In an isotropic, time-invariant and homogeneous medium, they are given by:. The wave equations are hyperbolic partial differential equation in space coordinates and time, which must be solved subject to the proper initial and boundary conditions. In certain applications, it is advantageous to cast Maxwell's equations in an integral form.
These can be done using the theorems of vector calculus. In an isotropic, time-invariant and homogeneous medium, the integral forms of Maxwell's equations are given by:. In that case, the electric and magnetic fields can be fully expressed in terms of these two vector potentials:.
This is usually an infinitesimally small vectorial point source. The total electric E field and total magnetic H field can be expressed in terms of the volume electric current source J and volume magnetic current source M in the following way:. The incident or impressed fields represent the source terms and provide the excitation of the structure. This can be a mathematical expression or a more complex recursive process.
Among EM. Cube 's computational modules, EM. Libera is based on the free-space Green's functions, whereas EM. Picasso is based on the dyadic Green's functions of an arbitrary multilayer planar structure. This implies that electromagnetic waves propagate in free space in a spherical form away from the source.
Assuming electric and magnetic surface current sources J and M residing on surfaces S J and S M , respectively, the near-field equations reduce to:.
Back to the Top of the Page. Back to EM. For example, a finite-difference modelling computer code developed for solving the equations in 13 and 14 can also be used to solve the electromagnetic eqs 20 and 21 by simply adjusting the code for the sign of the unit tensor in eq. However, the reverse is not possible; that is, a finite-difference modelling computer code developed for solving eqs 20 and 21 can be directly used for solving eqs 13 and The problem in the reverse operation comes from the fact that tensors and contain zero components that are not necessarily zero in or in e.
This problem stems from the fact that we have fewer equations in the electromagnetic case i.
However, there are some particular forms of the elastic wave equation, where we have fewer equations than in the electromagnetic system. The acoustic case is one example.
Table 3 summarizes the SH -elastic-electromagnetic equivalence. This observation implies that there exist alternative forms of the elastic-electromagnetic mathematical equivalences described in Tables 1—3. These alternative forms can be obtained by replacing the electric field and electric-displacement tensors with the magnetic ones. In Table 5 , we provide an alternative SH -elastic-electromagnetic equivalence to the one in Table 3.
The existence of alternative equivalences to those in Tables 1—3 reinforces the notion that these equivalences are primarily mathematical, although some physical interpretation can be used to prefer one to another.
http://lavifruits.wecan-group.com/liz-crnicas-de-la-ciudad-n-1.php Such preferences are likely to be driven by the applications under investigation. However, their equivalences are different from those in Table 3 but identical to those in Table 5 for all the quantities, except for the volume density of the material magnetic current, which is considered zero in their derivations. Our focus in this review has been on mathematical equivalences between elastic and electromagnetic systems, and the potential use of these equivalences for translating numerical solutions and techniques designed for electromagnetic systems to elastic systems, or vice versa.
Let us draw attention to the fact that these types of equivalences, especially acoustic-electromagnetic equivalences, have been studied in physics and optics for a long time e. MacCullagh ; Thomson ; Maxwell , for a different purpose. The motivation of these studies was that the electromagnetic theory is too abstract, especially the concepts of electric and magnetic fields.
By developing analogies between elasticity and electromagnetism, we can gain some understanding of these concepts and of the electromagnetic wave propagation. The famous ether also known as aether medium was born of these studies.
Third edition, with a foreword by Sir Edmund Whittaker. Oliver Heaviside continued active scientific work for more than twenty years after the publication of the. deocomdudistio.cf: Electromagnetic Theory (3 Volumes) (v. 3) () by Oliver Heaviside and a great selection of similar New, Used and Collectible.
Ether is a hypothetical linear incompressible elastic medium for transmitting light and heat radiation , filling all unoccupied space. In 19th-century physics, all waves are propagated through a medium; for example, water waves through water, sound waves through air. When Maxwell developed his electromagnetic theory of light, the 19th century physicists postulated ether as the medium that transmitted electromagnetic waves.
Ether was held to be invisible, without odour, and of such a nature that it did not interfere with the motions of bodies through space. In other words, the theory of relativity eliminated the need for a light-transmitting medium, so that today the term ether is used only in a historical context. Despite the failure of the ether model, the analogy between the propagation of electromagnetic waves in a vacuum and the propagation of acoustic waves in ether is known to have played a crucial role in the origins of the electromagnetic theory of light Whittaker The term electric displacement in electromagnetism actually has its origin in this analogy.
Although more efforts to modify the ether models is continuing e. This modern focus is motivated by the need to take advantage of recent developments of powerful techniques in the electromagnetic field, especially in the areas of the optics of composite materials and in electromagnetic microwave engineering, for the design and development of electro-mechanical transducers, acoustic waveguides and filters, and other applications which are equally useful for systems involving elastic waves.
An example of this new focus that a number of seismologists are aware of is the extension of the so-called perfect-match-layer absorbing boundary conditions, which was originally designed for electromagnetic systems, to elastic systems. There are numerical modelling techniques, such as finite-difference modelling, based on eqs 20 and 21 , can be used to model electromagnetic synthetic data for a given electromagnetic model.
The perfectly matched layer PML absorbing conditions described in Berenger is probably the most powerful way of implementing the absorbing boundaries today. Because of the equivalences in Table 2 , it is straightforward to use the PML solution of electromagnetic systems for acoustic systems. However, if the original formulation of PML was designed for elastic systems, the extension to EM systems will have been made straightforward by using Table 1.
We have shown how equivalences between an elastic system and an electromagnetic system can be derived. These mathematical equivalences can be used to take advantage of formulations and techniques from one area to the other one. For example, one can use these equivalences to model electromagnetic wave propagation from the computer code originally designed to model elastic wave propagation.
Similarly, a computer code designed for modelling code electromagnetic waves can be used to model acoustic wave propagation when the problem is limited to a 2-D medium. We would like to thank the sponsors of the CASP Project for their comments and suggestions during the review process.
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Close mobile search navigation Article Navigation. Volume Article Contents. On elastic-electromagnetic mathematical equivalences Luc T. E-mail: ikelle geos. Oxford Academic. Google Scholar. Cite Citation. Permissions Icon Permissions. Electromagnetic theory , Marine electromagnetics , Controlled source seismology , Theoretical seismology , Wave propagation , Acoustic properties.
In this paper, the position is specified by the coordinates.