the solution is constructed from a Langer uniform approximation in terms of Airy functions supplemented by a series of terms representing multifold wave interaction with the parameter gradients. This procedure allows an efficient treatment of turning point problems and in general it is an adequate approximation to retain only the leading order term in the interaction series. At an interface between varying media the solution may be expressed in terms of generalized interface coefficients or alternatively recast into a form which separates the effects of gradients and the interface itself. In the latter case the plane wave reflection and transmission coefficients are to be used. The resulting calculation scheme for the reflection matrices is an extension of the recursive scheme for uniform layers. The simple phase delay transmission effect of a uniform region is replaced by the reflection and transmission from a gradient zone sandwiched between two uniform media. This recursive scheme gives good results for crustal and upper mantle models, and only about a dozen subdivisions of the stratification are required down to 950 km. Checks on the accuracy of the computation from unitarity relations between reflection coefficients show that for periods less than 20 s, in the upper mantle, the error associated with neglect of the interaction series is less than 0.1 per cent except for a few cases. When a turning point is just at a structural boundary the error can increase to 2 per cent. For a low-velocity zone where close turning points occur for slownesses within the range of the inversion, rather larger errors may occur but these do not affect waves which turn well above or well below the zone. The leading order approximation allows no conversion of wave types except at a discontinuity in elastic properties and the error introduced by neglecting conversions at steep gradient zones of limited vertical extent (as at upper mantle discontinuities) may reach a few per cent.

On coupled seismic waves

Summary. The response of a stratified elastic medium can be conveniently characterized by the Green's tensor for the medium. For coupled seismic wave propagation (P—SV or fully anisotropic), the Green's tensor may be constructed directly from two matrices of linearly independent displacement solutions. Rather simple forms for the Green's tensor can be found if each displacement matrix satisfies one of the boundary conditions on the seismic field. This approach relates directly to ‘reflection matrix’ representations of the seismic field. For a stratified elastic half space the Green's tensor is used to give a spectral representation for coupled seismic waves. By means of a contour integration a general completeness relation is obtained for the ‘body wave’ and 'surface wave’ parts of the seismic field. This relation is appropriate for SH and P–SV waves in an isotropic medium and also for full anisotropy.

Maximum entropy and the relationship of the partial autocorrelation to the reflection coefficients of a layered system

A horizontally stratified half space bounded by a perfect reflector at the top gives rise to a seismogram which, when completed by the direct downgoing pulse at zero time and by symmetry about the origin for negative time, produces an autocorrelation function. If this autocorrelation is convolved with the corresponding prediction error operators of increasing length, we obtain a "gapped function," which deviates more and more from the perfect symmetry exhibited by the autocorrelation. This gapped function consists of the downgoing and upgoing waveforms at the top of each layer. The gap separates the two waveforms, and the gapwidth increases as deeper and deeper layers are reached. In particular, the width of the gap is a measure of the entropy of the seismogram at a given depth level-the deeper we go into the sub-surface, the higher the entropy of the corresponding gapped function. We explore the nature of the gapped function as it relates to the Toeplitz recursion generating the prediction error operators, and we re-derive the synthetic seismogram in terms of wave motion measured in units proportional to the square root of energy. We obtain an explicit relationship between the partial autocorrelation function on the one hand, and the reflection coefficient sequence on the other. This formulation allows us to generalize earlier results, so that we can treat the case for which the surface reflection coefficient is less than unity in magnitude. We investigate both the physical as well as the mathematical foundations of the stratified model, and the relation that this model bears to maximum entropy spectral analysis [1], [6]. In particular, we discuss Burg's fundamental result on the relationship of the maximum entropy spectrum to the reflection coefficient sequence characterizing the given subsurface model.

Seismic waves in a stratified half space

The response of a stratified elastic half space to a general source may be represented in terms of the reflection and transmission properties of the regions above and below the source. For P-SV and SH waves and both buried sources and receivers, convenient forms of the response may be found in which no loss of precision problems arise from growing exponential terms in the evanescent regime. These expressions have a ready physical interpretation and enable useful approximations to the response to be developed. The reflection representation leads to efficient computational procedures for models composed of uniform layers, which may be extended in an asymptotic development to piecewise smooth models.

Scattering of electromagnetic waves by a perfectly conducting half plane below a stratified overburden

By expanding the electromagnetic field into plane wave eigenfunctions, an integral equation is obtained which describes the coupling between an overburden, consisting of an arbitrary number of plane layers and an arbitrary object beneath a stratified half space. As an example of the technique, a perfectly conducting half plane inclined to the strata in a substratum is discussed. In this case the kernel of the integral equation may be expressed in terms of elementary functions. Numerical results are discussed for a plane wave incident on the ground.

Electromagnetic Wave Propagation in a Moving Sinusoidal Stratified Turbulent Plasma Medium

In this paper the propagation of electromagnetic waves is considered to be obliquely incident on a sinusoidally turbulent stratified half face, both horizontally and vertically polarization are taken for waves.The reflection and transmission of electromagnetic waves by a moving turbulent dispersive dielectric plasma medium half face or slab are studied theoretically. The propagation of electromagnetic wave is assumed to be polarized in a direction that is (a) normal, and (b) parallel to the plane of incidence. For this study we consider case; the medium is moving parallel to the interface. The dependence of the reflection and transmission coefficient on the motion is discussed.

Dispersion ofSH body waves in a continuously stratified elastic half space

The dispersion ofSH body waves in a continously stratified isotropic medium is studied theoretically. Limiting forms of the displacement are given for zero, small and large values of the horizontal component of the wave number. It has been shown that below a certain characteristic frequency the dependence of the displacement on the distance from the free surface is monotonic instead of periodic.

Propagation of sound pulses in a homogeneous layer under an inhomogeneous half space

We consider the propagation of two-dimensional sound pulses in a homogeneous layer −y 1⩽y⩽0. It is bounded by a plane stratified inhomogeneous half spacey⩾0 on one side and a perfectly reflecting boundary on the other. A line source is situated in the layer. The boundary condition isφ=0 or∂φ/∂y=0 aty=−y 1, whereφ is the acoustic velocity potential. We suppose that the velocity of wave propagationc is given byc −2=p−qe −αy iny>0, wherep, q, α are real and positive andp>q. It is equal to C′ in the layer where C′ is a constant. The method of dual integral transformation is used and the velocity potentialφ is obtained after using asymptotic expressions for some of the functions which are in the integrand. We obtain the incident, reflected, multiply-reflected and diffracted pulses in the layer.

THE VERTICAL MAGNETIC FIELD OF AN ALTERNATING CURRENT DIPOLE FOR HORIZONTALLY STRATIFIED MEDIA *

ABSTRACTFor the computation of the vertical component Hz of the magnetic field of a horizontal A.C. dipole lying on the earth's surface, a recurrence formula is presented for a horizontally stratified half space, to obtain the (n+ 1)‐layer case from the w‐layer case.By means of several computed diagrams for the two‐layer case, Hz can be determined for different ratios of conductivity of the subsoil and that of the overburden. Thereby the distance from the dipole as well as the layer thickness h are expressed in terms of the wave length A of a plain wave in the overburden. Assuming a sufficiently large conductivity difference, the results show that evidence about the subsurface conditions can be obtained if the distance between the measuring coil and the dipole is of the order of A/3, and if the thickness h of the layer varies within the range A/100 < h < A/6.As an example for the 3‐layer case, a nonconducting intermediate layer is assumed.