Static and propagating exponential solutions of Maxwell’s equations for crystals

1992 ◽  
Vol 438 (1904) ◽  
pp. 485-510
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Plane Waves ◽  
Electric Permittivity ◽  
Complex Vector ◽  
Time Harmonic ◽  
Inhomogeneous Plane ◽  
Isotropic Media ◽  
Special Case ◽  
Inhomogeneous Plane Waves

Solutions of Maxwell’s equations are considered for anisotropic media for which the electric permittivity κ and magnetic permeability µ are assumed to be arbitrary real positive definite symmetric second order tensors. The propagation of time-harmonic electromagnetic inhomogeneous plane waves or ‘propagating exponential solutions’ in such media has been presented previously. These solutions were systematically obtained by prescribing an ellipse - the directional ellipse associated with a bivector (complex vector) C - and finding the corresponding slowness bivectors. Here, it is shown that for some prescribed directional ellipses, not only propagating exponential solutions (PES), but also static exponential solutions (SES), may be obtained. There are a variety of possibilities. For example, for one choice of directional ellipse it is found that two SES and one PES are possible, whereas, for some other choices, only one SES or only one PES is possible. By a systematic use of bivectors and their associated ellipses, all the possible SES and PES are classified. To complete the classification it is necessary to examine special elliptical sections of the ellipsoids associated with the tensors κ , µ , κ -1 , µ -1 . In particular, sections by planes orthogonal to special directions called ‘generalized optic axes’ and ‘generalized ray axes’ play a major role. These axes reduce to the standard optic axes and ray axes in the special case of magnetically isotropic media.

Transient representation of the Sommerfeld‐Weyl integral with application to the point source response from a planar acoustic interface

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1495-1505 ◽  
Author(s):  
Martin Tygel ◽  
Peter Hubral
Keyword(s):  
Point Source ◽  
Plane Waves ◽  
Transient Waves ◽  
Layer Boundary ◽  
Spherical Source ◽  
Time Harmonic ◽  
Starting Point ◽  
Inhomogeneous Plane ◽  
Acoustic Interface ◽  
Inhomogeneous Plane Waves

Point source responses from a planar acoustic and/or elastic layer boundary (as well as from a stack of planar parallel layers) are generally obtained by using as a starting point the Sommerfeld‐Weyl integral, which can be viewed as decomposing a time‐harmonic spherical source into time‐harmonic homogeneous and inhomogeneous plane waves. This paper gives a powerful extension of this integral by providing a direct decomposition of an arbitrary transient spherical source into homogeneous and inhomogeneous transient plane waves. To demonstrate with an example the usefulness of this new point source integral representation, a transient solution is formulated for the reflected/transmitted response from a planar acoustic reflector. The result is obtained in the form of a relatively simple integral and essentially corresponds to the solution obtained by Bortfeld (1962). It, however, is arrived at in a physically more transparent way by strictly superimposing the reflected/transmitted transient waves leaving the interface in response to the incident transient homogeneous and inhomogeneous plane waves coming from the center of the point source.

Quality factor Q in dissipative anisotropic media

Geophysics ◽  
2008 ◽  
Vol 73 (4) ◽  
pp. T63-T75 ◽  
Author(s):  
Vlastislav Červený ◽  
Ivan Pšenčík
Keyword(s):  
Quality Factor ◽  
Plane Waves ◽  
Anisotropic Media ◽  
Exact Expression ◽  
Dissipated Energy ◽  
S Wave ◽  
Time Harmonic ◽  
Inhomogeneous Plane ◽  
Quality Factor Q ◽  
Inhomogeneous Plane Waves

In an isotropic dissipative medium, the attenuation properties of rocks are usually specified by quality factor [Formula: see text], a positive, dimensionless, real-valued, scalar quantity, independent of the direction of wave propagation. We propose a similar, scalar, but direction-dependent quality [Formula: see text]-factor (also called [Formula: see text]) for time-harmonic, homogeneous or inhomogeneous plane waves propagating in unbounded homogeneous dissipative anisotropic media. We define the [Formula: see text]-factor, as in isotropic viscoelastic media, as the ratio of the time-averaged complete stored energy and the dissipated energy, per unit volume. A solution of an algebraic equation of the sixth degree with complex-valued coefficients is necessary for the exact determination of [Formula: see text]. For weakly inhomogeneous plane waves propagating in arbitrarily anisotropic, weakly dissipative media, we simplify the exact expression for [Formula: see text] con-siderably using the perturbation method. The solution of the equation of the sixth degree is no longer required. We obtain a simple, explicit perturbation expression for the quality factor, denoted as [Formula: see text]. We prove that the direction-dependent [Formula: see text] is related to the attenuation coefficient [Formula: see text] measured along a profile in the direction of the energy-velocity vector (ray direction). The quality factor [Formula: see text] does not depend on the inhomogeneity of the plane wave under consideration and thus is a convenient measure of the intrinsic dissipative properties of rocks in the ray direction. In all other directions, the quality factor is influenced by the inhomogeneity of the wave under consideration. We illustrate the peculiarities in the behavior of [Formula: see text] and its accuracy on a model of anisotropic, weakly dissipative sedimentary rock. Examples show interesting inner loops in polar diagrams of [Formula: see text] in regions of S-wave triplications.

scholarly journals A UNIQUENESS THEOREM FOR AN INVERSE ELECTROMAGNETIC SCATTERING PROBLEM IN INHOMOGENEOUS ANISOTROPIC MEDIA

2003 ◽  
Vol 46 (2) ◽  
pp. 293-314 ◽  
Author(s):  
Fioralba Cakoni ◽  
David Colton
Keyword(s):  
Anisotropic Medium ◽  
Electromagnetic Scattering ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Scattering Problem ◽  
Plane Waves ◽  
Field Patterns ◽  
Regularity Properties ◽  
Time Harmonic ◽  
Secondary 35P25

AbstractWe show that the support of a (possibly) coated anisotropic medium is uniquely determined by the electric far-field patterns corresponding to incident time-harmonic electromagnetic plane waves with arbitrary polarization and direction. Our proof avoids the use of a fundamental solution to Maxwell’s equations in an anisotropic medium and instead relies on the well-posedness and regularity properties of solutions to an interior transmission problem for Maxwell’s equations.AMS 2000 Mathematics subject classification: Primary 35R30; 35Q60. Secondary 35P25; 78A45

scholarly journals Abstract Nonconforming Error Estimates and Application to Boundary Penalty Methods for Diffusion Equations and Time-Harmonic Maxwell’s Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 451-475 ◽  
Author(s):  
Alexandre Ern ◽  
Jean-Luc Guermond
Keyword(s):  
Error Analysis ◽  
Error Estimates ◽  
Material Properties ◽  
Maxwell’S Equations ◽  
Vector Fields ◽  
Maxwell's Equations ◽  
Heterogeneous Material ◽  
Test Functions ◽  
Time Harmonic ◽  
New Formulation

AbstractWe devise a novel framework for the error analysis of finite element approximations to low-regularity solutions in nonconforming settings where the discrete trial and test spaces are not subspaces of their exact counterparts. The key is to use face-to-cell extension operators so as to give a weak meaning to the normal or tangential trace on each mesh face individually for vector fields with minimal regularity and then to prove the consistency of this new formulation by means of some recently-derived mollification operators that commute with the usual derivative operators. We illustrate the technique on Nitsche’s boundary penalty method applied to a scalar diffusion equation and to the time-harmonic Maxwell’s equations. In both cases, the error estimates are robust in the case of heterogeneous material properties. We also revisit the error analysis framework proposed by Gudi where a trimming operator is introduced to map discrete test functions into conforming test functions. This technique also gives error estimates for minimal regularity solutions, but the constants depend on the material properties through contrast factors.

scholarly journals Two basic systems of maxwell’s equations in a rotating frame: application in theory of ring laser gyro

2020 ◽  
pp. 64-73
Author(s):  
Evgen Bondarenko
Keyword(s):  
Ring Laser ◽  
Electromagnetic Waves ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Plane Waves ◽  
Rotating Frame ◽  
Cylindrical Coordinates ◽  
Laser Gyro ◽  
Component Form ◽  
Velocity Approximation

In the paper, using a linear in angular velocity approximation, two basic well-known systems of Maxwell’s equations in a uniformly rotating frame of reference are considered. The first system of equations was first obtained in the work [L. I. Schiff, Proc. Natl. Acad. Sci. USA 25, 391 (1939)] on the base of use of the formalism of the theory of general relativity, and the second one – in the work [W. M. Irvine, Physica 30, 1160 (1964)] on the base of use of the method of orthonormal tetrad in this theory. In the paper, in the approximation of plane waves, these two vectorial systems of Maxwell’s equations are simplified and rewritten in cylindrical coordinates in scalar component form in order to find the lows of propagation of transversal components of electromagnetic waves in a circular resonator of ring laser gyro in the case of its rotation about sensitivity axis. On the base of these two simplified systems of Maxwell’s equations, the well-known wave equation and its analytical solutions for the named transversal components are obtained. As a result of substitution of these solutions into the first and second simplified systems of Maxwell’s equations, it is revealed that they satisfy only the second one.  On this basis, the conclusion is made that the second system of Maxwell’s equations is more suitable for application in the theory of ring laser gyro than the first one.

scholarly journals Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

2021 ◽  
Author(s):  
Rainer Mandel ◽  
Robert Schippa
Keyword(s):  
Gaussian Curvature ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Smooth Surfaces ◽  
Negative Index ◽  
Elliptic Surfaces ◽  
Conical Singularities ◽  
Fourier Restriction ◽  
Time Harmonic ◽  
Homogeneous Media

AbstractWe solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

scholarly journals New capabilities of Chetaev´s model

2016 ◽  
Vol 2 (2) ◽  
pp. 104-114
Author(s):  
Михаил Савин ◽  
Mihail Savin ◽  
Юрий Израильский ◽  
Yuriy Izrailsky
Keyword(s):  
Experimental Data ◽  
Plane Waves ◽  
Reflection Coefficients ◽  
Geomagnetic Pulsations ◽  
Energy Fluxes ◽  
Field Recordings ◽  
Inhomogeneous Plane ◽  
Inhomogeneous Plane Wave ◽  
S Model ◽  
Inhomogeneous Plane Waves

This paper considers anomalies in the magnetotelluric field in the Pc3 range of geomagnetic pulsations. We report experimental data on Pc3 field recordings which show negative (from Earth’s surface to air) energy fluxes Sz<0 and reflection coefficients |Q|>1. Using the model of inhomogeneous plane wave (Chetaev’s model), we try to analytically interpret anomalies of energy fluxes. We present two three-layer models with both electric and magnetic modes satisfying the condition |Qh|>1. Here we discuss a possibility of explaining observable effects by the resonance interaction between inhomogeneous plane waves and layered media.

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