Green’s tensors and associated potentials for electromagnetic waves in inhomogeneous material media

1989 ◽  
Vol 426 (1870) ◽  
pp. 79-106 ◽  
Keyword(s):  
Electromagnetic Waves ◽  
Mathematical Formulation ◽  
Electric Quadrupole ◽  
Source Zone ◽  
Scattering Problems ◽  
Inhomogeneous Dielectric ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
Constitutive Parameters ◽  
Finite Domains

The explicit form of the Green’s tensor in unbounded inhomogeneous dielectric, magnetic and partly conducting material is obtained. The tensor is given explicitly in terms of two scalar Green’s function potentials, each of which obeys its own wave equation. It is found that the gradients of the constitutive parameters at the source of the radiating currents may have a strong effect on the multipolar field. For example, an electric dipole embedded in a uniaxially inhomogeneous source-zone is found to produce additional fields with the characteristics of electric quadrupole ( l = 2, m = ± 1) and a magnetic dipole ( l = 1, m = ± 1). I present a mathematical formulation of Huygens’ principle for inhomogeneous media in terms of the Green’s tensor and its associated tensor potentials, thus extending the use of the Green’s tensor to scattering problems in inhomogeneous finite domains.

Scattering of a Plane Elastic Wave From Objects Near an Interface

1972 ◽  
Vol 39 (4) ◽  
pp. 1019-1026 ◽  
Author(s):  
Stephen B. Bennett
Keyword(s):  
Elastic Wave ◽  
Electromagnetic Waves ◽  
Three Dimensions ◽  
Circular Cylinders ◽  
Scattering Problems ◽  
Reflection And Refraction ◽  
Time Harmonic ◽  
Incident Plane ◽  
Boundary Curves ◽  
Consistent Formulation

The displacement field generated by the reflection and refraction of plane (time harmonic) elastic waves by finite obstacles of arbitrary shape, in the neighborhood of a plane interface between two elastic media, is investigated. The technique employed allows a consistent formulation of the problem for both two and three dimensions, and is not limited either to boundary shapes which are level surfaces in appropriate coordinate systems, i.e., circular cylinders, spheres, etc., or to closed boundary curves or surfaces. The approach is due to Twersky, and has been applied to many problems of the scattering of electromagnetic waves. The method consists of expressing the net field due to all multiple scattering in terms of the field reflected from each boundary in isolation when subjected to an incident plane elastic wave. Thus the technique makes use of more elemental scattering problems whose solutions are extant. By way of illustration, a numerical solution to the scattering of a plane elastic wave by a rigid circular cylindrical obstacle adjacent to a plane free surface is considered.

SEISMIC RECIPROCITY

Geophysics ◽  
1959 ◽  
Vol 24 (4) ◽  
pp. 681-691 ◽  
Author(s):  
Leon Knopoff ◽  
Anthony F. Gangi
Keyword(s):  
Scalar Product ◽  
Point Sources ◽  
Simple Experiment ◽  
Reciprocity Relations ◽  
Green's Tensor ◽  
Vector Case ◽  
Green’S Tensor

The reciprocity relationship describing the relations among the fields resulting from the interchange of point sources and receivers may be extended to the seismic case. Seismic reciprocity can be described either in terms of the scalar product of the vectors representing the excitation of the source and the field at the receiver, or in terms of a Green’s tensor describing these two quantities. Theoretical reciprocity relations give no information concerning reciprocity in the cases for which the scalar product vanishes. A simple experiment in the vector case demonstrates that reciprocity is not obtained when the scalar product of the two vectors vanishes.

Computation of Green’s tensor integrals for three‐dimensional electromagnetic problems using fast Hankel transforms

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1754-1759 ◽  
Author(s):  
Walter L. Anderson
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Hankel Transform ◽  
Electromagnetic Modeling ◽  
Electromagnetic Problems ◽  
Hankel Transforms ◽  
Homogeneous Half Space ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
The Many

A new method is presented that rapidly evaluates the many Green’s tensor integrals encountered in three‐dimensional electromagnetic modeling using an integral equation. Application of a fast Hankel transform (FHT) algorithm (Anderson, 1982) is the basis for the new solution, where efficient and accurate computation of Hankel transforms are obtained by related and lagged convolutions (linear digital filtering). The FHT algorithm is briefly reviewed and compared to earlier convolution algorithms written by the author. The homogeneous and layered half‐space cases for the Green’s tensor integrals are presented in a form so that the FHT can be easily applied in practice. Computer timing runs comparing the FHT to conventional direct convolution methods are discussed, where the FHT’s performance was about 6 times faster for a homogeneous half‐space, and about 108 times faster for a five‐layer half‐space. Subsequent interpolation after the FHT is called is required to compute specific values of the tensor integrals at selected transform arguments; however, due to the relatively small lagged convolution interval used (same as the digital filter’s), a simple and fast interpolation is sufficient (e.g., by cubic splines).

Reply by the author to R. Ignetik

Geophysics ◽  
1989 ◽  
Vol 54 (11) ◽  
pp. 1501-1502
Author(s):  
Art P. Raiche
Keyword(s):  
Loop Modeling ◽  
Magnetic Dipoles ◽  
Layered Earth ◽  
Starting Point ◽  
Green's Tensor ◽  
Green’S Tensor ◽  
Receiver Point

The failure of Ignetik to recognize the logic underlying my approach to polygonal‐loop modeling demonstrates a need to present some points more clearly. The starting point was that I wanted to represent the transmitter as an array of vertical magnetic dipoles rather than horizontal linear dipoles. The reason for this was to minimize the computation needed for downhole receivers and multiple transmitter loops. The layered‐earth Green’s tensor elements for horizontal linear dipoles for a general “receiver” point are very complicated for the N‐layer case. Those for the vertical dipole are very simple and require much less computation.

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