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 = ∇ × H − q ∂t equation µ ¶ ∂ ∂A = ∇ × (∇ × A ) − q ∇µ − 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 xˆ = (x, y, z), and each time t, by the vectorial functions 2 ∂ ∂ A0 = ∇(∇ · A0) − ∇ A0 − q (∇µ0) + pq . (4) E = E(t, xˆ), the electric field (EF), and H = H(t, xˆ), the ∂t ∂t2 magnetic field (MF). When assuming as known the variables From the previous relation, it is obtained that J = J(t, xˆ), the current density (vectorial field), σ = σ(t, xˆ), the charge density (scalar function), q, the permittivity con- ∂2A ∂ ∇2A − pq 0 = −J + ∇(∇ · A ) − q (∇µ ) 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 ¶ ∂µ0 equations: ∇ · E = σ/q, the Gauss law for EF, ∇ · H = 0, = −J + ∇(∇ · A0) − ∇ q ∇ × E + p(∂H/∂t) = 0ˆ ∂t the Gauss law for MF, , the Faraday µ ¶ law, and ∇ × H − q(∂E/∂t) = J, the Ampere` law. Here, ∂µ = −J + ∇ ∇ · A − q 0 . (5) ∇ := (∂/∂x, ∂/∂y, ∂/∂z), 0ˆ = (0, 0, 0), and the media is 0 ∂t assumed free of magnetics sources. In this case, operations ∇ · F and ∇ × 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, xˆ). Hence, a σ ∂ method to find E and H can be constructed from the previous ∇2µ = + p (∇ · 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 ∂µ0 ∇2f − pq = − ∇ · A − q , (7) points. Then, F is the gradient of some scalar function f (that ∂t2 0 ∂t is, F = ∇f), if and only if ∇ × F = 0ˆ. Theorem 2: If F : R3 → R3 is a C1-vectorial field such and µ, as the function µ := p(∂f/∂t) + µ0. From these that ∇ · F = 0, then, there is a C1-vectorial field G such that definitions we obtain the relation F = ∇ × G. ∂f (µ − µ ) = 0 . (8) These theorems are demonstrated in [21] for real valued ∂t p functions dependent on variable xˆ (where F = F (ˆx)), however, they can be generalized to complex valued functions of Thus, it can be noticed that the form F : R3 → C3. Moreover, these theorems are independent of variable t, so, they are also true for complex H = ∇ × A, (9) fields [29] of the form F : R4 → C3 where F = F (t, xˆ). Now, by assuming E and H as C2-fields (that means, C1- with A = A0 + ∇f. The result in Eq. (9) is logical because ˆ fields with continuous second order partial derivatives), from ∇ × (∇f) = 0 for all differentiable scalar field f; in other words, ∇ × A = ∇ × 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 µ1 that satisfies H = ∇ × A0. (1) ∂A E + p = ∇µ1, (10) Additionally, from Faraday law and Eq. (1), we have that ∂t in analogy to the steps given for deriving Eq. (3). However, ∂ 0ˆ = ∇ × E + p (∇ × A ) ∇µ = ∇µ1 due to the fact that ∂t 0 . . (2) ∂ . µ ¶ ∇µ1 = E + p (A0 + ∇f) ∂A ∂t = ∇ × E + p 0 , µ ¶ ∂t ∂A0 ∂ = E + p + p (∇f) ∂t ∂t which, from Theorem 1, implies the existence of a function µ ¶ µ0 satisfying ∂f ∂A = ∇µ0 + p∇ = ∇µ, (11) E + p 0 = ∇µ . (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 ∇µ1 by ∇µ in Eq. (10) in order to obtain the vectorial identity ∇ × (∇ × A0) = ∇(∇ · A0) − ∇ A0 2 2 2 2 2 2 2 (where ∇ := (∂ /∂x )+(∂ /∂y )+(∂ /∂z ) is the Lapla- ∂A E + p = ∇µ. (12) cian operator), it follows that ∂t

Rev. Mex. Fis. E 64 (2018) 26–38 28 A. TELLEZ-QUI´ NONES,˜ J.C. VALDIVIEZO-NAVARRO, A. SALAZAR-GARIBAY AND A.A. LOPEZ-CALOCA´

Thus, by using Eqs. (9) and (12), this A must satisfy is denoted by µ1. In opposite form, the object, assumed to be like a non-emitting electric pulse source (ζ = 0), reflects 2 µ ¶ 2 ∂ A ∂µ or backscatters the incident wave. So, the reflected wave or ∇ A − pq 2 = −J + ∇ ∇ · A − q (13) ∂t ∂t backscattered field is denoted by µ0 (details of this concep- tion are explained in [5]). Consequently, the general solution and σ ∂ µ is known as total field. Since the total field can be found ∇2µ = + p (∇ · A), (14) q ∂t by solving first the homogeneous wave equation, we are go- µ in analogy to the steps for concluding Eqs. (5) and (6) from ing to focus our attention in the construction of 0. Thus, in µ µ Eqs. (1) and (3). However, we have now a simplification in this section, variable 0 will be denote as for simplifying our calculations because notation. Moreover, it is important to remark that any com- ponent of the vectorial function A in Eq.(18), with J = 0ˆ, ∂µ ∇ · A − q = 0, (15) can be found just as solving the scalar case for µ in Eq.(18), ∂t with ζ = 0. In this sense, it will be sufficient to establish the as the reader can confirm. theory for solving this scalar case. Since Eq. (15) takes place, then Eq. (13) reduces to Let us consider a scalar wave with spatial period or wave- length λ0, refractive index media n = 1 (air) and angular- 2 ∂ A temporal frequency ω0. Such wave can be represented by ∇2A − pq = −J. (16) ∂t2 the function µ˜(t, xˆ) = a(ˆx) cos (ω0t + φ(ˆx)), where a(ˆx) is the amplitude, and φ(ˆx) is the phase. This cosinu- In a similar fashion, from Eq. (15), the expression in Eq. (14) soidal form is well known from the classic theory for solv- is rewritten as ∂2µ σ ing the homogeneous wave equation; specifically speaking, ∇2µ − pq = . (17) ∂t2 q the method called separation of variables, and the Fourier series [22]. However, in an equivalent way, the form of Therefore, by considering the wave propagation velocity √ the wave can be generalized to the complex representation c0 := 1/ pq (as defined in [30]), and introducing the func- µ(t, xˆ) = f(ˆx)eiω0t, where µ˜ is the real part of µ, f(ˆx) = tion ζ := −σ/q, Eqs. (16) and (17) are correspondingly ex- a(ˆx)eiφ(ˆx) is the spatial part of µ (the so-called phasor or pressed as complex perturbation [4]), and i is the imaginary unit. Since 1 ∂2A this complex representation must satisfy the homogeneous 2 2 2 2 2 2 2 − ∇ A = J, (18a) wave equation, we have that ∇ µ − (1/c0)(∂ µ/∂t ) = 0, c0 ∂t where c0 = λ0/T0 is the wave propagation velocity and 2 1 ∂ µ 2 T0 denotes the temporal period of such wave. In this case 2 2 − ∇ µ = ζ. (18b) c0 ∂t τ0 = 1/T0 would be called simply as the temporal frequency, where ω0 = 2πτ0. Thus, when substituting the complex Then, when assuming J and ζ as known functions, the so- representation of µ in the homogeneous wave equation, we lutions of Eqs. (18) for A and µ, permit us to find H and iω0t 2 2 2 2 iω0t obtain that e ∇ f(ˆx) − (i ω0/c0)f(ˆx)e = 0, which E, from Eqs. (9) and (12), respectively. Equation (18) for implies µ, and analogously for A, is known as the non-homogeneous (∇2 + k2)f(ˆx) = 0. (19) wave equation, or d’Alembert equation [5] in the distribu- 0 tional sense. Due to its undulatory nature, the solution of this This last expression is called Helmholtz equation, where equation is called scalar wave in the case of µ, or vectorial k0 = ω0/c0 = (2πτ0)/(λ0τ0) = (2π)/λ0 is the wave num- wave, in the case of A. ber or the angular-spatial frequency. On the other hand, a well-known result in vectorial calculus is the divergence Gauss theorem [21], which can be 3. The Helmholtz equation, the divergence written as Gauss theorem, and the Green identities Z3 Z2 ∇ · F dV = F · ndA,ˆ (20) For constructing a solution of the non-homogeneous wave equation, from the methods applied for solving differential Ω ∂Ω equations [22, 31, 32], it is typically proposed a general solu- where F = F (ˆx) is a C1-vectorial field on Ω such that 3 3 3 tion of the form µ = µ0 + µ1, where µ0 is solution of the F : R → R , and Ω is an elemental region in R with homogeneous version of the wave equation (when ζ = 0), positive parametrized boundary ∂Ω. The surface ∂Ω can be and µ1 is solution of the original non-homogeneous equa- a sphere, an ellipsoid, a parallelepiped, etc. In this theorem, R 3 R 2 tion (when ζ 6= 0). In radar language [33–35], function ζ is symbol Ω denotes the triple integral over the region Ω, ∂Ω interpreted as the source or the electric pulse emitted by an denotes the double integral over the surface ∂Ω, dV is a vol- antenna. In this context, when ζ 6= 0, it is understood that ume differential element, dA is an area differential element, the pulse induces a propagating wave which reaches an ob- and symbol · denotes dot product. In addition, vectorial func- ject. This wave is known as emitting or incident field and it tion nˆ represents the unitary normal vector with respect to the

Rev. Mex. Fis. E 64 (2018) 26–38 THE RETARDED POTENTIAL OF A NON-HOMOGENEOUS WAVE EQUATION: INTRODUCTORY ANALYSIS.. . 29 surface ∂Ω, in such a way that, it points towards the outside key result for supporting the scalar diffraction theory and of the surface. So, if we take F = f∇g, where f = f(ˆx) it is related in part with the solution of a particular distri- and g = g(ˆx) are two differentiable scalar fields from R3 to butional non-homogeneous wave equation, the so-called re- R, then tarded potential of the d’Alembert equation [5]. Now, let L be a linear operator applied to scalar functions depending on 2 ∇ · F = ∇ · (f∇g) = ∇f · ∇g + f∇ g. (21) xˆ. Formally, a distribution G(ˆx, yˆ) is said to be a Green function with respect to L, if it satisfies: a) G(ˆx, yˆ) = G(ˆy, xˆ) When substituting this equation in the divergence Gauss re- for all xˆ = (x, y, z) and yˆ = (u, v, w) in R3, and b) lation, and introducing the directional derivative (∂g/∂n) := L[G(ˆx, yˆ)] = δ(ˆy − xˆ), where δ(ˆy − xˆ) is the Dirac delta ∇g · nˆ, we obtain distribution. However, every distribution D is said to be a Z3 Z3 Z2 Dirac delta, if it satisfies: 1) D(ˆy − xˆ) = 0, for all yˆ 6=x ˆ, ∂g ∇f · ∇gdV + f∇2gdV = f dA. (22) and 2) ∂n Z3 Ω Ω ∂Ω D(ˆy − xˆ)dV (ˆy) = 1.

Equation (22) is known as the first Green identity. Analo- R3 gously, when considering F = g∇f, the equality In this definition dV (ˆy) refers to a volume differential ele- Z3 Z3 Z2 ment with respect to the integration variable yˆ. Thus, when ∂f ∇g · ∇fdV + g∇2fdV = g dA, (23) an arbitrary distribution D fulfills properties 1) and 2), we ∂n write δ := D. Ω Ω ∂Ω Let us consider G(ˆx, yˆ) := gyˆ(ˆx), where gyˆ(ˆx) = is deduced. By subtracting Eq. (23) from Eq. (22), it is con- eik0||xˆ−yˆ||/||xˆ − yˆ||. Here, || · || denotes the Euclidean norm 3 cluded that for vectors in R . This function satisfies gyˆ(ˆx) = gxˆ(ˆy) by symmetry, then G(ˆx, yˆ) fulfills property a). Is this G a Green Z3 Z2 µ ¶ ∂g ∂f function? What linear operator could be related to this G (f∇2g − g∇2f)dV = f − g dA, (24) ∂n ∂n to declare it as a Green function? Well, the obvious answer Ω ∂Ω is that such operator must be involved with the Helmholtz 2 2 equation. So, if we consider L := [−1/(4π)](∇ + k0), then a relation called second Green identity. It is important to 2 2 L[f(ˆx)] = [−1/(4π)](∇ + k0)f(ˆx) for any smooth func- remark that the divergence Gauss theorem is also valid for 3 1 3 3 tion f : R → C. Moreover, D(ˆy − xˆ) := L[G(ˆx, yˆ)] = C -complex vectorial fields F : R → C that can be ex- 2 2 pressed in terms of complex scalar fields f, g : R3 → C. [−1/(4π)](∇ + k0)gyˆ(ˆx) = 0 for all xˆ 6=y ˆ, from Propo- This last includes the possibility of considering fields of the sition 1 in Appendix A. This also implies that property 1) is form F = F (t, xˆ), f = f(t, xˆ), and g = g(t, xˆ), which satisfied by D(ˆy − xˆ), every time that xˆ 6=y ˆ. Thus, assum- means that F : R4 → C3, and f, g : R4 → C. The va- ing this D as a simple function depending on yˆ, which is a lidity of this theorem is a consequence of the linearity of the discontinuous function in yˆ =x ˆ, we rigorously have that integrals for complex valued expressions that can be denoted Z3 as F = Re(F ) + iIm(F ), where Re(F ) and Im(F ), are the D(ˆy − xˆ)dV (ˆy) = 0 real and the complex parts of F , respectively. Thus, the only 3 required condition is to have Re(F ) and Im(F ), as C1-real R valued functions. by using improper integrals. Even in the case of more complex integrals, if a function is zero almost everywhere in 4. The Green functions: language of distribu- certain domain, then its Lebesgue integral on such domain should be zero [27]. Thus, since property 2) is not achieved tions by this D, it implies that G(ˆx, yˆ) = gyˆ(ˆx) is not a Green Informally, any function g that satisfies the Helmholtz equa- function. But, why so many references [2–4,8] declare gyˆ(ˆx) tion almost everywhere [26,27] on R3, could be called Green as a Green function? Well, may be the answer is in the in- function [4], however, such g requires to satisfy another prop- terpretation of D, and consequently G, as distributions. In erties in the context of distributions [26]. In this sense, ex- the spirit of considering a sequence of well-behaved approxi- pression almost everywhere refers to a property which is sat- mating functions (see the Remarks in Appendix A), G can be isfied at all points of a domain with the exception of the points rewritten to the equivalent form in a zero volume subset of the domain. For this particular ik0r G(ˆx, yˆ) := lim e fγ (r), (25) case, the property would be the fulfilling of the Helmholtz γ→0+ equation. On the other hand, the Green functions and the where r = ||xˆ − yˆ|| and lim fγ (r) = 1/r for r > 0. In this Green identities are important and useful, specially in op- γ→0+ tics, for establishing a transcendental theorem: the integral case for each γ > 0, fγ is a smooth function for all r > 0 theorem of Helmholtz and Kirchhoff. This theorem is the and right continuous at r = 0. In other words, dfγ /dr exists

Rev. Mex. Fis. E 64 (2018) 26–38 30 A. TELLEZ-QUI´ NONES,˜ J.C. VALDIVIEZO-NAVARRO, A. SALAZAR-GARIBAY AND A.A. LOPEZ-CALOCA´

2 · ¸ for all r > 0, and lim fγ (r) = fγ (0), respectively. Now, Z + ik0r − 1 r→0 ... = − eik0r (ˆx − yˆ) it is evident that G in Eq. (25) satisfies properties a) and 1), r3 however, if property 2) is required for D with this new G, we ∂Bε(ˆx) need to construct function f conveniently. There are many γ Z2 · ¸ forms to do this construction, but we are going to propose (ˆy − xˆ) ik0r − 1 · dA(ˆy) = eik0r a particular one. Before starting with this construction, let ||yˆ − xˆ|| r3 us think again that G(ˆx, yˆ) is gyˆ(ˆx), and consider the next ∂Bε(ˆx) µ ¶ argumentation: r2 × dA(ˆy) = 4πeik0ε(ik ε − 1), (29) From property 1), it is inferred that the integral of D r 0 on R3 is the same that the integral of D on any open ball B (ˆx) = {yˆ ∈ R3 : ||yˆ − xˆ|| < ε} ε . Then, where the values r = ||xˆ − yˆ|| in the double integrals are all 3 equal to ε, because yˆ ∈ ∂Bε(ˆx) = {yˆ ∈ R : ||yˆ − xˆ|| = ε}. Z3 µ ¶ −1 When considering Eqs. (27), (28), and (29) in connection D(ˆy − xˆ)dV (ˆy) = 4π with Eq. (26), we get property 2). Therefore, function gyˆ(ˆx) R3 results to be a Green function. Z3 µ ¶ The last argumentation, although desirable, is false and its −1 ik0r 3 × (∇2 + k2)g (ˆx)dV (ˆy) = main fail is: function ∇gyˆ(ˆx) = e (ik0r −1)(ˆx−yˆ)/r is 0 yˆ 1 4π not a C -function (a smooth function) on Bε(ˆx), specifically Bε(ˆx) at yˆ =x ˆ where r = 0. Then, we have incorrectly applied   Z3 Z3 the divergence Gauss theorem in Eq. (28), which requires  2 2  smoothness for the vector field ∇gyˆ(ˆx). However, this mis- ×  ∇ gyˆ(ˆx)dV (ˆy) + k gyˆ(ˆx)dV (ˆy) , (26) 0 take motivates us to think about an appropriate election of Bε(ˆx) Bε(ˆx) function fγ in Eq. (25). The sequence of functions fγ satisfy ik0r ik0r lim e fγ (r) = e /r = gxˆ(ˆy) for all yˆ 6=x ˆ (r > 0), for all ε > 0 fixed. Thus, when considering a translation to γ→0+ the origin, a change to spherical coordinates, and the use of but that is not enough. For each γ > 0, we also need smooth- ik0r improper integrals, we have that ness for the terms ∇e fγ (r) for all yˆ ∈ Bε(ˆx). However, since the term eik0r is smooth for all r ≥ 0, then a sequence Z3 Z3 of functions of the form

gyˆ(ˆx)dV (ˆy) = gxˆ(ˆy)dV (ˆy) ( p0(r) r ≥ γ Bε(ˆx) Bε(ˆx) fγ (r) := (30) pγ (r) 0 ≤ r < γ, 4π = [eik0ε(1 − ik ε) − 1], (27) k2 0 0 where p0(r) := 1/r, could be useful. For each γ > 0, pγ (t) could be a convenient polynomial function, in such a way independently of the discontinuity of gxˆ(ˆy) at yˆ =x ˆ. Now, 0 0 00 00 that pγ (γ) = p0(γ), pγ (γ) = p0(γ), and pγ (γ) = p0 (γ). Here, symbols 0 and 00 denote, correspondingly, the ﬁrst and Z3 Z3 the second derivatives with respect to r. Hence, these con- 2 ∇ gyˆ(ˆx)dV (ˆy) = ∇ · ∇gyˆ(ˆx)dV (ˆy) ditions would warrant the smoothness of fγ for all r > 0.

Bε(ˆx) Bε(ˆx) However, if pγ (r) is a linear combination of r-powers and ik0r we want to analyze the behavior of ∇e fγ (r) (particularly 3 Z at r = 0), we ﬁrst need to calculate the resultant expressions = − ∇yˆ · ∇gyˆ(ˆx)dV (ˆy) of the partial derivatives of eik0rrn, with respect to x, y, and

Bε(ˆx) z. For instance, from the chain rule we have Z2 ∂ (x − u) = − ∇g (ˆx) · nˆ(ˆy)dA(ˆy), (28) [eik0rrn] = eik0r[nrn−1 + ik rn] , (31) yˆ ∂x 0 r ∂Bε(ˆx) for xˆ 6=y ˆ (or r > 0). This expression is not necessarily where nˆ(ˆy) is the unitary normal vector to the surface ∂Bε(ˆx) right continuous at r = 0 when considering the definition and dA(ˆy) is an area differential element with respect to yˆ. of continuity by lateral limits. However, if we achieved to This is a consequence of Proposition 2 in Appendix A and avoid divisions by r (removing the discontinuity at r = 0), the divergence Gauss theorem in variable yˆ. Following the the formula in Eq. (31) would be better behaved. Thus, by calculus of the previous triple integral (now a double inte- considering integer powers of r such that n ≥ 2 and defining gral) we have that the vector î := (1, 0, 0), we get

Rev. Mex. Fis. E 64 (2018) 26–38 THE RETARDED POTENTIAL OF A NON-HOMOGENEOUS WAVE EQUATION: INTRODUCTORY ANALYSIS.. . 31

function for all yˆ by construction. From this construction and ¯ Eq. (25), we can calculate again ik0r n ik0r n ∂ ¯ e r |xˆ=ˆy+hî − e r |xˆ=ˆy [eik0rrn]¯ := lim ¯ 3 ∂x xˆ=ˆy h→0 h Z D(ˆy − xˆ)dV (ˆy) eik0|h||h|n = lim = 0, (32) R3 h→0 h  Z3 due to the fact that 1  2 ik0r = − lim  ∇ e fγ (r)dV (ˆy) 4π γ→0+ eik0|h||h|n eik0|h||h|n lim = lim = 0, Bε(ˆx) h→0+ h h→0− h  Z3 as the reader can confirm. Moreover, from Eq.(31) it follows 2 ik0r  +k e fγ (r)dV (ˆy) , (37) that 0 Bε(ˆx) ∂ lim [eik0rrn] xˆ→yˆ ∂x for any ε > 0 arbitrary and fixed. But this time we have ½ ¾ (x − u) ik0r n−1 n Z3 Z3 = lim e [nr + ik0r ] ik0r x→u e r ik0r r→0+ e fγ (r)dV (ˆy) = dV (ˆy) © ª r ik0r n−2 n−1 = lim e [nr + ik r ](x − u) = 0. (33) Bε(ˆx) Bε(ˆx)\Bγ (ˆx) x→u 0 r→0+ Z3 ik0r In consequence, Eqs. (31)-(33) warrant continuity for the + e pγ (r)dV (ˆy) term ∂[eik0rrn]/∂x for all xˆ 6=y ˆ, and also for xˆ =y ˆ, B (ˆx) when considering this term as function of xˆ. Nevertheless, γ ik0r n−1 n 4π £ ¤ due to the radial symmetry of e [nr + ik0r ]/r in ik0ε ik0γ = e (1 − ik0ε) − e (1 − ik0γ) ik0r n 2 Eq. (31), term ∂[e r ]/∂x, interpreted as function of yˆ, k0 must be also a continuous function. The same reasoning ap- 4πγ3 plies to ∂[eik0rrn]/∂y and ∂[eik0rrn]/∂z; therefore, if f (r) + eik0r• p (r ) , (38) γ γ • 3 is a polynomial of powers n ≥ 2 for r values such that 0 ≤ ik0r r < γ, then ∇e fγ (r) will be continuous for all points by definition of fγ when considering γ < ε. In the last yˆ corresponding to those r values. Moreover, without loss of expression, the integral of (eik0r/r) can be calculated by a 2 3 4 5 generality, we can assume that pγ (r) = ar +br +cr +dr translation to the origin and spherical coordinates, while the ik0r with pγ (γ/2) = p0(γ/2), which implies that integral of e pγ (r) is obtained from the mean value theorem for integrals [21,36] due to the integrand’s continuity. In p (r) = (42/γ3)r2 − (111/γ4)r3 γ this case r• = ||xˆ − yˆ•|| with yˆ• ∈ Bγ (ˆx) ∪ ∂Bγ (ˆx), and 3 + (102/γ5)r4 − (32/γ6)r5. (34) the factor (4πγ /3) is the volume of the ball Bγ (ˆx). On the other hand, when taking the complex module of the integral So, from Eqs. (30) and (34) we get in Eq. (38), we have that ½ ¯ ¯ ik0r ¯ 3 ¯ ik0r e A(r)(ˆx − yˆ) r ≥ γ ¯ Z ¯ ∇e fγ (r)= ik r (35) ¯ ¯ 4π e 0 B(r)(ˆx − yˆ) 0 ≤ r < γ, ¯ eik0rf (r)dV (ˆy)¯ ≤ |eik0ε(1 − ik ε) ¯ γ ¯ k2 0 ¯ ¯ 0 3 Bε(ˆx) where A(r) = [ik0r − 1]/r and 4πγ3 2 3 ik0γ 84 333r 408r 160r − e (1 − ik0γ)| + |pγ (r•)|. (39) B(r) = − + − 3 γ3 γ4 γ5 γ6 · ¸ Additionally, from Eq. (34), the triangle inequality, and the 42r 111r2 102r3 32r4 + (ik ) − + − , (36) fact that r• ≤ γ, it follows that 0 γ3 γ4 γ5 γ6 4πγ3 1148πγ2 are two functions such that lim A(r) = A(γ) = B(γ) = |pγ (r•)| ≤ . (40) r→γ+ 3 3 lim B(r), lim A0(r) = A0(γ) = B0(γ) = lim B0(r), r→γ− r→γ+ r→γ− 3 2 0 0 In an extreme case (4πγ /3)|pγ (r•)| ' (4πγ /3) for r•- lim B(r) = B(0), and lim B (r) = B (0). Equation (35) 3 r→0+ r→0+ values close or equal to γ, but the term (4πγ /3)|pγ (r•)| is now a smooth function (C1-class) for all yˆ 6=x ˆ and also for is always dominated by proportional factors to γ2. Then, ik0r yˆ =x ˆ. In the same way, term e fγ (r) is another smooth when considering very small values of γ (the limit context

Rev. Mex. Fis. E 64 (2018) 26–38 32 A. TELLEZ-QUI´ NONES,˜ J.C. VALDIVIEZO-NAVARRO, A. SALAZAR-GARIBAY AND A.A. LOPEZ-CALOCA´ of Eq. (37)), the inequalities in Eqs. (39) and (40) permit us almost everywhere. Something similar happens when think- ik0r• 3 to establish that e pγ (r•)(4πγ /3) ≈ 0 and ing in the one dimensional step function, also known as the Heaviside step. The derivative of the Heaviside step is zero Z3 at all points, except in the jump discontinuity where it is not 2 ik0r k0 e fγ (r)dV (ˆy) deﬁned. The integral of this derivative along the real line is

Bε(ˆx) identically zero. However, if the Heaviside step is understood £ ¤ as a distribution, then its weak derivative [26] is a Dirac delta ik0ε ≈ 4π e (1 − ik0ε) − 1 , (41) (see the remarks in Appendix A). As well known, the integral of this weak derivative along the real line is identically one. ik0r from Eq. (38). Now, due to the form of term ∇e fγ (r) in These affirmations may seem contradictory, but they are only 2 ik0r Eq. (35) and Proposition 2, it follows that ∇ e fγ (r) = a question of abstract interpretation. ik0r ik0r ∇ · ∇e fγ (r) = −∇yˆ · ∇e fγ (r). Therefore, Z3 5. The integral theorem of Helmholtz and 2 ik0r ∇ e fγ (r)dV (ˆy) Kirchhoff and the retarded potential B (ˆx) ε Let us consider an elemental region Ω in R3 as a closed set, Z3 in such a way that its boundary is conformed by two in- ik0r = − ∇yˆ · ∇e fγ (r)dV (ˆy) dependent surfaces. This means that ∂Ω = S ∪ ∂Bε(ˆx0) with S ∩ ∂Bε(ˆx0) = ∅, where S is a surface that bounds Bε(ˆx) ∂Bε(ˆx0) and ∅ denotes the empty set. Also, we are going Z2 to assume that S is smooth by parts, where S ⊂ Ω and ik r (ˆy − xˆ) = − e 0 A(r)(ˆx − yˆ) · dA(ˆy) ∂B (ˆx ) ⊂ Ω, due to the fact that Ω is closed. So, re- ||yˆ − xˆ|| ε 0 ∂Bε(ˆx) gion Ω can be assumed as a glass ovoid with an inside air sphere, where the elliptical surface of the ovoid corresponds = ··· = 4πeik0ε(ik ε − 1), (42) 0 to S, and the surface of the air sphere is ∂Bε(ˆx0). It should be understood that ∂Ω limits two exterior zones and one in- as a consequence of the divergence Gauss theorem in vari- ner zone (see Fig. 1): the big exterior zone given by the ik0r able yˆ, applied to the field ∇e fγ (r), and the formula for c 3 open set (Ω ∪ Bε(ˆx0)) = {xˆ ∈ R :x ˆ ∈ / Ω ∪ Bε(ˆx0)}, ik0r ∇e fγ (r) when γ < ε = r. Naturally, the reduction of the small exterior zone given by Bε(ˆx0), and the inner zone the calculations in the double integral of Eq. (42) is implied Ω\∂Ω = Ω\(S ∪∂B (x )) = {xˆ ∈ Ω :x ˆ ∈ / S ∪∂B (x )}. yˆ ∈ ∂B (ˆx) ε 0 ε 0 from the fact that ε . So, since the smoothness of Therefore, when considering possible parametrizations of ∇eik0rf (r) B (ˆx) the field γ on ε is warranted in this case, then the surfaces S and ∂B (ˆx ), we must think that the uni- G ε 0 function in Eq. (25) satisfies property 2), when considering tary normals nˆ(ˆx) should point towards the big exterior zone G(ˆx, yˆ) Eqs. (41), (42), and (37). In conclusion, in Eq. (25), when xˆ ∈ S, and point towards the small exterior zone when f p with γ in Eq. (30) and γ in Eq. (34), is a true Green func- xˆ ∈ ∂B (ˆx ), respectively. So, if we apply the second Green G ε 0 tion. And of course, this is understood as the limit of a identity to a C1-phasor f(ˆx) = a(ˆx)eiφ(ˆx), corresponding to g (ˆx) sequence of smooth functions that converge to yˆ almost some µ solution of the homogeneous wave equation on Ω, everywhere. In mathematical terms, G(ˆx, yˆ) = g (ˆx) for all yˆ and the function g(ˆx) = eik0r/r with r = ||xˆ − xˆ ||, then we xˆ 6=y ˆ xˆ =y ˆ g (ˆy) 0 , except in , where yˆ is not defined and get G(ˆy, yˆ) = 0. Informally speaking, that is why gyˆ(ˆx) inherits the name of its equivalent G(ˆx, yˆ). Finally, we could be attempted to believe that G is an ordinary function like   eik0||xˆ−yˆ||/||xˆ − yˆ|| for xˆ 6=y ˆ G (ˆx, yˆ) = (43) •  0 for xˆ =y, ˆ and that is true, in some way, with respect to the image sets induced by both expressions. Although formally

L[G•]dV (ˆy) = 0,

R3 there is no obstacle to call G• as a Green function because FIGURE 1. Sketch of the region Ω. In this example, it is assumed G is also a generalization of G• or, equivalently, G• = gyˆ as an ovoid with an inside sphere of radius ε.

Rev. Mex. Fis. E 64 (2018) 26–38 THE RETARDED POTENTIAL OF A NON-HOMOGENEOUS WAVE EQUATION: INTRODUCTORY ANALYSIS.. . 33

by using the formulas in Proposition 1. Owing to the fact Z2 µ ¶ Z3 that vector nˆ points towards the small exterior zone in this ∂g ∂f f − g dA = (f∇2g − g∇2f)dV. (44) particular case, we have that ∂n ∂n ∂Ω Ω ∂g (ˆx∗) = ∇g · nˆ| , (51a) However, ∂n xˆ∗ Z3 Z3 xˆ0 − xˆ∗ xˆ∗ − xˆ0 (f∇2g − g∇2f)dV = (−k2fg + k2gf)dV = 0, (45) nˆ(ˆx∗) = = − , (51b) 0 0 ||xˆ0 − xˆ∗|| ε Ω Ω because both the phasor f and function g satisfy the then ik0ε Helmholtz equation for all points in Ω. This implies that ∂g e (1 − ik0ε) (ˆx∗) = , (52) Z2 µ ¶ Z2 µ ¶ ∂n ε2 ∂g ∂f ∂g ∂f f − g dA = f − g dA ∂n ∂n ∂n ∂n from Eqs. (50) and (51). Hence ∂Ω S Z2 µ ¶ Z2 µ ¶ ∂g ∂f ∂g ∂f + f − g dA = 0, (46) f − g dA ∂n ∂n ∂n ∂n ∂B (ˆx ) ∂Bε(ˆx0) ε 0 · ¸ ∂f or, equivalently, ik0ε ik0ε = 4π f(ˆx∗)e (1 − ik0ε) − εe (ˆx∗) , (53) Z2 µ ¶ ∂n ∂g ∂f f − g dA ∂n ∂n after substituting Eqs. (50) and (52) into Eq. (49). If ε → 0+, ∂Bε(ˆx0) then xˆ∗ → xˆ0 and Z2 µ ¶ ∂g ∂f = − f − g dA. (47) Z2 µ ¶ ∂n ∂n ∂g ∂f lim f − g dA = 4πf(ˆx0), (54) S ε→0+ ∂n ∂n Since the double integral on the right hand side of Eq.(47) is ∂Bε(ˆx0) a constant, independently of the ε value, it follows that from Eq. (53) and the fact that f and ∂f/∂n are continuous Z2 µ ¶ ∂g ∂f functions (f is C1-function). The limit in Eq. (54) represents lim f − g dA = ε→0+ ∂n ∂n the same to write ∂Bε(ˆx0) Z2 µ ¶ Z2 µ ¶ 1 ∂f ∂g ∂g ∂f f(ˆx ) = g − f dA, (55) − f − g dA, (48) 0 4π ∂n ∂n ∂n ∂n S S including the possibility that ∂Ω = S ∪ {xˆ0} for the limit by considering Eqs. (48) and (54). Equation (55) is valid 1 3 ik r case. Now, the integral on ∂Bε(ˆx0) in Eqs.(47) and (48) can for any C -phasor f : R → C, where g(ˆx) = e 0 /r and be expressed as r = ||xˆ − xˆ0||. In this case S ∪ {xˆ0} = ∂Ω is the union Z2 µ ¶ of a smooth-by-parts surface S with the frontier point xˆ0. ∂g ∂f f − g dA Examples of smooth-by parts surfaces could be a sphere, an ∂n ∂n ellipsoid, the three faces of triangular pyramid, the six faces ∂Bε(ˆx0) of a parallelepiped, etc. So, without loss of generality and µ ¶¯ ∂g ∂f ¯ in the context of distributions, the result in Eq. (55) is also = 4πε2 f − g ¯ , (49) ∂n ∂n ¯ valid when simply assuming S = ∂Ω in such a way that xˆ∗ xˆ0 ∈ Ω\S. Equation (55) corresponds to the integral theorem from the mean value theorem of integrals, where 4πε2 is of Helmholtz and Kirchhoff, which can be generalized to any the area of ∂Bε(ˆx0) and xˆ∗ is some ﬁxed point in ∂Bε(ˆx0). g function satisfying the Helmholtz equation in an equivalent Thus, if xˆ∗ ∈ ∂Bε(ˆx0), then sense. In other words, g in Eq. (55) could be any Green func- eik0ε tion. In this argumentation we have assumed that L[f] = 0 g(ˆx∗) = , (50a) xˆ ∈ Ω ε for all , but such hypothesis can be modiﬁed to an µ ¶ equivalent case as follows: Let G be a Green function with ik0ε − 1 ∇g(ˆx ) = eik0ε (ˆx − xˆ ), (50b) respect to the linear operator L, and let h be a distribution ∗ ε3 ∗ 0 similar to h0δ(ˆx−xˆ1) with a constant factor (an independent

Rev. Mex. Fis. E 64 (2018) 26–38 34 A. TELLEZ-QUI´ NONES,˜ J.C. VALDIVIEZO-NAVARRO, A. SALAZAR-GARIBAY AND A.A. LOPEZ-CALOCA´ term with respect to variable xˆ) h0, and xˆ1 ∈/ Ω \ ∂Ω. Thus, in Eq. (59), we get a potential solution for the problem of if f is a function such that L[f(ˆx)] = h(ˆx), then solving L[f(ˆx)] = h(ˆx) for all xˆ ∈ Ω \ ∂Ω. Such a problem would be solved with the next hypothesis: the knowl- Z2 1 ³ ∂f edge of functions f and ∂f/∂n on ∂Ω, a given function h, f(ˆy) = G(ˆx, yˆ) (ˆx) 4π ∂n preferable smooth and defined on Ω, and a possible and con- ∂Ω venient Green function G, defined with respect to the opera- ∂G ´ tor L. The inferred solution from Eq. (59) can be expressed − f(ˆx) (ˆx, yˆ) dA(ˆx), (56) as f(ˆx) = f (ˆx) + f (ˆx) with ∂n 0 1 Z2 µ ¶ for all yˆ ∈ Ω \ ∂Ω. Equation (56) can be simply deduced, for 1 ∂f ∂G f (ˆx) := G − f dA(ˆy) (60) instance, from the second Green identity we have 0 4π ∂n ∂n ∂Ω Z2 µ ¶ Z3 ∂f ∂G 2 2 and G −f dA(ˆx) = (G∇ f − f∇ G)dV (ˆx) 3 ∂n ∂n Z ∂Ω Ω f1(ˆx) := G(ˆx, yˆ)h(ˆy)dV (ˆy), (61) Z3 Z3 Ω 2 2 2 2 = G(∇ + k0)fdV (ˆx)− f(∇ + k0)GdV (ˆx), (57) where Ω Ω Z3 but this is the same that L [f1(ˆx)] = L[G(ˆx, yˆ)]h(ˆy)dV (ˆy) Ω Z3 Z3 3 ... = −4π GL[f]dV (ˆx) + 4π fL[G]dV (ˆx) Z = δ(ˆy − xˆ)h(ˆy)dV (ˆy) = h(ˆx). (62) Ω Ω Ω Z3 = −4π G(ˆy, xˆ)h(ˆx)dV (ˆx) Since L[f(ˆx)] = h(ˆx) by hypothesis, it follows that L[f0(ˆx)] = 0. In other words, f0 and f1 are particu- Ω lar solutions of the homogeneous and the non-homogeneous 3 Z Helmholtz equation, respectively. We say that f0 and f1 are + 4π f(ˆx)δ(ˆx − yˆ)dV (ˆx) “particular” functions, because both of them depend on the Ω G chosen. Eventually, beyond of considering a bounded set as Ω, if we think in solving L[f] = h on R3, the result in Z3 Eq. (59) would be equivalent to consider = −4π G(ˆy, xˆ)h(ˆx)dV (ˆx) + 4πf(ˆy), (58) Z3 Ω f(ˆx) = G(ˆx, yˆ)h(ˆy)dV (ˆy), (63) from definition of G. Then, from Eqs.(57) and (58) we get R3 Z2 µ ¶ 1 ∂f ∂G as the potential solution. f(ˆy) = G − f dA(ˆx) On the other hand, the theory of Green functions, ex- 4π ∂n ∂n ∂Ω posed in a previous section, was described with respect to the operator [−1/(4π)][∇2 + k2]. However, this theory Z3 0 does not change if we consider other similar operators as + G(ˆy, xˆ)h(ˆx)dV (ˆx). (59) 2 2 2 2 (∇ + k0) or [∇ + (k0/c0) ]. Thus, when considering 2 2 Ω L := [−1/(4π)][∇ + (k0/c0) ], function

So, if h(ˆx) = h0δ(ˆx − xˆ1) with xˆ1 ∈/ Ω \ ∂Ω, then the triple −ei(−k0/c0)||xˆ−yˆ|| g (ˆx) := , (64) integral on Ω in Eq. (59) is identically zero from the proper- 0 ||xˆ − yˆ|| ties of the Dirac delta. This is because satisﬁes that µ ¶ 3 2 Z 2 k0 ∇ + 2 g0(ˆx) = 0, (65) G(ˆy, xˆ)δ(ˆx − xˆ1)dV (ˆx) = G(ˆy, xˆ1), c0 Ω when assuming yˆ as a constant vector and xˆ 6=y ˆ. To verify Eq. (65), it is only required to replace k and xˆ in Proposi- only if xˆ ∈ Ω \ ∂Ω. Moreover, for all xˆ ∈ Ω \ ∂Ω, the 0 0 1 tion 1 by −k /c and yˆ, respectively. Now, we want to solve integrand G(ˆx, yˆ)h(ˆx) = h G(ˆx, yˆ)δ(ˆx − xˆ ) = 0 does 0 0 0 1 µ ¶ not have any discontinuity in Ω \ ∂Ω when considering G 1 ∂2 − ∇2 µ(t, xˆ) = ζ(t, xˆ), (66) in Eq. (25). Thus, by interchanging xˆ by yˆ and vice-versa 2 2 c0 ∂t

Rev. Mex. Fis. E 64 (2018) 26–38 THE RETARDED POTENTIAL OF A NON-HOMOGENEOUS WAVE EQUATION: INTRODUCTORY ANALYSIS.. . 35 on a domain Ω, where ζ (considered only as function of instance, if we simply assume that ζ is a smooth function on xˆ) behaves as a Dirac delta translated to some point out- Ω, then Eq. (68) can be expressed as side of Ω \ ∂Ω. So, by proposing a solution of the form µ ¶ ik t ||xˆ − yˆ|| µ(t, xˆ) = f(ˆx)e 0 and substituting this solution in Eq. (66), L[f(ˆx)] = h(ˆx) := H t − , xˆ , (72) c we get µ ¶ 0 k2 ik0t 2 0 −ik0t e −∇ − 2 f(ˆx) = ζ(t, xˆ). (67) where H(t, xˆ) = (e /(4π))ζ(t, xˆ). From Eq. (61), by c0 interchanging xˆ by yˆ, and considering G = −g0, it follows Therefore, when Eq.(67) is evaluated at (t−(||xˆ−yˆ||/c0), xˆ), that it is obtained that Z3 µ ¶ µ ¶ e−i(k0/c0)||xˆ−yˆ|| ||xˆ − yˆ|| k2 ik0(t−(||xˆ−yˆ||/c0)) 2 0 f(ˆy) = H t − , xˆ dV (ˆx) e −∇ − 2 f(ˆx) ||xˆ − yˆ|| c0 c0 Ω

= ζ(t − (||xˆ − yˆ||/c0), xˆ). (68) Z3 µ ¶ e−ik0t 1 ||xˆ − yˆ|| = ζ t − , xˆ dV (ˆx), (73) In some way, the equality in Eq. (67) suggests that ζ could 4π ||xˆ − yˆ|| c0 be interpreted as a function with separable variables, time Ω t, and position xˆ. This is similar to conveniently think that is a potential solution of the equation L[f] = h given in ζ(t, xˆ) = P (t)Q(ˆx), then Eq. (67) can be expressed as Eq. (72). Indeed, from the form assumed for µ, Eq. (73) L[f(ˆx)] = h(ˆx), where h(ˆx) = h0Q(ˆx) and h0 = h0(t) = induces again the formula given in Eq. (71). Finally, if we P (t)e−ik0t/(4π) is a constant term with respect to variable want to solve L[f] = h on R3, we could use Eq.(63) and xˆ. Consequently, if the spatial part of ζ is a function like conclude that Q(ˆx) = δ(ˆx−xˆ ) with xˆ ∈/ Ω\∂Ω, then the phasor f can be 1 1 Z3 µ ¶ calculated by the integral theorem of Helmholtz and Kirch- 1 1 ||xˆ − yˆ|| hoff, the second Green identity, and any convenient Green µ(t, yˆ)= ζ t − , xˆ dV (ˆx), (74) 4π ||xˆ − yˆ|| c0 function G, as R3 Z2 µ ¶ in analogy to the previous formulas that preserve the name 1 ∂f ∂G f(ˆy) = G − f dA(ˆx) of retarded potential. Nevertheless, Eq. (74) is also valid for 4π ∂n ∂n the case when ζ is a Dirac delta δ, specially in the case of ap- ∂Ω proximating this δ with a sequence of smooth functions like Z3 1 the Gaussians. = (G∇2f − f∇2G)dV (ˆx), (69) 4π Ω 6. Discussion and Conclusions for all yˆ ∈ Ω \ ∂Ω. In addition, when taking G = g , it is 0 We have derived the retarded potential of a non- concluded that homogeneous wave equation by considering certain subtle Z3 mathematical details. These details refer to the use of dis- 1 e−ik0||xˆ−yˆ||/c0 f(ˆy) = tributions, in our own and simpliﬁed interpretation of these 4π ||xˆ − yˆ|| generalized functions, and in what sense it is said that a given Ω function is a Green function. According to our analysis, we µ 2 ¶ 2 k0 obtained a distributional solution f for Eq. (72), which per- × −∇ − 2 f(ˆx)dV (ˆx), (70) c0 mits us to construct a solution µ for Eq. (66). When considering Eq. (72) in a bounded set Ω, the solution can take from Eqs.(64), (65), and (69). Furthermore, Eq.(70) implies place by establishing boundary conditions in the frontier ∂Ω, that as exposed by Eq. (59). These conditions may be imposed in Z3 µ ¶ f, or in ∂f/∂n, depending on the problem. For instance, in 1 1 ||xˆ−yˆ|| µ(t, yˆ)= ζ t− , xˆ dV (ˆx), (71) the simple case of h = 0 for all points in Ω, it follows that 4π ||xˆ−yˆ|| c0 f = f0 from Eq. (60). In this particular case, if we only Ω know f on the frontier ∂Ω (Dirichlet problem), then a desir- from the form assumed for µ, and Eqs. (68) and (70). The able election of G could be a Green function that vanishes on expression in Eq. (71) is known as the retarded potential of this boundary. ζ, as mentioned in [5]. Its name reveals that the signal µ is In a general perspective, the base idea of the potential so- recovered from the source ζ with a delay in time. This poten- lutions is manifested by Eqs. (59)-(61) and (63), where the tial represents a standard solution of the d’Alembert equation election of an appropriate Green function is crucial to obtain displayed in Eq. (66). Moreover, the formula in Eq. (71) is speciﬁc results. For example, the formula given by Eq. (71) independent of the fact that ζ behaves as a Dirac delta. For in the bounded case, or by Eq. (74) in the unbounded case,

Rev. Mex. Fis. E 64 (2018) 26–38 36 A. TELLEZ-QUI´ NONES,˜ J.C. VALDIVIEZO-NAVARRO, A. SALAZAR-GARIBAY AND A.A. LOPEZ-CALOCA´ respectively. The retarded potentials allow us to build so- the set of C∞-functions φ such that, φ, with all its deriva- lutions of classic problems in electrodynamics, wave prop- tives vanish at infinity as fast as ||xˆ||−N when ||xˆ|| → ∞, agators, radar, among others. For instance, potential theory and independently of how long is the positive integer N. Any can be applied for modeling equations related with the emis- function φ in the set D is said to be particularly well-behaved sion and detection of a SAR signal [5, 6]. In a SAR config- and such set would correspond to the space of test functions uration, the main equation that involves the recovered val- defined in [26]. According to [28], a sequence of functions ues of the signal and the scattering object density is a conse- {hγ } (considering values γ > 0) is said to be regular, if and R 3 quence of using Eqs. (71) or (74), in connection with the first + only if limγ→0 R3 hγ φdV exists, for all φ ∈ D. Thus, a Born approximation. Such an approximation reduces the ill- distribution H is a regular sequence of functions in D given posed problem of recovering the scattering object to a simple by {hγ }, where symbol convolution-filtering problem. About the mathematical rigor found in some references, Z3 an explicit and formal explanation about Green functions, by HφdV using operator theory, is found in [1]. Although in this ref- R3 erence there is no mention on that g (ˆx) = eik0r/r is Green yˆ means function with respect to the Helmholtz operator, an exten- Z3 sive analysis to derive Green functions from many different lim hγ φdV. (A1) linear operators is given. Nevertheless, such analysis is out γ→0+ 3 of the scope of these notes. On the other hand, it is inter- R Of course, this symbol is not a true integral, because H is a esting to notice that expression gyˆ(ˆx) is declared as a Green function in many books (for example [2–4, 8]), without any sequence. Even in the case when H is interpreted as the limit formal proof of that fact. Whereas in some other books and of this sequence, such limit could not be an ordinary function for some other kind of Green functions, a proof is provided and that is why H is declared as a symbolic function [28]. but with drawbacks. For instance, the argumentation in [21] In a more strict sense, a distribution, is a continuous linear when justifying f(ˆx, yˆ) = −1/(4π||xˆ−yˆ||) as a Green func- functional TH defined on the space of functions D [1, 26], as tion with respect to a distributional Poisson equation. In that the symbolic notation reference, the drawback is exactly the same that was exposed Z3 in Sec. 4 on the illegal use of the divergence Gauss theorem. T (φ) := HφdV Of course, we do not pretend to criticize that books because H they are actually excellent references and we are far from ex- R3 posing a formal proof. However, we expect at least to mo- suggests. tivate the reader on the importance of considering sequences II) In this discussion, we have relaxed the hypothesis by con- of approximating functions [28], to have a more clear idea sidering D, as the set conformed by functions that are at about the handling of distributions. Such as made in Eq. (25), least of C1-class and that vanish at infinity in the next weak by considering these kind of sequences, an equivalence re- sense [26]: function h vanishes at infinity, if and only if lation could be established between G(ˆx, yˆ), defined in Eq. V [St,h] is finite, for all constant t > 0. Here, St,h := {xˆ : (25), and function g (ˆx). Since G(ˆx, yˆ) = g (ˆx) is valid yˆ yˆ |h(ˆx)| > t} and V [St,h] is the volume (or Borel measure) almost everywhere with respect to the variable yˆ ∈ R3 (or of the set St,h. Moreover, in this work, a distribution H is xˆ ∈ R3) [26, 27], there is no doubt now of calling g (ˆx) yˆ understood as the limit of a sequence {hγ }, conformed by el- as a Green function. In a similar manner, the argumenta- ements in the set D, our particular set of well-behaved or test tion in [21] when justifying f(ˆx, yˆ) as a Green function could functions. However, for practical purposes, our test functions be improved when considering sequences of smooth approx- do not require to satisfy more properties, like the regularity imating functions. Just as Jackson explains in his analysis condition imposed in [28]. Conceptually, we are assuming on Poisson and Laplace equations in reference [3], we want that to emphasize a very good footnote which refers to the vol- H(ˆx) := lim hγ (ˆx). (A2) ume integral of Eq. (1.36) in that reference: “The reader may γ→0+ complain that (1.36) has been obtained in an illegal fashion Therefore, when an ordinary function h is such that h(ˆx) = since 1/|x − x0| is not well-behaved inside the volume V . H(ˆx) almost everywhere, we say that h is the distribution H. ik0r Rigor can be restored by using a limiting process. . . ” Well, III) When considering {e fγ (r)} with r = ||xˆ|| in Eq. (25) such limiting process has been exemplified in this work. (the simple case when yˆ = 0ˆ), we clearly have that functions ik0r 1 hγ (ˆx) := e fγ (r) are of C -class, as described in Sec. 4. |h (ˆx)| = |f (r)| = f (r) S = S := Appendix A Now, since γ γ γ , then t,hγ t,γ {xˆ : fγ (r) > t}. Hence, given a pair of constants t, γ > 0, Remarks: it is not difficult to observe that V [St,γ ] is finite in any case: 3 I) Based on [28], definition of distribution for the general a) For 2/γ ≤ t, there is no xˆ ∈ R such that fγ (r) > t be- 3 case of functions h : R → C is given as follows: Let D be cause the maximum value of fγ is achieved at r = γ/2, then

Rev. Mex. Fis. E 64 (2018) 26–38 THE RETARDED POTENTIAL OF A NON-HOMOGENEOUS WAVE EQUATION: INTRODUCTORY ANALYSIS.. . 37

ik0r St,γ = ∅ and V [∅] = 0. b) For 1/γ ≤ t < 2/γ, we have Proposition 1 The function g(ˆx) = e /r, where r = ˆ ˆ ||xˆ − xˆ0|| > 0 and xˆ0 is a constant position, satisfies St,γ = Br2 (0) \ Br1 (0) for some fixed values r1 and r2 such µ ¶ that 0 < r1 < r2 ≤ γ. Here, Br (0)ˆ = Br (0)ˆ ∪ ∂Br (0)ˆ 1 1 1 ik0r − 1 ˆ ˆ ∇g(ˆx) = eik0r (ˆx − xˆ ). (A4) and that implies St,γ ⊂ Br2 (0) ⊂ Bγ (0), which means that 3 0 ˆ ˆ 3 r V [St,γ ] ≤ V [Br2 (0)] ≤ V [Bγ (0)] = (4πγ /3). c) For ˆ ˆ ˆ t < 1/γ, we have St,γ = B1/t(0) \ Br1 (0) ⊂ B1/t(0) Moreover, such function satisfies the Helmholtz equation 2 2 for some fixed value r1 such that 0 < r1 < 1/t, then (∇ + k0)g(ˆx) = 0 for all xˆ 6=x ˆ0. V [S ] ≤ V [B (0)]ˆ = (4π/3t3). t,γ 1/t Proposition 2 Let us consider the variables xˆ = (x, y, z), Therefore, remark III) establishes that the sequence of yˆ = (u, v, w), and the operator ∇ := (∂/∂u, ∂/∂v, ∂/∂w). functions used in Eq. (25) is a sequence of well-behaved yˆ Thus, for any vectorial field of the form H(r)(ˆx − yˆ), with functions, a sequence in the set D defined in remark II). r = ||xˆ − yˆ|| and H(r) as a smooth function for all r > 0, D φ IV) According to our definition of , any function in we have that this set has at least a continuous partial derivative ∂φ/∂x (it could be also with respect to y, or z). Then, a sequence of ∇yˆ · H(r)(ˆx − yˆ) = −∇ · H(r)(ˆx − yˆ). (A5) 2 C -class functions in D given by {hγ }, could induce a se- ik0||xˆ−yˆ|| quence {∂hγ /∂x} contained in D. If so, we can talk about In particular, function gyˆ(ˆx) = e /||xˆ − yˆ|| satis- ik0r 3 ∂wH/∂x = limγ→0+ ∂hγ /∂x and H = limγ→0+ hγ , inde- fies that ∇gyˆ(ˆx) = e [(ik0r − 1)/r ](ˆx − yˆ), as a direct pendently if these limits represent functions or not. There- consequence of Proposition 1 (when xˆ0 =y ˆ). Therefore, fore, if the sequences {∂hγ /∂x} and {hγ } are such that ∇yˆ · ∇gyˆ(ˆx) = −∇ · ∇gyˆ(ˆx), (A6) Z3 Z3 xˆ 6=y ˆ (∂wH/∂x)φdV = − H(∂φ/∂x)dV, (A3) for .

R3 R3 The last two propositions can be demonstrated by a care- ful calculation of partial derivatives and an adequate use of for all φ ∈ D (by considering the symbolic representation in the chain rule. Eq. (A1)), then ∂wH/∂x is said to be a weak derivative of H in a distributional sense. Of course, we have used the notation ∂wH/∂x, to distinguish this limit (or generalized function) Acknowledgment from the usual partial derivative of H, which is ∂H/∂x every The authors thank Dr. Gerardo Garc´ıa-Almeida for their time that H is interpreted as an ordinary function (H = h). valuable comments on this paper.

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