Half-Spaces

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This... 

In this section, two illustrative numerical results, obtained by means of the described reconstruction algorithm, are presented. Input data are calculated in the frequency range of 26 to 38 GHz using matrix formulas [8], describing the reflection of a normally incident plane wave from the multilayered half-space. 

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

Figure 2 a) A deep sub-surface crack in a half-space conductor b) the utilized transducer... 

Fig. 7 Error of measuring crack depth in plate by using calibration curve for crack in half space.

Here we have carried out the velocity integral over the required half-space and used the explicit fonn of the Maxwell-Boltzmaim distribution fiinction, given by equation (A3.1.3). 

The constant, C, depends on the dimensions of the anode bed. It follows from line 4 of Table 24-1 for a horizontal ground in the half space... 

Current and voltage distribution remain completely the same if a plane cuts the sphere through its center and the upper half is removed (half space). Since only half the current flows from the hemisphere, its grounding resistance is obtained by substituting 1/2 for 7 in Eq. (24-8) (see Table 24-1, column 1) ... 

If the spherical anode is situated at a finite depth, f, the resistance is higher than for t and lower than for t = 0 (hemisphere at the surface of the electrolyte). Its value is obtained by the mirror image of the anode at the surface (f = 0), so that the sectional view gives an equipotential line distribution similar to that shown in Fig. 24-4 for the current distribution around a pipeline. This remains unchanged if the upper half is removed (i.e., only the half space is considered). 

The formula for the grounding resistance of a circular plate in half space [2] which can be used as an approximation for a defect in the pipe coating gives double the resistance... 

For an elliptical plate in half space with the axes a and b, the resistance is approximately... 

The derivation for the general case of the horizontal rod anode in half space with the notation in Table 24-1, line 9 [7] gives ... 

It has been also shown that when a thin polymer film is directly coated onto a substrate with a low modulus ( < 10 MPa), if the contact radius to layer thickness ratio is large (afh> 20), the surface layer will make a negligible contribution to the stiffness of the system and the layered solid system acts as a homogeneous half-space of substrate material while the surface and interfacial properties are governed by those of the layer [32,33]. The extension of the JKR theory to such layered bodies has two important implications. Firstly, hard and opaque materials can be coated on soft and clear substrates which deform more readily by small surface forces. Secondly, viscoelastic materials can be coated on soft elastic substrates, thereby reducing their time-dependent effects. 

Boussinesq and Cerruti made use of potential theory for the solution of contact problems at the surface of an elastic half space. One of the most important results is the solution to the displacement associated with a concentrated normal point load P applied to the surface of an elastic half space. As presented in Johnson [49]... 

Similarly, the assumption that the contact area is small enough that the particle can be represented by an elastic half space allows the radii of the two contacting particles to be combined into a single effective radius that represents how the contacting shapes interact. 

To the best of our knowledge, there was only one attempt to consider inhomogeneous fluids adsorbed in disordered porous media [31] before our recent studies [32,33]. Inhomogeneous rephca Ornstein-Zernike equations, complemented by either the Born-Green-Yvon (BGY) or the Lovett-Mou-Buff-Wertheim (LMBW) equation for density profiles, have been proposed to study adsorption of a fluid near a plane boundary of a disordered matrix, which has been assumed uniform in a half-space [31]. However, the theory has not been complemented by any numerical solution. Our main goal is to consider a simple model for adsorption of a simple fluid in confined porous media and to solve it. In this section we follow our previously reported work [32,33]. 

As an adsorption geometry one considers a semi-infinite system with an impenetrable wall at z = 0, such that monomer positions are restricted to the positive half-space z > 0. At the wall acts a short-range attractive potential, either as a square well... 

Since the interface behaves like a capacitor, Helmholtz described it as two rigid charged planes of opposite sign [2]. For a more quantitative description Gouy and Chapman introduced a model for the electrolyte at a microscopic level [2]. In the Gouy-Chapman approach the interfacial properties are related to ionic distributions at the interface, the solvent is a dielectric medium of dielectric constant e filling the solution half-space up to the perfect charged plane—the wall. The ionic solution is considered as formed... 

Diffraction of detonation from a tube (0=52 mm) to half-space through an annular orifice with central conical obstacle ( = 15°) in a C2H2 + 2.5O2 mixture at 33 mbar initial pressure. Comparison of experimental soot patterns with numerical simulations. (Courtesy of B. A. Khasainov.)... 

We have derived a formula for calculation of the potential of the field everywhere in the upper half space if both the potential U and the vertical component of the attraction field are known at the earth s surface, iS o- Having taken the derivative from both sides of Equation (1.110) we obtain for the vertical component of the attraction field at the point p ... 

Equation (1.116) allows us to calculate the vertical component of the field in the upper half space when it is known at the earth s surface. Correspondingly, this transformation is called upward continuation and it is used to reduce an infiuence of geological noise, caused by heterogeneities with relatively small dimensions. 

Suppose that we know the field at points of some plane and that it is caused by an arbitrary body, located beneath this surface of observation. Also, the field g, Fig. 1.12e, vanishes far away from the body. The lower half space is a volume where all the sources are located and it is surrounded by the observation plane S and a hemi-spherical surface Sq with relatively large radius where the field can be treated as that of a point source. Correspondingly, the flux through this surface is... 

The solid angle under which the plane is seen from the front and back sides does not depend on the position of the point p and is equal to + 2n, respectively. In other words, planar surface masses with infinite extension and constant density create a uniform field in each half space ... 

A half space with density 5q z) that creates a background for a secondary field. 

The role of a boundary in a manifold with boundary can be interpreted with reference to a hyperplane within a Euclidean space E using the concept of halfspace, where the hyperplane is in fact the boundary of the half-space. By appropriate reordering of the coordinates, a half-space Hn becomes the subset of a Euclidean space En containing all points of En with non-negative value for the last coordinate. 

A space M where each point x e M has an open neighborhood homeomorphic to a set open within a Euclidean half-space Hn, is an K-dimensional manifold with boundary. 

Consider a plane metal electrode situated at z = 0, with the metal occupying the half-space z < 0, the solution the region z > 0. In a simple model the excess surface charge density a in the metal is balanced by a space charge density p(z) in the solution, which takes the form p(z) = Aexp(—kz), where k depends on the properties of the solution. Determine the constant A from the charge balance condition. Calculate the interfacial capacity assuming that k is independent of a. 

The feasible region lies within the unshaded area of Figure 7.1 defined by the intersections of the half spaces satisfying the linear inequalities. The numbered points are called extreme points, comer points, or vertices of this set. If the constraints are linear, only a finite number of vertices exist. 

Two half-spaces with uniform initial concentrations... 

The infinite medium with one-dimensional diffusion and constant diffusion coefficient can be treated easily with the point source theory. Let us first assume that two half-spaces with uniform initial concentrations C0 for x < 0 and 0 for x > 0 are brought into contact with each other. The amount of substance distributed per unit surface between x and x + dx is just C0dx. From the previous result, at time t the effect of the point source C0 dx located at x on the concentration at x will be... 

Summing over all the point sources at x from — oo to -t-oo and noting that the contribution from the half-space x > 0 is zero will give the concentration distribution... 

Figure 8.14 Two half-spaces in contact with each other at x = 0 have the concentration C0 for x<0 and 0 for x>0. Interface concentration is constant and equal to C0/2. The curves are labeled for different values of the parameter On.

Let us consider the influence of a solid-liquid interface advancing at a constant velocity on the solid-liquid fractionation of an element i. In the case of unidirectional solidification, it is convenient to consider that liquid crosses the immobile interface with an absolute constant velocity v, while a solid-liquid fractionation coefficient K is applied to the fractionation of element i. Let us assume that the interface is at x=0, the medium being solid for x<0. Liquid fills the half-space 0[Pg.442]

---