The general solution of the three‐dimensional acoustic equation and of Maxwell’s equations in the infinite domain in terms of the asymptotic solution in the wave zone

1992 ◽  
Vol 33 (1) ◽  
pp. 86-101 ◽  
Author(s):  
Harry E. Moses ◽  
Raymond J. Nagem ◽  
Guido v. H. Sandri
Keyword(s):  
General Solution ◽  
Asymptotic Solution ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Three Dimensional ◽  
Wave Zone ◽  
Infinite Domain ◽  
Acoustic Equation

Exact solutions of the three-dimensional scalar wave equation and Maxwell's equations from the approximate solutions in the wave zone through the use of the Radon transform

1989 ◽  
Vol 422 (1863) ◽  
pp. 351-365 ◽  
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Radon Transform ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Three Dimensional ◽  
Asymptotic Solutions ◽  
Scalar Wave ◽  
Wave Zone ◽  
One Dimensional

In our earlier paper we have shown that the solutions of both the three-dimensional scalar wave equation, which is also the three-dimensional acoustic equation, and Maxwell’s equations have forms in the wave zone, which, except for a factor 1/ r , represent one-dimensional wave motions along straight lines through the origin. We also showed that it is possible to reconstruct the exact solutions from the asymptotic forms. Thus we could prescribe the solutions in the wave zone and obtain the exact solutions that would lead to them. In the present paper we show how the exact solutions can be obtained from the asymptotic solutions and conversely, through the use of a refined Radon transform, which we introduced in a previous paper. We have thus obtained a way of obtaining the exact three-dimensional solutions from the essentially one-dimensional solutions of the asymp­totic form entirely in terms of transforms. This is an alternative way to obtaining exact solutions in terms of initial values through the use of Riemann functions. The exact solutions that we obtain through the use of the Radon transform are causal and therefore physical solutions. That is, these solutions for time t > 0 could have been obtained from the initial value problem by prescribing the solution and its time-derivative, in the acoustic case, and the electric and magnetic fields, in the case of Maxwell’s equations, at time t = 0. The role of time in the relation between the exact solutions and in the asymptotic solutions is made very explicit in the present paper.

scholarly journals On the Coefficients of the Singularities of the Solution of Maxwell’s Equations near Polyhedral Edges

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
Boniface Nkemzi
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Numerical Approximation ◽  
Three Dimensional ◽  
Physical Domain ◽  
Two Dimensional ◽  
Explicit Formulas ◽  
Nonsmooth Domains ◽  
Crack Tips ◽  
Domains With Corners

The solution fields of Maxwell’s equations are known to exhibit singularities near corners, crack tips, edges, and so forth of the physical domain. The structures of the singular fields are well known up to some undetermined coefficients. In two-dimensional domains with corners and cracks, the unknown coefficients are real constants. However, in three-dimensional domains the unknown coefficients are functions defined along the corresponding edges. This paper proposes explicit formulas for the computation of these coefficients in the case of two-dimensional domains with corners and three-dimensional domains with straight edges. The coefficients of the singular fields along straight edges of three-dimensional domains are represented in terms of Fourier series. The formulas presented are aimed at the numerical approximation of the coefficients of the singular fields. They can also be used for the construction of adaptiveH1-nodal finite-element procedures for the efficient numerical treatment of Maxwell’s equations in nonsmooth domains.

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