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So far, we have only solved the wave equation for the special case of an ideal dielectric of zero conductivity y. We shall now extend our considerations to media of finite conductivity.

In the first part a system of equations for an inductive charged relativistic fluid with finite conductivity is written in a space time with given metric, taking into account thermodynamic phenomena. Speeds of propagation of various types of waves are determined under a restrictive hypothesis concerning the heat currentq: thatq depends only on the thermodynamical quantities and the gradient of one function of these quantities.

In the present paper, we formulate the equations of magnetovisco-elasticity and appropriate boundary conditions for a linearly viscoelastic medium with finite conductivity. In particular, we shall consider the boundary conditions for an ideal medium, that is perfectly elastic and perfectly conductive.

It has been verified experimentally that the value of the back switched ferroelectric polarization in PVDF during short circuiting depends on the preceding poling time, as was predicted earlier by analyzing a model suggested for initial poling. The back switched polarization exhibits a maximum in samples poled for about 100 /spl mu/s, while it decreases to zero for much shorter and much longer poling times. The experimentally found build-up of a slow polarization component, as well as the high stability of polarization in long time poled PVDF are explained by the charge accumulation at the interfacial boundaries between ferroelectric and non-ferroelectric components of PVDF due to a finite conductivity. The suggested model is applicable for all two-component ferroelectrics.

Practical planar circuits involve metallic edges with finite conductivity and non-ideal dielectrics: it is usually more or less implicitly assumed that fields and induced currents behave as if conductors and dielectrics were ideal. In this paper we show such an assumption to be erroneous and that the presence of real conductors and dielectrics leads to a simpler and more physical picture.

Hydraulic fracturing is important to the development of low permeability reservoirs. Recent advances in the technology of hydraulic fracturing, coupled with its economic success, have led to its application in high permeability reservoirs. Consequently, fracturing is now applied in all types of formations including highly faulted systems. The existence of a fault near a well may impede or contribute to the fracture extension process. Various models and techniques have been introduced in the literature for the interpretation of transient pressure behavior of vertically fractured wells. However, none of the current models and techniques has taken into account the case of a fracture intersecting a fault. This paper presents a theoretical model for analyzing the pressure behavior of a finite-conductivity vertical hydraulic fracture intersected by a sealing-finite fault, considering different angles of intersection and neglecting in all cases wellbore and fracture storage effects. The point at which fracture and fault conjoint has also been a matter of analysis in this study. The model was solved numerically using PEBI grids. For the case where the fracture-fault system form an angle of 90 or 75°, the pressure response is characterized by a maximum point or "hump" in the pressure derivative which takes place once the transient response of the fracture has vanished. This is because the fault delays the development of the pseudoradial flow regime. For smaller angles, a peak is not observed; however, there is a transition period that causes an apparent pseudoradial flow to develop earlier. This is because the hump tries to develop but the fault is not long enough for that to happen. When the fault is very long (LD > re), pseudoradial flow does not develop because the fault acts as a boundary. A horizontal line on the pressure derivative curve is observed which can be easily confused with that of the actual pseudoradial flow regime. Correlations to estimate both fault length and fracture-fault angles are presented. A simulated example is presented to illustrate the proposed step-by-step procedure for interpreting a post-fracture pressure test of well near a sealing-fault.

Summary This paper presents a hybrid numerical/analytical model for the pressure-transient response of a finite-conductivity fracture intercepted by a horizontal well. The model dynamically couples a numerical fracture model with an analytical reservoir model. This approach allows us to include finer details of the fracture characteristics while keeping the computational work manageable. For example, the fracture may have irregular shape, nonuniform width, and variable conductivity, and the well may not intersect the fracture at its geometric center. In this paper, we use the hybrid model to investigate the effects of fracture properties on the pressure-transient characteristics of a single, finite-conductivity horizontal-well fracture. The single horizontal-well-fracture model can be extended easily to multiply fractured horizontal wells by superposition. The model also can be used to compute the pseudoskin caused by the effects of nonideal fracture geometry, variable conductivity, and flow choking around the wellbore and to investigate the influence of fracture properties on the performance of horizontal wells.

We present a fundamental and accurate approach to compute the attenuation of electromagnetic waves propagating in rectangular waveguides with finite conductivity walls. The wavenumbers kx and ky in the x and y directions respectively, are obtained as roots of a set of transcendental equations derived by matching the tangential component of the electric field (E) and the magnetic field (H) at the surface of the waveguide walls. The electrical properties of the wall material are determined by the complex permittivity ε, permeability μ, and conductivity σ. We have examined the validity of our model by carrying out measurements on the loss arising from the fundamental TE10 mode near the cutoff frequency. We also found good agreement between our results and those obtained by others including Papadopoulos' perturbation method across a wide range of frequencies, in particular in the vicinity of cutoff. In the presence of degenerate modes however, our method gives higher losses, which we attribute to the coupling between modes as a result of dispersion.

The errors in phase-difference measurement which arise when using the Decca Navigator system, operating at frequencies near 100 kc/s, in the immediate neighbourhood of obstacles such as trees and telegraph wires, have been examined; the results of the measurements are described and discussed. Vertical obstacles show some uniformity of behaviour, but long horizontal conductors produce widely varying effects. A tentative theory of the phase errors due to a single tree is given in an Appendix, and fairly good agreement with observations is obtained. In Section 2 of the paper, errors obtained at good sites on the base line extensions of the red and green lattices are shown graphically. The effects of the near-field components and of the finite conductivity of the ground are apparent. Good agreement is obtained between the mean of the observations and a composite theoretical curve, based on the use of Norton's plane-earth theory and Bremmer's curved-earth theory of ground-wave propagation. Rapid fluctuations of error about the mean curve are observed, indicating that the phase velocity of the ground wave may show variations from point to point of up to 1% of the mean value.

The possibility of making perfectly conducting rectangular-groove gratings perfectly blazed simultaneously in both polarizations is investigated and numerous efficiency curves are given. Attention is given to the fact that these curves are unfortunately greatly modified if finite conductivity is taken into account.

The force on a plasma which is moving relative to a finite conductivity wall is evaluated assuming that the plasma is distorted by a saturated magnetohydrodynamic mode.

The response of a finite conductivity plasma to resonant magnetic perturbations is studied. The equations, which are derived for the time development of magnetic islands, help one to interpret the singular currents which occur under the assumption of perfect plasma conductivity. The relation to the Rutherford regime of resistive instabilities is given.

Apparent thermal conductivity ka, measured with a line source lying between two dissimilar media, equals the mean conductivity of the two media, ka = (k1+k2)/2. This theoretically derived formula is substantiated by data in the literature. With reference to a medium of known conductivity k1, the relation justifies the development of an inexpensive, simple, rapid, and reliable method to determine conductivity k2 of samples which cannot be penetrated with a conventional needle probe. Measurements by methods of half-space needle probe or half-space line source have been made in the past by discarding the contribution of k1, resulting in overestimation of conductivity if correction through calibration is not made. For example, with respect to an epoxy half-space probe, a sample of fused silica (k2 = 1.36 W m−1 K−1) could be overestimated by 6.5% if the finite conductivity of epoxy (k1 = 0.087 W m−1 K−1) is neglected.

A wave-theoretical analysis of MESFET electrode structures is undertaken, incorporating losses of both the semiconducting layer and the finite conductivity of the electrodes. It is shown that the relevant microwave FET mode has slow-wave properties with a slowing factor of about 6.

Results described here show that infinite conductivity theory matches experimental grating efficiencies very closely when wavelengths exceed 4 µm, in fact often serves adequately above 1 µm. At shorter wavelengths, i.e. the visible region, the influence of finite conductivity on the theory must be included, in which case once again excellent agreement with experiment can be demonstrated. The nature of anomalies, especially as affected by different metal surfaces, is a highly sensitive test for theory. In the vacuum ultraviolet region correspondence between theory and experiment could still be observed, although some new practical limitations intervene.

The theory of cylindrical probes for measuring thermal conductivity is extended to the case of a probe of finite conductivity containing a line source at its centre. This provides a more realistic approximation to most actual probes than the theory for a probe of infinite conductivity developed by other authors. New experimental results are presented which are in complete agreement with theory It is shown how an estimate can be obtained of the magnitude of a possible thermal contact resistance between the probe and the medium and how its influence on the measured conductivity can be assessed.

The velocity of guided electromagnetic (e.m.) waves in a stripline of GaAs is strongly reduced by the finite conductivity in the strip. Travelling-domain formation is prevented if 2c becomes smaller than the domain drift velocity, and a static high-field domain may form at the anode. The slow e.m. wave is expected to influence strongly the transmission-amplification gain.

The height dependence of the field strength of VLF radio waves is considered. Using previously developed theory, the height-gain function of the first two modes is calculated in terms of Airy functions of complex argument. It is indicated, for frequencies of the order of 25 kc/s, that the height-gain function reaches a maximum value at a height of the order of 40 km when the reflecting layer is about 70 km. The form of the height-gain function is also shown to be dependent on the finite conductivity of the ground. An experimental curve for 18.0 kc/s based on a rocket measurement shows some agreement with the theory.

A method is described of calculating the losses in interdigital and combline filters due to the finite conductivity of the materials. Factors which degrade the theoretical unloaded Q factor of the individual elements are mentioned, and a practical example is given, showing excellent agreement between theoretical and measured results.

A prolate spheroidal dipole antenna embedded in a larger dielectric confocal prolate spheroid with finite conductivity is analized as a boundary value problem in electromagnetic theory; the expressions for the fields, antenna current distribution and input impedance are obtained.