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The retarded potential of a non-homogeneous wave equation: introductory

analysis through the Green functions

A. T´ellez-Qui˜nonesa,∗, J.C. Valdiviezo-Navarroa, A. Salazar-Garibaya, and A.A. L´opez-Calocab aCONACYT-Centro de Investigaci´on en Geograf´ıa y Geom´atica, Ing. J.L. Tamayo, A.C. (Unidad M´erida),

Carretera Sierra Papacal-Chuburna Puerto Km 5, Sierra Papacal-Yucat´an, 97302, M´exico. bCentro de Investigaci´on en Geograf´ıa y Geom´atica, Ing. J.L. Tamayo, A.C., Contoy 137-Lomas de Padierna, Tlalpan-Ciudad de M´exico, 14240, M´exico.

e-mail: atellezq@gmail.com

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

Potential theory can be simply understood as the art of solving a linear distributional non-homogeneous differential equation through the Green functions [1–3]. In the context of this study, our interest resides in the construction of an integral solution that derives from the divergence Gauss the-orem, certain Green identities, and the handling of the Green functions. In the construction of this potential solution we also find other results of great importance, such as the inte-gral theorem of Helmholtz and Kirchhoff, which is the main result that supports the scalar diffraction theory in optics [4]. However, Green functions are not properly functions in the usual sense, since they are formally defined as distributions. Distributional theory, Green functions, and the use of Green identities have been successfully implemented in many theo-retical and applied works, e.g., SAR theory [5–7], scattering and wave propagation [8–12], wave diffraction and electro-dynamics [13–16], phase unwrapping [17–20], etc.

The concept of distribution is not easy to explain and in most of the references where distributions are mentioned, such as basic courses in calculus [21], differential equa-tions [22], linear systems [23], or Fourier analysis [24], a detailed explanation of such abstractions is not usually given. Hence, in order to have a more solid notion of the concept of distributions, or generalized functions, specialized litera-ture should be consulted. This literalitera-ture is specifically re-lated to a coarse field of mathematics called functional anal-ysis [25, 26] and some notions on Lebesgue measure [27]. Certainly, in this work we will not provide specific and de-tailed information about distributions. Instead of that, we are

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2.

Derivation of the non-homogeneous wave

equation

It is well known that, an electromagnetic wave, such as a radar wave, is characterized in each point of the space

ˆ

x = (x, y, z), and each timet, by the vectorial functions E = E(t,xˆ), the electric field (EF), andH = H(t,xˆ), the magnetic field (MF). When assuming as known the variables J =J(t,xˆ), the current density (vectorial field),σ=σ(t,xˆ), the charge density (scalar function),q, the permittivity con-stant, and p, the permeability constant, the fieldsE andH can be found. These fields are determined by the Maxwell equations: ∇ ·E =σ/q, the Gauss law for EF,∇ ·H = 0, the Gauss law for MF,∇ ×E+p(∂H/∂t) = ˆ0, the Faraday law, and∇ ×H −q(∂E/∂t) = J, the Amp`ere law. Here,

:= (∂/∂x, ∂/∂y, ∂/∂z),ˆ0 = (0,0,0), and the media is assumed free of magnetics sources. In this case, operations

∇ ·F and∇ ×F refer, respectively, to the divergence and the rotational of any vectorial fieldF =F(t,xˆ). Hence, a method to findEandHcan be constructed from the previous Maxwell equations and the next theorems:

Theorem 1: Let F : R3 R3 be a real valued vectorial field of class C1 (continuous function with continuous first order partial derivatives), exceptionally in a finite number of points. Then,Fis the gradient of some scalar functionf(that is,F=∇f), if and only if∇ ×F = ˆ0.

Theorem 2: If F : R3 R3 is aC1-vectorial field such that∇ ·F = 0, then, there is aC1-vectorial fieldGsuch that F =∇ ×G.

These theorems are demonstrated in [21] for real valued functions dependent on variablexˆ(whereF =Fx)), how-ever, they can be generalized to complex valued functions of the formF : R3 C3. Moreover, these theorems are

in-dependent of variablet, so, they are also true for complex fields [29] of the formF : R4 C3whereF = F(t,xˆ).

Now, by assuming E andH as C2-fields (that means, C1

-fields with continuous second order partial derivatives), from the Gauss law for MF, and Theorem 2, there exists a fieldA0

such that

H=∇ ×A0. (1)

Additionally, from Faraday law and Eq. (1), we have that

ˆ0 =∇ ×E+p∂

∂t(∇ ×A0) ..

.

=∇ ×

µ

E+p∂A0 ∂t

,

(2)

which, from Theorem 1, implies the existence of a function µ0satisfying

E+p∂A0

∂t =∇µ0. (3) On the other hand, from the Amp`ere law, Eqs.(1) and (3), and the vectorial identity∇ ×(∇ ×A0) =(∇ ·A0)− ∇2A0

(where2:= (2/∂x2)+(2/∂y2)+(2/∂z2)is the Lapla-cian operator), it follows that

J=∇ ×H−q∂E ∂t

=∇ ×(∇ ×A0)−q∂

∂t

µ

∇µ0−p∂A0

∂t

=(∇ ·A0)− ∇2A0−q∂

∂t(∇µ0) +pq 2A

0

∂t2 . (4)

From the previous relation, it is obtained that

2A0−pq∂ 2A

0

∂t2 =−J+(∇ ·A0)−q

∂t(∇µ0)

=−J+(∇ ·A0)− ∇

µ

q∂µ0 ∂t

=−J+

µ

∇ ·A0−q

∂µ0

∂t

. (5)

Now, by using the Gauss law for EF and Eq. (3), we get

2µ0=

σ q +p

∂t(∇ ·A0). (6) Let us definef as a scalar solution of the equation

2fpq2f

∂t2 =

µ

∇ ·A0−q

∂µ0

∂t

, (7)

and µ, as the functionµ := p(∂f /∂t) +µ0. From these

definitions we obtain the relation ∂f

∂t =

(µ−µ0)

p . (8)

Thus, it can be noticed that

H =∇ ×A, (9)

withA=A0+∇f. The result in Eq. (9) is logical because

∇ ×(∇f) = ˆ0for all differentiable scalar fieldf; in other words,∇ ×A =∇ ×A0for suchA. SinceH can be

cal-culated, as suggested in Eq. (9), there is a functionµ1 that

satisfies

E+p∂A

∂t =∇µ1, (10) in analogy to the steps given for deriving Eq. (3). However,

∇µ=∇µ1due to the fact that

∇µ1=E+p∂

∂t(A0+∇f)

=

µ

E+p∂A0 ∂t

+p∂ ∂t(∇f)

=∇µ0+p∇

µ

∂f ∂t

=∇µ, (11)

a consequence of Eqs. (3), (8), and (10). In this case, we can replace∇µ1by∇µin Eq. (10) in order to obtain

E+p∂A

(3)

Thus, by using Eqs. (9) and (12), thisAmust satisfy

2A−pq∂

2A

∂t2 =−J+

µ

∇ ·A−q∂µ ∂t

(13) and

2µ= σ

q +p

∂t(∇ ·A), (14) in analogy to the steps for concluding Eqs. (5) and (6) from Eqs. (1) and (3). However, we have now a simplification in our calculations because

∇ ·A−q∂µ

∂t = 0, (15) as the reader can confirm.

Since Eq. (15) takes place, then Eq. (13) reduces to

2Apq2A

∂t2 =−J. (16)

In a similar fashion, from Eq. (15), the expression in Eq. (14) is rewritten as

2µpq2µ

∂t2 =

σ

q. (17)

Therefore, by considering the wave propagation velocity c0 := 1/√pq(as defined in [30]), and introducing the

func-tionζ :=−σ/q, Eqs. (16) and (17) are correspondingly ex-pressed as

1

c2 0

2A

∂t2 − ∇

2A=J, (18a)

1

c2 0

2µ

∂t2 − ∇

2µ=ζ. (18b)

Then, when assumingJ andζas known functions, the so-lutions of Eqs. (18) forA andµ, permit us to findH and E, from Eqs. (9) and (12), respectively. Equation (18) for µ, and analogously forA, is known as the non-homogeneous wave equation, or d’Alembert equation [5] in the distribu-tional sense. Due to its undulatory nature, the solution of this equation is called scalar wave in the case ofµ, or vectorial wave, in the case ofA.

3.

The Helmholtz equation, the divergence

Gauss theorem, and the Green identities

For constructing a solution of the non-homogeneous wave equation, from the methods applied for solving differential equations [22, 31, 32], it is typically proposed a general solu-tion of the formµ = µ0+µ1, whereµ0 is solution of the

homogeneous version of the wave equation (whenζ = 0), andµ1 is solution of the original non-homogeneous

equa-tion (whenζ 6= 0). In radar language [33–35], functionζis interpreted as the source or the electric pulse emitted by an antenna. In this context, whenζ 6= 0, it is understood that the pulse induces a propagating wave which reaches an ob-ject. This wave is known as emitting or incident field and it

is denoted byµ1. In opposite form, the object, assumed to

be like a non-emitting electric pulse source (ζ= 0), reflects or backscatters the incident wave. So, the reflected wave or backscattered field is denoted byµ0 (details of this

concep-tion are explained in [5]). Consequently, the general soluconcep-tion µis known as total field. Since the total field can be found by solving first the homogeneous wave equation, we are go-ing to focus our attention in the construction ofµ0. Thus, in

this section, variableµ0will be denote asµfor simplifying

notation. Moreover, it is important to remark that any com-ponent of the vectorial functionA in Eq.(18), withJ = ˆ0, can be found just as solving the scalar case forµin Eq.(18), withζ= 0. In this sense, it will be sufficient to establish the theory for solving this scalar case.

Let us consider a scalar wave with spatial period or wave-lengthλ0, refractive index median = 1(air) and

angular-temporal frequency ω0. Such wave can be represented by

the function µ˜(t,xˆ) = ax) cos (ω0t+φx)), where ax)

is the amplitude, and φx) is the phase. This cosinu-soidal form is well known from the classic theory for solv-ing the homogeneous wave equation; specifically speaksolv-ing, the method called separation of variables, and the Fourier series [22]. However, in an equivalent way, the form of the wave can be generalized to the complex representation µ(t,xˆ) = fx)eiω0t, whereµ˜ is the real part ofµ,fx) = ax)eiφx) is the spatial part of µ (the so-called phasor or

complex perturbation [4]), andiis the imaginary unit. Since this complex representation must satisfy the homogeneous wave equation, we have that2µ(1/c2

0)(2µ/∂t2) = 0,

where c0 = λ0/T0 is the wave propagation velocity and

T0 denotes the temporal period of such wave. In this case

τ0= 1/T0would be called simply as the temporal frequency,

whereω0 = 2πτ0. Thus, when substituting the complex

representation ofµin the homogeneous wave equation, we obtain thateiω0t∇2fx)(i2ω2

0/c20)fx)eiω0t = 0, which

implies

(2+k2

0)fx) = 0. (19)

This last expression is called Helmholtz equation, where k0=ω0/c0= (2πτ0)/(λ0τ0) = (2π)0is the wave

num-ber or the angular-spatial frequency.

On the other hand, a well-known result in vectorial cal-culus is the divergence Gauss theorem [21], which can be written as

3

Z

∇ ·F dV =

2

Z

ˆndA, (20)

where F = Fx) is a C1-vectorial field on such that

F : R3 R3, and is an elemental region in R3 with

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surfaceΩ, in such a way that, it points towards the outside of the surface. So, if we takeF = f∇g, wheref = fx) andg =gx)are two differentiable scalar fields fromR3to

R, then

∇ ·F =∇ ·(f∇g) =∇f · ∇g+f∇2g. (21)

When substituting this equation in the divergence Gauss re-lation, and introducing the directional derivative(∂g/∂n) :=

∇g·n, we obtainˆ

3

Z

∇f· ∇gdV +

3

Z

f∇2gdV = 2

Z

f∂g

∂ndA. (22) Equation (22) is known as the first Green identity. Analo-gously, when consideringF =g∇f, the equality

3

Z

∇g· ∇f dV +

3

Z

g∇2f dV = 2

Z

g∂f

∂ndA, (23) is deduced. By subtracting Eq. (23) from Eq. (22), it is con-cluded that

3

Z

(f∇2g−g∇2f)dV =

2

Z

µ

f∂g ∂n−g

∂f ∂n

dA, (24)

a relation called second Green identity. It is important to remark that the divergence Gauss theorem is also valid for C1-complex vectorial fieldsF : R3 C3 that can be

ex-pressed in terms of complex scalar fields f, g : R3 C. This last includes the possibility of considering fields of the form F = F(t,xˆ), f = f(t,xˆ), andg = g(t,ˆx), which means that F : R4 C3, andf, g : R4 C. The

va-lidity of this theorem is a consequence of the linearity of the integrals for complex valued expressions that can be denoted asF =Re(F) +iIm(F), where Re(F)and Im(F), are the real and the complex parts ofF, respectively. Thus, the only required condition is to have Re(F)and Im(F), as C1-real

valued functions.

4.

The Green functions: language of

distribu-tions

Informally, any functiongthat satisfies the Helmholtz equa-tion almost everywhere [26, 27] onR3, could be called Green function [4], however, suchgrequires to satisfy another prop-erties in the context of distributions [26]. In this sense, ex-pression almost everywhere refers to a property which is sat-isfied at all points of a domain with the exception of the points in a zero volume subset of the domain. For this particular case, the property would be the fulfilling of the Helmholtz equation. On the other hand, the Green functions and the Green identities are important and useful, specially in op-tics, for establishing a transcendental theorem: the integral theorem of Helmholtz and Kirchhoff. This theorem is the

key result for supporting the scalar diffraction theory and it is related in part with the solution of a particular distri-butional non-homogeneous wave equation, the so-called re-tarded potential of the d’Alembert equation [5]. Now, letL be a linear operator applied to scalar functions depending on

ˆ

x. Formally, a distributionGx,yˆ)is said to be a Green func-tion with respect toL, if it satisfies: a)Gx,yˆ) = Gy,xˆ)

for all xˆ = (x, y, z) and yˆ = (u, v, w) in R3, and b)

L[Gx,yˆ)] = δy−xˆ), whereδy −xˆ)is the Dirac delta distribution. However, every distributionD is said to be a Dirac delta, if it satisfies: 1)Dy−xˆ) = 0, for allyˆ 6= ˆx, and 2)

3

Z

R3

Dy−xˆ)dVy) = 1.

In this definitiondVy)refers to a volume differential ele-ment with respect to the integration variabley. Thus, whenˆ

an arbitrary distribution D fulfills properties 1) and 2), we writeδ:=D.

Let us consider Gx,yˆ) := gyˆ(ˆx), where gyˆ(ˆx) =

eik0||ˆx−y||ˆ /||xˆyˆ||. Here,|| · ||denotes the Euclidean norm for vectors inR3. This function satisfiesgyˆ(ˆx) = gxˆ(ˆy)by

symmetry, thenGx,yˆ)fulfills property a). Is thisGa Green function? What linear operator could be related to this G to declare it as a Green function? Well, the obvious answer is that such operator must be involved with the Helmholtz equation. So, if we considerL:= [1/(4π)](2+k02), then L[fx)] = [1/(4π)](2+k2

0)fx)for any smooth

func-tion f : R3 C. Moreover,Dy−xˆ) := L[Gx,yˆ)] =

[1/(4π)](2+k2

0)gyˆ(ˆx) = 0for allxˆ 6= ˆy, from

Propo-sition 1 in Appendix A. This also implies that property 1) is satisfied byDy−xˆ), every time thatxˆ 6= ˆy. Thus, assum-ing thisDas a simple function depending on y, which is aˆ

discontinuous function inyˆ= ˆx, we rigorously have that

3

Z

R3

Dy−xˆ)dVy) = 0

by using improper integrals. Even in the case of more com-plex integrals, if a function is zero almost everywhere in certain domain, then its Lebesgue integral on such domain should be zero [27]. Thus, since property 2) is not achieved by this D, it implies that Gx,yˆ) = gyˆ(ˆx) is not a Green

function. But, why so many references [2–4, 8] declaregyˆ(ˆx)

as a Green function? Well, may be the answer is in the in-terpretation ofD, and consequentlyG, as distributions. In the spirit of considering a sequence of well-behaved approxi-mating functions (see the Remarks in Appendix A),Gcan be rewritten to the equivalent form

Gx,yˆ) := lim

γ→0+e

ik0r(r), (25)

wherer=||xˆ−yˆ||and lim

(5)

for allr >0, and lim

r→0+(r) = (0), respectively. Now, it is evident thatGin Eq. (25) satisfies properties a) and 1), however, if property 2) is required forDwith this newG, we need to construct function conveniently. There are many forms to do this construction, but we are going to propose a particular one. Before starting with this construction, let us think again thatGx,yˆ) isgyˆ(ˆx), and consider the next

argumentation:

From property 1), it is inferred that the integral of D onR3 is the same that the integral of D on any open ball x) ={yˆR3:||yˆxˆ||< ε}. Then,

3

Z

R3

Dy−ˆx)dVy) =

µ

1 4π

×

3

Z

x)

(2+k02)gyˆ(ˆx)dVy) =

µ

1 4π

×

  

3

Z

x) 2g

ˆ

yx)dVy) +k20 3

Z

x)

gyˆ(ˆx)dVy)

  , (26)

for allε > 0fixed. Thus, when considering a translation to the origin, a change to spherical coordinates, and the use of improper integrals, we have that

3

Z

x)

gyˆ(ˆx)dVy) = 3

Z

x)

gxˆ(ˆy)dVy)

=4π

k2 0

[eik0ε(1ik

0ε)1], (27)

independently of the discontinuity ofgxˆ(ˆy)atyˆ= ˆx. Now, 3

Z

x) 2g

ˆ

yx)dVy) =

3

Z

x)

∇ · ∇gyˆ(ˆx)dVy)

=

3

Z

x)

∇yˆ· ∇gyˆ(ˆx)dVy)

=

2

Z

∂Bεx)

∇gyˆ(ˆx)·nˆ(ˆy)dAy), (28)

wherenˆ(ˆy)is the unitary normal vector to the surface∂Bεx)

anddAy)is an area differential element with respect toy.ˆ

This is a consequence of Proposition 2 in Appendix A and the divergence Gauss theorem in variabley. Following theˆ

calculus of the previous triple integral (now a double inte-gral) we have that

. . .=

2

Z

∂Bεx)

eik0r

·

ik0r−1

r3

¸

x−yˆ)

·y−xˆ)

||yˆ−xˆ||dAy) =

2

Z

∂Bεx)

eik0r

·

ik0r−1

r3

¸

×

µ

r2

r

dAy) = 4πeik0ε(ik

0ε−1), (29)

where the valuesr=||xˆ−yˆ||in the double integrals are all equal toε, becauseyˆ∈∂Bεx) ={yˆR3:||yˆxˆ||=ε}.

When considering Eqs. (27), (28), and (29) in connection with Eq. (26), we get property 2). Therefore, functiongyˆ(ˆx)

results to be a Green function.

The last argumentation, although desirable, is false and its main fail is: function∇gyˆ(ˆx) =eik0r(ik0r−1)(ˆx−yˆ)/r3is

not aC1-function (a smooth function) onx), specifically atyˆ = ˆxwherer = 0. Then, we have incorrectly applied the divergence Gauss theorem in Eq. (28), which requires smoothness for the vector field∇gyˆ(ˆx). However, this

mis-take motivates us to think about an appropriate election of functionin Eq. (25). The sequence of functionssatisfy

lim

γ→0+e

ik0r(r) = eik0r/r =g

ˆ

xy)for allyˆ6= ˆx(r > 0), but that is not enough. For eachγ >0, we also need smooth-ness for the terms∇eik0r(r)for allyˆx). However, since the termeik0ris smooth for allr0, then a sequence of functions of the form

(r) :=

(

p0(r) r≥γ

(r) 0≤r < γ, (30)

wherep0(r) := 1/r, could be useful. For eachγ >0,(t)

could be a convenient polynomial function, in such a way that(γ) = p0(γ),p0γ(γ) = p00(γ), andp00γ(γ) = p000(γ).

Here, symbols0and00denote, correspondingly, the first and the second derivatives with respect tor. Hence, these con-ditions would warrant the smoothness of for allr > 0. However, if(r)is a linear combination of r-powers and we want to analyze the behavior of∇eik0r(r)(particularly atr= 0), we first need to calculate the resultant expressions of the partial derivatives ofeik0rrn, with respect tox,y, and z. For instance, from the chain rule we have

∂x[e

ik0rrn] =eik0r[nrn−1+ik

0rn](x−u)

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∂x[e

ik0rrn]

¯ ¯ ¯ ¯

ˆ

xy

:= lim

h→0

eik0rrn|

ˆ

xy+hˆi−eik0rrn|xˆ=ˆy h

= lim

h→0

eik0|h||h|n

h = 0, (32)

due to the fact that

lim

h→0+

eik0|h||h|n

h = limh→0

eik0|h||h|n h = 0,

as the reader can confirm. Moreover, from Eq.(31) it follows that

lim

ˆ

x→ˆy ∂x[e

ik0rrn]

= limxu

r→0+

½

eik0r[nrn−1+ik

0rn]

(x−u)

r

¾

= limxu

r→0+

©

eik0r[nrn−2+ik

0rn−1](x−u)

ª

= 0. (33)

In consequence, Eqs. (31)-(33) warrant continuity for the term [eik0rrn]/∂x for all xˆ 6= ˆy, and also for xˆ = ˆy, when considering this term as function ofx. Nevertheless,ˆ due to the radial symmetry of eik0r[nrn−1 +ik0rn]/r in Eq. (31), term [eik0rrn]/∂x, interpreted as function ofy,ˆ must be also a continuous function. The same reasoning ap-plies to[eik0rrn]/∂yand[eik0rrn]/∂z; therefore, if(r) is a polynomial of powersn≥2forrvalues such that0

r < γ, then ∇eik0r(r)will be continuous for all points

ˆ

ycorresponding to thoservalues. Moreover, without loss of generality, we can assume that(r) =ar2+br3+cr4+dr5

with(γ/2) =p0(γ/2), which implies that

(r) = (423)r2(1114)r3

+ (1025)r4(326)r5. (34)

So, from Eqs. (30) and (34) we get

∇eik0r(r)=

½

eik0rA(r)(ˆxyˆ) rγ

eik0rB(r)(ˆxyˆ) 0r < γ, (35) whereA(r) = [ik0r−1]/r3and

B(r) = 84

γ3

333r γ4 +

408r2

γ5

160r3

γ6

+ (ik0)

·

42r γ3

111r2

γ4 +

102r3

γ5

32r4

γ6

¸

, (36)

are two functions such that lim

r→γ+A(r) = A(γ) = B(γ) =

lim

r→γ−B(r),r→γlim+A

0(r) = A0(γ) = B0(γ) = lim r→γ−B

0(r),

lim

r→0+B(r) =B(0), andr→lim0+B

0(r) =B0(0). Equation (35) is now a smooth function (C1-class) for allyˆ6= ˆxand also for

ˆ

y = ˆx. In the same way, termeik0rfγ(r)is another smooth

function for allyˆby construction. From this construction and Eq. (25), we can calculate again

3

Z

R3

Dy−xˆ)dVy)

= 1

4πγ→lim0+

 

3

Z

x)

2eik0r(r)dVy)

+k2 0

3

Z

x)

eik0r(r)dVy)

 

, (37)

for anyε >0arbitrary and fixed. But this time we have

3

Z

x)

eik0r(r)dVy) =

3

Z

x)\Bγx)

eik0r r dVy)

+

3

Z

x)

eik0r(r)dVy)

= 4π

k2 0

£

eik0ε(1ik

0ε)−eik0γ(1−ik0γ)

¤

+eik0r•(r•)4πγ 3

3 , (38)

by definition of when considering γ < ε. In the last expression, the integral of (eik0r/r)can be calculated by a translation to the origin and spherical coordinates, while the integral ofeik0r(r)is obtained from the mean value theo-rem for integrals [21, 36] due to the integrand’s continuity. In this caser• = ||xˆ−y•ˆ ||withy•ˆ x)∪∂Bγx), and the factor(4πγ3/3)is the volume of the ballx). On the other hand, when taking the complex module of the integral in Eq. (38), we have that

¯ ¯ ¯ ¯ ¯ ¯ ¯

3

Z

x)

eik0rf

γ(r)dVy)

¯ ¯ ¯ ¯ ¯ ¯ ¯

4π

k2 0

|eik0ε(1ik

0ε)

−eik0γ(1ik

0γ)|+4πγ 3

3 |pγ(r•)|. (39)

Additionally, from Eq. (34), the triangle inequality, and the fact thatr•≤γ, it follows that

4πγ3

3 |pγ(r•)| ≤

1148πγ2

3 . (40)

In an extreme case (4πγ3/3)|(r•)| ' (4πγ2/3) for r•

-values close or equal to γ, but the term(4πγ3/3)|(r•)|

(7)

of Eq. (37)), the inequalities in Eqs. (39) and (40) permit us to establish thateik0r•(r•)(4πγ3/3)0and

k20 3

Z

x)

eik0r(r)dVy)

4π£eik0ε(1ik

0ε)1

¤

, (41) from Eq. (38). Now, due to the form of term∇eik0r(r)in Eq. (35) and Proposition 2, it follows that2eik0r(r) =

∇ · ∇eik0r(r) =−∇

ˆ

y· ∇eik0rfγ(r). Therefore,

3

Z

x)

2eik0r(r)dVy)

=

3

Z

x)

∇yˆ· ∇eik0rfγ(r)dVy)

=

2

Z

∂Bεx)

eik0rA(r)(ˆxyˆ)·y−xˆ)

||yˆ−xˆ||dAy)

=· · ·= 4πeik0ε(ik

0ε−1), (42)

as a consequence of the divergence Gauss theorem in vari-abley, applied to the fieldˆ ∇eik0r(r), and the formula for

∇eik0r(r)whenγ < ε = r. Naturally, the reduction of the calculations in the double integral of Eq. (42) is implied from the fact thatyˆ ∂Bεx). So, since the smoothness of the field∇eik0rfγ(r)onx)is warranted in this case, then functionGin Eq. (25) satisfies property 2), when considering Eqs. (41), (42), and (37). In conclusion,Gx,yˆ)in Eq. (25), within Eq. (30) and in Eq. (34), is a true Green func-tion. And of course, thisGis understood as the limit of a sequence of smooth functions that converge togyˆ(ˆx)almost

everywhere. In mathematical terms,Gx,yˆ) =gyˆ(ˆx)for all

ˆ

x 6= ˆy, except inxˆ = ˆy, where gyˆ(ˆy) is not defined and

Gy,yˆ) = 0. Informally speaking, that is whygyˆ(ˆx)inherits

the name of its equivalentGx,yˆ).

Finally, we could be attempted to believe thatGis an or-dinary function like

G•x,yˆ) =

 

eik0||x−ˆ y||ˆ /||xˆ−yˆ|| for xˆ6= ˆy

0 for xˆ= ˆy,

(43)

and that is true, in some way, with respect to the image sets induced by both expressions. Although formally

3

Z

R3

L[G•]dVy) = 0,

there is no obstacle to callG• as a Green function because Gis also a generalization ofG• or, equivalently,G• = gyˆ

almost everywhere. Something similar happens when think-ing in the one dimensional step function, also known as the Heaviside step. The derivative of the Heaviside step is zero at all points, except in the jump discontinuity where it is not defined. The integral of this derivative along the real line is identically zero. However, if the Heaviside step is understood as a distribution, then its weak derivative [26] is a Dirac delta (see the remarks in Appendix A). As well known, the integral of this weak derivative along the real line is identically one. These affirmations may seem contradictory, but they are only a question of abstract interpretation.

5.

The integral theorem of Helmholtz and

Kirchhoff and the retarded potential

Let us consider an elemental regionΩinR3as a closed set,

in such a way that its boundary is conformed by two in-dependent surfaces. This means that Ω = S ∪∂Bεx0)

withS ∩∂Bεx0) = , whereS is a surface that bounds

∂Bεx0) and denotes the empty set. Also, we are going

to assume that S is smooth by parts, where S Ω and ∂Bεx0) Ω, due to the fact that Ω is closed. So,

re-gionΩcan be assumed as a glass ovoid with an inside air sphere, where the elliptical surface of the ovoid corresponds toS, and the surface of the air sphere is∂Bεx0). It should

be understood thatΩlimits two exterior zones and one in-ner zone (see Fig. 1): the big exterior zone given by the open set(Ω∪Bεx0))c = {xˆ R3 : ˆx /∈∪Bεx0)},

the small exterior zone given byx0), and the inner zone

\∂Ω = Ω\(S∪∂Bε(x0)) ={xˆΩ : ˆx /∈S∪∂Bε(x0)}.

Therefore, when considering possible parametrizations of the surfaces S and ∂Bεx0), we must think that the

uni-tary normalsˆnx)should point towards the big exterior zone whenxˆ∈S, and point towards the small exterior zone when

ˆ

x∈∂Bεx0), respectively. So, if we apply the second Green

identity to aC1-phasorfx) =ax)eiφx), corresponding to someµsolution of the homogeneous wave equation onΩ, and the functiongx) =eik0r/rwithr=||xˆ−xˆ0||, then we get

(8)

2

Z

µ

f∂g ∂n−g

∂f ∂ndA= 3 Z Ω

(f∇2gg2f)dV. (44)

However,

3

Z

(f∇2gg2f)dV = 3

Z

(−k2

0f g+k20gf)dV = 0, (45)

because both the phasor f and function g satisfy the Helmholtz equation for all points inΩ. This implies that

2

Z

µ

f∂g ∂n−g

∂f ∂ndA= 2 Z S µ f∂g ∂n−g

∂f ∂ndA + 2 Z

∂Bεx0)

µ

f∂g ∂n−g

∂f ∂n

dA= 0, (46)

or, equivalently,

2

Z

∂Bεx0)

µ

f∂g ∂n−g

∂f ∂ndA = 2 Z S µ f∂g ∂n−g

∂f ∂n

dA. (47)

Since the double integral on the right hand side of Eq.(47) is a constant, independently of theεvalue, it follows that

lim

ε→0+

2

Z

∂Bεx0)

µ

f∂g ∂n−g

∂f ∂ndA= 2 Z S µ f∂g ∂n−g

∂f ∂n

dA, (48)

including the possibility thatΩ = S∪ {xˆ0} for the limit

case. Now, the integral on∂Bεx0)in Eqs.(47) and (48) can

be expressed as

2

Z

∂Bεx0)

µ

f∂g ∂n−g

∂f ∂n

dA

= 4πε2

µ

f∂g ∂n−g

∂f ∂n ¶¯ ¯ ¯ ¯ ˆ x∗ , (49) from the mean value theorem of integrals, where 4πε2 is

the area of∂Bεx0)andx∗ˆ is some fixed point in∂Bεx0).

Thus, ifx∗ˆ ∈∂Bεx0), then

gx∗) =e

ik0ε

ε , (50a)

∇gx∗) =eik0ε

µ

ik0ε−1

ε3

x∗−xˆ0), (50b)

by using the formulas in Proposition 1. Owing to the fact that vectornˆ points towards the small exterior zone in this particular case, we have that

∂g

∂nx∗) =∇g·nˆ|xˆ∗, (51a)

ˆ

nx∗) = xˆ0−x∗ˆ

||xˆ0−x∗ˆ || =

ˆ

x∗−xˆ0

ε , (51b)

then

∂g ∂nx∗) =

eik0ε(1ik

0ε)

ε2 , (52)

from Eqs. (50) and (51). Hence

2

Z

∂Bεx0)

µ

f∂g ∂n−g

∂f ∂n

dA

= 4π

·

fx∗)eik0ε(1−ik0ε)−εeik0ε

∂f ∂nx∗)

¸

, (53)

after substituting Eqs. (50) and (52) into Eq. (49). Ifε→0+, thenx∗ˆ →xˆ0and

lim

ε→0+

2

Z

∂Bεx0)

µ

f∂g ∂n−g

∂f ∂n

dA= 4πfx0), (54)

from Eq. (53) and the fact thatf and∂f /∂nare continuous functions (f isC1-function). The limit in Eq. (54) represents the same to write

fx0) =

1 4π 2 Z S µ g∂f ∂n−f

∂g ∂n

dA, (55)

by considering Eqs. (48) and (54). Equation (55) is valid for anyC1-phasorf :R3 C, wheregx) =eik0r/rand r = ||xˆ−xˆ0||. In this caseS∪ {xˆ0} = Ωis the union

of a smooth-by-parts surface S with the frontier point xˆ0.

Examples of smooth-by parts surfaces could be a sphere, an ellipsoid, the three faces of triangular pyramid, the six faces of a parallelepiped, etc. So, without loss of generality and in the context of distributions, the result in Eq. (55) is also valid when simply assuming S = Ω in such a way that

ˆ

x0\S. Equation (55) corresponds to the integral theorem

of Helmholtz and Kirchhoff, which can be generalized to any gfunction satisfying the Helmholtz equation in an equivalent sense. In other words,gin Eq. (55) could be any Green func-tion. In this argumentation we have assumed thatL[f] = 0

(9)

term with respect to variablex)ˆ h0, andxˆ1∈/\∂Ω. Thus,

iffis a function such thatL[fx)] =hx), then

fy) = 1 4π

2

Z

³

Gx,yˆ)∂f

∂nx)

−fx)∂G

∂nx,yˆ)

´

dAx), (56) for allyˆ\∂Ω. Equation (56) can be simply deduced, for instance, from the second Green identity we have

2

Z

µ

G∂f ∂n−f

∂G ∂n

dAx) =

3

Z

(G∇2f−f∇2G)dVx)

=

3

Z

G(2+k20)f dVx) 3

Z

f(2+k02)GdVx), (57)

but this is the same that

. . .=4π

3

Z

GL[f]dVx) + 4π

3

Z

f L[G]dVx)

=4π

3

Z

Gy,xˆ)hx)dVx)

+ 4π

3

Z

fx)δx−yˆ)dVx)

=4π

3

Z

Gy,xˆ)hx)dVx) + 4πfy), (58)

from definition ofG. Then, from Eqs.(57) and (58) we get

fy) = 1 4π

2

Z

µ

G∂f ∂n−f

∂G ∂n

dAx)

+

3

Z

Gy,xˆ)hx)dVx). (59)

So, ifhx) =h0δx−xˆ1)withxˆ1∈/\∂Ω, then the triple

integral onΩin Eq. (59) is identically zero from the proper-ties of the Dirac delta. This is because

3

Z

Gy,xˆ)δx−xˆ1)dVx) =Gy,xˆ1),

only ifxˆ1 \∂Ω. Moreover, for all xˆ \∂Ω, the

integrand Gx,yˆ)hx) = h0Gx,yˆ)δx−xˆ1) = 0 does

not have any discontinuity inΩ\∂Ωwhen consideringG in Eq. (25). Thus, by interchangingxˆbyyˆand vice-versa

in Eq. (59), we get a potential solution for the problem of solvingL[fx)] = hx)for allxˆ \∂Ω. Such a prob-lem would be solved with the next hypothesis: the knowl-edge of functionsf and∂f /∂nonΩ, a given functionh, preferable smooth and defined onΩ, and a possible and con-venient Green functionG, defined with respect to the opera-torL. The inferred solution from Eq. (59) can be expressed asfx) =f0(ˆx) +f1(ˆx)with

f0(ˆx) := 1

4π

2

Z

µ

G∂f ∂n−f

∂G ∂n

dAy) (60)

and

f1(ˆx) := 3

Z

Gx,yˆ)hy)dVy), (61)

where

L[f1(ˆx)] = 3

Z

L[Gx,yˆ)]hy)dVy)

=

3

Z

δy−xˆ)hy)dVy) =hx). (62)

Since L[fx)] = hx) by hypothesis, it follows that L[f0(ˆx)] = 0. In other words, f0 and f1 are

particu-lar solutions of the homogeneous and the non-homogeneous Helmholtz equation, respectively. We say thatf0andf1are

“particular” functions, because both of them depend on the Gchosen. Eventually, beyond of considering a bounded set asΩ, if we think in solvingL[f] = honR3, the result in Eq. (59) would be equivalent to consider

fx) =

3

Z

R3

Gx,yˆ)hy)dVy), (63)

as the potential solution.

On the other hand, the theory of Green functions, ex-posed in a previous section, was described with respect to the operator [1/(4π)][2 +k02]. However, this theory does not change if we consider other similar operators as

(2 +k2

0) or [2 + (k0/c0)2]. Thus, when considering

L:= [1/(4π)][2+ (k

0/c0)2], function

g0(ˆx) :=

−ei(−k0/c0)||x−ˆ y||ˆ

||xˆ−yˆ|| , (64)

satisfies that µ

2+k02

c2 0

g0(ˆx) = 0, (65)

when assumingyˆas a constant vector andxˆ 6= ˆy. To verify Eq. (65), it is only required to replacek0andxˆ0in

Proposi-tion 1 by−k0/c0andy, respectively. Now, we want to solveˆ

µ

1

c2 0

2

∂t2 − ∇ 2

(10)

on a domain Ω, where ζ (considered only as function of

ˆ

x) behaves as a Dirac delta translated to some point out-side of Ω\∂Ω. So, by proposing a solution of the form µ(t,xˆ) =fx)eik0tand substituting this solution in Eq. (66), we get

eik0t

µ

−∇2k02

c2 0

fx) =ζ(t,xˆ). (67) Therefore, when Eq.(67) is evaluated at(t−(||xˆ−yˆ||/c0),xˆ),

it is obtained that

eik0(t−(||x−ˆ y||/cˆ 0))

µ

−∇2k02

c2 0

fx)

=ζ(t−(||xˆ−yˆ||/c0),xˆ). (68)

In some way, the equality in Eq. (67) suggests thatζcould be interpreted as a function with separable variables, time t, and positionx. This is similar to conveniently think thatˆ

ζ(t,xˆ) = P(t)Qx), then Eq. (67) can be expressed as L[fx)] =hx), wherehx) =h0Qx)andh0 =h0(t) =

P(t)e−ik0t/(4π)is a constant term with respect to variable

ˆ

x. Consequently, if the spatial part ofζ is a function like Qx) =δx−xˆ1)withxˆ1∈/\∂Ω, then the phasorfcan be

calculated by the integral theorem of Helmholtz and Kirch-hoff, the second Green identity, and any convenient Green functionG, as

fy) = 1 4π

2

Z

µ

G∂f ∂n−f

∂G ∂n

dAx)

= 1 4π

3

Z

(G∇2f−f∇2G)dVx), (69)

for allyˆ \∂Ω. In addition, when taking G= g0, it is

concluded that

fy) = 1 4π

3

Z

e−ik0||ˆx−y||/cˆ 0

||xˆ−yˆ||

×

µ

−∇2k02

c2 0

fx)dVx), (70)

from Eqs.(64), (65), and (69). Furthermore, Eq.(70) implies that

µ(t,yˆ)= 1 4π

3

Z

1

||xˆ−yˆ||ζ

µ

t−||xˆ−yˆ||

c0 ,xˆ

dVx), (71)

from the form assumed forµ, and Eqs. (68) and (70). The expression in Eq. (71) is known as the retarded potential of ζ, as mentioned in [5]. Its name reveals that the signalµis recovered from the sourceζwith a delay in time. This poten-tial represents a standard solution of the d’Alembert equation displayed in Eq. (66). Moreover, the formula in Eq. (71) is independent of the fact thatζbehaves as a Dirac delta. For

instance, if we simply assume thatζis a smooth function on

Ω, then Eq. (68) can be expressed as L[fx)] =hx) :=H

µ

t−||xˆ−yˆ||

c0 ,xˆ

, (72)

whereH(t,xˆ) = (e−ik0t/(4π))ζ(t,xˆ). From Eq. (61), by interchangingxˆbyy, and consideringˆ G= −g0, it follows

that

fy) =

3

Z

e−i(k0/c0)||x−ˆ y||ˆ

||xˆ−yˆ|| H

µ

t−||xˆ−yˆ||

c0 ,xˆ

dVx)

= e

−ik0t

4π

3

Z

1

||ˆx−yˆ||ζ

µ

t−||xˆ−yˆ||

c0 ,xˆ

dVx), (73)

is a potential solution of the equation L[f] = h given in Eq. (72). Indeed, from the form assumed for µ, Eq. (73) induces again the formula given in Eq. (71). Finally, if we want to solve L[f] = hon R3, we could use Eq.(63) and conclude that

µ(t,yˆ)= 1 4π

3

Z

R3

1

||xˆ−yˆ||ζ

µ

t−||ˆx−yˆ||

c0 ,xˆ

dVx), (74)

in analogy to the previous formulas that preserve the name of retarded potential. Nevertheless, Eq. (74) is also valid for the case whenζis a Dirac deltaδ, specially in the case of ap-proximating thisδwith a sequence of smooth functions like the Gaussians.

6.

Discussion and Conclusions

We have derived the retarded potential of a non-homogeneous wave equation by considering certain subtle mathematical details. These details refer to the use of dis-tributions, in our own and simplified interpretation of these generalized functions, and in what sense it is said that a given function is a Green function. According to our analysis, we obtained a distributional solutionf for Eq. (72), which per-mits us to construct a solution µfor Eq. (66). When con-sidering Eq. (72) in a bounded setΩ, the solution can take place by establishing boundary conditions in the frontierΩ, as exposed by Eq. (59). These conditions may be imposed in f, or in∂f /∂n, depending on the problem. For instance, in the simple case ofh = 0for all points inΩ, it follows that f = f0 from Eq. (60). In this particular case, if we only

knowf on the frontierΩ(Dirichlet problem), then a desir-able election ofGcould be a Green function that vanishes on this boundary.

(11)

respectively. The retarded potentials allow us to build so-lutions of classic problems in electrodynamics, wave prop-agators, radar, among others. For instance, potential theory can be applied for modeling equations related with the emis-sion and detection of a SAR signal [5, 6]. In a SAR config-uration, the main equation that involves the recovered val-ues of the signal and the scattering object density is a conse-quence of using Eqs. (71) or (74), in connection with the first Born approximation. Such an approximation reduces the ill-posed problem of recovering the scattering object to a simple convolution-filtering problem.

About the mathematical rigor found in some references, an explicit and formal explanation about Green functions, by using operator theory, is found in [1]. Although in this ref-erence there is no mention on thatgyˆ(ˆx) =eik0r/ris Green

function with respect to the Helmholtz operator, an exten-sive analysis to derive Green functions from many different linear operators is given. Nevertheless, such analysis is out of the scope of these notes. On the other hand, it is inter-esting to notice that expressiongyˆ(ˆx)is declared as a Green

function in many books (for example [2–4, 8]), without any formal proof of that fact. Whereas in some other books and for some other kind of Green functions, a proof is provided but with drawbacks. For instance, the argumentation in [21] when justifyingfx,yˆ) =1/(4π||xˆ−yˆ||)as a Green func-tion with respect to a distribufunc-tional Poisson equafunc-tion. In that reference, the drawback is exactly the same that was exposed in Sec. 4 on the illegal use of the divergence Gauss theorem. Of course, we do not pretend to criticize that books because they are actually excellent references and we are far from ex-posing a formal proof. However, we expect at least to mo-tivate the reader on the importance of considering sequences of approximating functions [28], to have a more clear idea about the handling of distributions. Such as made in Eq. (25), by considering these kind of sequences, an equivalence re-lation could be established betweenGx,yˆ), defined in Eq. (25), and functiongyˆ(ˆx). SinceGx,yˆ) = gyˆ(ˆx) is valid

almost everywhere with respect to the variableyˆ R3 (or

ˆ

x R3) [26, 27], there is no doubt now of callingg ˆ

yx) as a Green function. In a similar manner, the argumenta-tion in [21] when justifyingfx,yˆ)as a Green function could be improved when considering sequences of smooth approx-imating functions. Just as Jackson explains in his analysis on Poisson and Laplace equations in reference [3], we want to emphasize a very good footnote which refers to the vol-ume integral of Eq. (1.36) in that reference: “The reader may complain that (1.36) has been obtained in an illegal fashion since1/|xx0|is not well-behaved inside the volume V. Rigor can be restored by using a limiting process. . . ” Well, such limiting process has been exemplified in this work.

Appendix A

Remarks:

I) Based on [28], definition of distribution for the general case of functionsh:R3 Cis given as follows: LetDbe

the set ofC∞-functionsφsuch that, φ, with all its deriva-tives vanish at infinity as fast as||xˆ||−N when ||xˆ|| → ∞, and independently of how long is the positive integerN. Any functionφin the setDis said to be particularly well-behaved and such set would correspond to the space of test functions defined in [26]. According to [28], a sequence of functions

{hγ}(considering valuesγ >0) is said to be regular, if and only iflimγ→0+

R3

R3hγφdV exists, for all φ ∈ D. Thus, a distributionHis a regular sequence of functions inDgiven by{hγ}, where symbol

3

Z

R3 HφdV

means

lim

γ→0+

3

Z

R3

hγφdV. (A1)

Of course, this symbol is not a true integral, becauseH is a sequence. Even in the case whenHis interpreted as the limit of this sequence, such limit could not be an ordinary function and that is whyH is declared as a symbolic function [28]. In a more strict sense, a distribution, is a continuous linear functionalTHdefined on the space of functionsD[1, 26], as the symbolic notation

TH(φ) :=

3

Z

R3 HφdV

suggests.

II) In this discussion, we have relaxed the hypothesis by con-sidering D, as the set conformed by functions that are at least ofC1-class and that vanish at infinity in the next weak sense [26]: function h vanishes at infinity, if and only if V[St,h]is finite, for all constantt >0. Here,St,h := {xˆ :

|hx)| > t} andV[St,h]is the volume (or Borel measure) of the setSt,h. Moreover, in this work, a distributionH is understood as the limit of a sequence{hγ}, conformed by el-ements in the setD, our particular set of well-behaved or test functions. However, for practical purposes, our test functions do not require to satisfy more properties, like the regularity condition imposed in [28]. Conceptually, we are assuming that

Hx) := lim

γ→0+x). (A2) Therefore, when an ordinary functionhis such thathx) =

Figure

FIGURE 1. Sketch of the region Ω. In this example, it is assumed as an ovoid with an inside sphere of radius ε.

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