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The Retarded Potential of a Non-Homogeneous Wave Equation: Introductory Analysis Through the Green Functions

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The Retarded Potential of a Non-Homogeneous Wave Equation: Introductory Analysis Through the Green Functions EDUCATION Revista Mexicana de F´ısica E 64 (2018) 26–38 JANUARY–JUNE 2018 The retarded potential of a non-homogeneous wave equation: introductory analysis through the Green functions A. Tellez-Qui´ nones˜ a;¤, J.C. Valdiviezo-Navarroa, A. Salazar-Garibaya, and A.A. Lopez-Caloca´ b aCONACYT-Centro de Investigacion´ en Geograf´ıa y Geomatica,´ Ing. J.L. Tamayo, A.C. (Unidad Merida),´ Carretera Sierra Papacal-Chuburna Puerto Km 5, Sierra Papacal-Yucatan,´ 97302, Mexico.´ bCentro de Investigacion´ en Geograf´ıa y Geomatica,´ Ing. J.L. Tamayo, A.C., Contoy 137-Lomas de Padierna, Tlalpan-Ciudad de Mexico,´ 14240, Mexico.´ e-mail: [email protected] Received 17 August 2017; accepted 11 September 2017 The retarded potential, a solution of the non-homogeneous wave equation, is a subject of particular interest in many physics and engineering applications. Examples of such applications may be the problem of solving the wave equation involved in the emission and reception of a signal in a synthetic aperture radar (SAR), scattering and backscattering, and general electrodynamics for media free of magnetic charges. However, the construction of this potential solution is based on the theory of distributions, a topic that requires special care and time to be understood with mathematical rigor. Thus, the goal of this study is to provide an introductory analysis, with a medium level of formalism, on the construction of this potential solution and the handling of Green functions represented by sequences of well-behaved approximating functions. Keywords: Mathematical methods in physics; mathematics; diffraction theory; backscattering; radar. PACS: 01.30.Rr; 02.30.Em; 41.20.Jb 1. Introduction going to use the artifice of working with sequences of well- behaved approximating functions [28], which permit us to Potential theory can be simply understood as the art of talk about distributions concisely and without too much com- solving a linear distributional non-homogeneous differential plexity. However, we hope to motivate the reader in the study equation through the Green functions [1–3]. In the context of distributions through one of their most important appli- of this study, our interest resides in the construction of an cations: the standard solution of the non-homogeneous wave integral solution that derives from the divergence Gauss the- equation, also known as the retarded potential. To understand orem, certain Green identities, and the handling of the Green the construction of this integral solution, we first need to ex- functions. In the construction of this potential solution we pose the main problem involved with the non-homogeneous also find other results of great importance, such as the inte- wave equation. Thus, in order to establish the context of such gral theorem of Helmholtz and Kirchhoff, which is the main a problem, we start in Sec. 2 with a typical deduction of the result that supports the scalar diffraction theory in optics [4]. non-homogeneous wave equation from the Maxwell equa- However, Green functions are not properly functions in the tions. Motivated in the perspective of SAR theory, in the sub- usual sense, since they are formally defined as distributions. sequent sections, we provide some descriptive guidelines for Distributional theory, Green functions, and the use of Green constructing the potential solution of this non-homogeneous identities have been successfully implemented in many theo- wave equation. The potential solution is supported by the di- retical and applied works, e.g., SAR theory [5–7], scattering vergence Gauss theorem and the Green identities, described and wave propagation [8–12], wave diffraction and electro- in Sec. 3. Of course, the definition of Green functions is also dynamics [13–16], phase unwrapping [17–20], etc. explained in Sec. 4, where we discuss about certain incon- The concept of distribution is not easy to explain and in gruities when working with Green functions, found in certain most of the references where distributions are mentioned, references (for example [21]). These incongruities refer to such as basic courses in calculus [21], differential equa- the question: How can be demonstrated that a function is tions [22], linear systems [23], or Fourier analysis [24], a a Green function with some of formalism? We emphasize detailed explanation of such abstractions is not usually given. that we do not want to criticize the descriptive style of refer- Hence, in order to have a more solid notion of the concept ences [21–23] or [24], which are indeed quite advisable. We of distributions, or generalized functions, specialized litera- simply want to point out that the treatment of distributions ture should be consulted. This literature is specifically re- should be made with more care. Additionally to this discus- lated to a coarse field of mathematics called functional anal- sion, a description of the integral theorem of Helmholtz and ysis [25, 26] and some notions on Lebesgue measure [27]. Kirchhoff, and its relation with the retarded potential is pre- Certainly, in this work we will not provide specific and de- sented in Sec. 5. Finally, Sec. 6 outlines our conclusions. tailed information about distributions. Instead of that, we are THE RETARDED POTENTIAL OF A NON-HOMOGENEOUS WAVE EQUATION: INTRODUCTORY ANALYSIS.. 27 @E 2. Derivation of the non-homogeneous wave J = r £ H ¡ q @t equation µ ¶ @ @A = r £ (r £ A ) ¡ q r¹ ¡ p 0 It is well known that, an electromagnetic wave, such as 0 @t 0 @t a radar wave, is characterized in each point of the space 2 x^ = (x; y; z), and each time t, by the vectorial functions 2 @ @ A0 = r(r ¢ A0) ¡ r A0 ¡ q (r¹0) + pq : (4) E = E(t; x^), the electric field (EF), and H = H(t; x^), the @t @t2 magnetic field (MF). When assuming as known the variables From the previous relation, it is obtained that J = J(t; x^), the current density (vectorial field), σ = σ(t; x^), the charge density (scalar function), q, the permittivity con- @2A @ r2A ¡ pq 0 = ¡J + r(r ¢ A ) ¡ q (r¹ ) stant, and p, the permeability constant, the fields E and H 0 @t2 0 @t 0 can be found. These fields are determined by the Maxwell µ ¶ @¹0 equations: r ¢ E = σ=q, the Gauss law for EF, r ¢ H = 0, = ¡J + r(r ¢ A0) ¡ r q r £ E + p(@H=@t) = 0^ @t the Gauss law for MF, , the Faraday µ ¶ law, and r £ H ¡ q(@E=@t) = J, the Ampere` law. Here, @¹ = ¡J + r r ¢ A ¡ q 0 : (5) r := (@=@x; @=@y; @=@z), 0^ = (0; 0; 0), and the media is 0 @t assumed free of magnetics sources. In this case, operations r ¢ F and r £ F refer, respectively, to the divergence and Now, by using the Gauss law for EF and Eq. (3), we get the rotational of any vectorial field F = F (t; x^). Hence, a σ @ method to find E and H can be constructed from the previous r2¹ = + p (r ¢ A ): (6) 0 q @t 0 Maxwell equations and the next theorems: Theorem 1: Let F : R3 ! R3 be a real valued vectorial Let us define f as a scalar solution of the equation field of class C1 (continuous function with continuous first 2 µ ¶ order partial derivatives), exceptionally in a finite number of @ f @¹0 r2f ¡ pq = ¡ r ¢ A ¡ q ; (7) points. Then, F is the gradient of some scalar function f (that @t2 0 @t is, F = rf), if and only if r £ F = 0^. Theorem 2: If F : R3 ! R3 is a C1-vectorial field such and ¹, as the function ¹ := p(@f=@t) + ¹0. From these that r ¢ F = 0, then, there is a C1-vectorial field G such that definitions we obtain the relation F = r £ G. @f (¹ ¡ ¹ ) = 0 : (8) These theorems are demonstrated in [21] for real valued @t p functions dependent on variable x^ (where F = F (^x)), how- ever, they can be generalized to complex valued functions of Thus, it can be noticed that the form F : R3 ! C3. Moreover, these theorems are in- dependent of variable t, so, they are also true for complex H = r £ A; (9) fields [29] of the form F : R4 ! C3 where F = F (t; x^). Now, by assuming E and H as C2-fields (that means, C1- with A = A0 + rf. The result in Eq. (9) is logical because ^ fields with continuous second order partial derivatives), from r £ (rf) = 0 for all differentiable scalar field f; in other words, r £ A = r £ A0 for such A. Since H can be cal- the Gauss law for MF, and Theorem 2, there exists a field A0 such that culated, as suggested in Eq. (9), there is a function ¹1 that satisfies H = r £ A0: (1) @A E + p = r¹1; (10) Additionally, from Faraday law and Eq. (1), we have that @t in analogy to the steps given for deriving Eq. (3). However, @ 0^ = r £ E + p (r £ A ) r¹ = r¹1 due to the fact that @t 0 . (2) @ . µ ¶ r¹1 = E + p (A0 + rf) @A @t = r £ E + p 0 ; µ ¶ @t @A0 @ = E + p + p (rf) @t @t which, from Theorem 1, implies the existence of a function µ ¶ ¹0 satisfying @f @A = r¹0 + pr = r¹; (11) E + p 0 = r¹ : (3) @t @t 0 On the other hand, from the Ampere` law, Eqs.(1) and (3), and a consequence of Eqs. (3), (8), and (10). In this case, we can 2 replace r¹1 by r¹ in Eq. (10) in order to obtain the vectorial identity r £ (r £ A0) = r(r ¢ A0) ¡ r A0 2 2 2 2 2 2 2 (where r := (@ =@x )+(@ =@y )+(@ =@z ) is the Lapla- @A E + p = r¹: (12) cian operator), it follows that @t Rev.
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