Reference

NATL INST. OF STAND & TECH

NBS

Publications

A111DS 1731^

\

Vau ov *

(0 NBS TECHNICAL NOTE 1157

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

Integral Equations for

Transient Electromagnetic Fields

luo

. U5753

No. 1157
NATIONAL BUREAU OF STANDARDS

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Integral Equations for FEB 2 4 1982

Transient Electromagnetic Fields

, u

no- ))£i

Egon Marx

Center for Manufacturing Engineerinj

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

^

0F oQl

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1982
National Bureau of Standards Technical Note 1157

Nat. Bur. Stand. (U.S.), Tech. Note 1157, 68 pages (Feb 1982)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1982
CONTENTS

Page

1. Introduction 2

2. Maxwell's Equations 5

3. Derivation of Integral Equations for the Fields . 10

h. Scattering by Perfect Conductors IT

5. Conclusions 27

6. Acknowledgements 28

References 29

Appendix A. The Electromagnetic Field Tensor and

Potential Vector 30

Appendix B. Vector Identities and Green's Theorems . 36

Appendix C. Green's Functions for the Scalar Wave

Equation 1+5

Appendix D. Dyadic Green Functions for Transient

Fields 53

in

INTEGRAL EQUATION FOR TRANSIENT ELECTROMAGNETIC FIELDS

Egon Marx

National Engineering Laboratory

National Bureau of Standards

Washington, D C 2023*+

Integral equations for the electric and magnetic fields in

free space are derived from Maxwell's equations. The fields are

expressed in terms of their initial values, boundary values, and

sources with the help of a retarded Green function for the scalar

wave equation. These equations are then used to derive integral

equations for the surface charge and current densities induced by

the scattering of a transient electromagnetic field by perfect

conductors. An alternative solution of Maxwell's equations with

the help of dyadic Green functions is also presented.

Key words: dyadic Green functions; electromagnetic scattering;

integral equations; perfect conductors; transient

electromagnetic fields; wave equations.
1. Introduction

The Wave Opt ics~~ program at the National Bureau of Standards is concerned

with the scattering of light by small objects and rough surfaces. The

most widely used approach to this problem involves (ideally) monochromatic

waves, and results are given or data are taken for many different angles.

We propose to consider the scattering of pulses finite in duration and

spatial extent for one or a few directions. The scattered wave for such a

broadband transient field contains information on the shape and the material

of the scatterer.

Monochromatic electromagnetic waves are represented by complex vector

functions of the space variables for a fixed frequency, and these functions

obey the Helmholtz equation. Although an arbitrary field can be decomposed

into a superposition of monochromatic waves by Fourier analysis, the

equations and solutions are not necessarily equivalent. Thus, we prefer to

derive formal solutions and integral equations directly from Maxwell's

equations, rather than backing into them starting with monochromatic waves.

The expressions we obtain and the method of derivation are not readily

available in the literature, and we present them in this note in considerable

detail for the interested reader who is not already familiar with the subject.

The basic Maxwell equations have to be complemented by the constitutive

relations for the medium or for vacuum, and by initial and boundary conditions.

These aspects of the theory are presented and analyzed in Section 2.

A set of linear partial differential equations can be formally solved

with the help of the appropriate Green functions. When the Green function

with the right boundary conditions can be found, this solution is expressed

in terms of integrals of known functions. Otherwise, the formal solution
in terms of an arbitrary Green function is in fact an integral equation,

which often can serve as a basis for numerical computations.

The numerical solution of equations for monochromatic fields normally

involves the inversion of large matrices. In the case of time-varying

fields, the consequences of causality allow for a stepping-in-time procedure

that does not involve such an operation and can be advantageous under some

circumstances

In Section 3 we derive the formal solutions of Maxwell' s equations in

terms of Green's functions for the scalar wave equation. We start from the

equations for the electromagnetic potentials in a Lorentz gauge, but the

final expressions contain only the fields and the sources. We also conclude

that unknown fields on the boundary cannot be simply eliminated from the

integrals, so that we really arrive at integral equations for the electric

and magnetic fields.

The integral equations for the fields can be reduced to simpler

equations for the surface charge and current densities in the special case

of scattering pulses by perfect conductors. We derive several equations in

Section k, and find that some of these differ from those found elsewhere

(Mittra [l]).* We include a detailed discussion of the limiting process

that takes the observation point to the boundary surface.

A large part of the mathematical derivations is relegated to four

appendices. Appendix A deals with the notation and concepts related to

Maxwell's equations in four-dimensional space-time, as well as a discussion

of potentials. In Appendix B, we list a number of vector identities we

use, and present the derivation of different forms of Green's theorem. In

*Numbers in square brackets indicate the literature references at the

end of this report.
Appendix C, we derive a number of properties of Green's functions for the

scalar wave equation, as well as the explicit form of the free-space retarded

Green function. In Appendix D we develop a solution of Maxwell's equations

by means of dyadic Green functions. This approach allows the expression

of formal solutions in terms of given boundary conditions alone, although

the actual derivation of dyadic Green functions is difficult. In this

Appendix we derive an explicit expression for the free-space dyadic Green

function for the vector wave equation.

The level of mathematical rigor in this presentation is probably

average for the subject. The functions we deal with are assumed to be

sufficiently well-behaved and the surfaces smooth enough so that mathematical

transformations such as those in Gauss's divergence theorem apply. We

recognize that Green's functions are not really well-behaved, and a proper

mathematical context for them would be the theory of distributions, but we

apply the formalism we develop to them regardless, as is customary in this

field. Special care has to be taken when problems appear, as discussed by

Yaghj ian [2] .
2. Maxwell's Equations

Four vector fields are usually defined to describe electromagnetic

-> phenomena; ->•-» they are the electric field intensity E, the displacement vector

+

D, the magnetic induction field B, and the magnetic field intensity H.

They obey Maxwell's equations

V-D = p, (1)

VxE = -3B/3t, (2)

V-B = 0, (3)

VxH = j+3D/3t, (4)

where the sources P and j represent the free charge and current densities.

These equations do not determine the fields unless additional relations are

given, the constitutive relations. They depend on the properties of the

medium, and in their simplest form they are

D = e E, (5) o

B = y QH, (6)

in free space. The same relationships apply to homogeneous, isotropic,

nondispersive media, which are characterized by constant permittivity e

and permeability y. In more general cases, e and y can be functions of

time and position, functions of frequency for monochromatic fields in

dispersive media, or tensors for anisotropic media such as crystals. Even

more complicated relationships are possible, such as those for ferromagnetic
materials where hysteresis occurs for time-varying fields.

For a conducting medium, it is often possible to relate the current

density to the electric field by a simple form of Ohm's law,

J = <>E, (7)

where o is the conductivity; for monochromatic fields the permittivity

then becomes a complex function of the frequency.

We restrict ourselves to the set of Maxwell's equations in free space,

eqs. (l) to (6), but it is only necessary to replace e and p by constants

e and y to generalize the results to simple dielectrics.

We also assume that the medium and the boundaries are at rest. When

this is not the case, Lorentz transformations can often be used to reduce

the problem to a rest frame, or a more general four-dimensional formalism

may be required, as introduced in Appendix A and used by Marx [3].

Actually, only eqs. (2) and {k) are true equations of motion. The

other two Maxwell equations are constraints on the fields and it is sufficient

to satisfy them at an initial time t Q

. They are then satisfied for all

times t, since

(3/3t)(V-D-p) = V- (3D/3t)-3p/9t = V- (VxH-J)-3p/3t

= -V-J-3p/3t = 0, (8)

(3/3t)V«B = V-(3B/3t) = V- (-Vxf) = 0, (9)

where we have used the conservation of charge, eq. (All), a requirement

that the sources have to satisfy for eqs. (l) and (k) to be compatible.
Also, if P is given at the initial time, we can find it at later

times from

p(x,t) = p(x,t

o )-/£ dt'vff(x,t'). (10)

For free space, eqs. (l) and (U) reduce t<

V'E = p/e ,

VxB = y T+y e 3E 3t,

o o o

(11)

(12)

and the wave solutions propagate with speed c given by

c = 1/e y

o o

1 (13)

When we restrict our attention to a re/ion V of space bounded by a

surface S with unit outward normal n, we h^e to give boundary conditions

in addition to the initial conditions to s/ecify the fields uniquely. We

follow Stratton [k] and use eqs. (B9), (2/ and (12) to write

• (ExB) = -£-3B/3t/E H it*3E/3t-vi J«E. (14)

We integrate over V and use eq. (B15)/° obtain

^dSn-ExB = -(l/2)(d/t)/v

dV(B 2+e

o

y o E

2 )-y

o /v

dVj".F:. (15)
We now show that the fields in the region V are uniquely determined by the

-> *-*

current density j , the initial values of E and B (subject to the constraints)

+ -> and either the tangential component of E or the tangential component of B

on S. If the fields are not unique, we could find two solutions that

satisfy Maxwell's equations and the other conditions. Since the equations

are linear, the difference of these two fields would also satisfy Maxwell's

equations with vanishing current density, initial values, and boundary

values. When the difference fields are substituted into eq. (15), the

last term vanishes because j does. If either n*E or n*B vanishes, the

surface integral in this equation is zero and these fields satisfy

(d/dt)j r dV(E 2+e y B

2

) = 0. (16)

•7 o o

Thus, the value of the integral is a constant that has to be zero because

these fields vanish at the initialtime. Since the integrand is non-negative,

both E and B have to be zero and tie two solutions of the original problem must

be the same, hence unique.

There are two simple relations between the fields on the surface S

that can be derived from Maxwell's euations with the help of eq. (B35).

The time derivatives of the normal ccyponents of the fields are given by

3E /3t = ft- [(17 y )VxB-(l/e )J] n o o o

-t = -(l/e

ov)V

s .(flxB)-(l/e

o

)fi-J, (17)

3B /3t = -fl'VxE V "(fixE). (18) n s
The normal component of E can then be found if we know its initial value.

* * the tangential component of B (or the surface divergence of n*B) and the

-v > normal component of j ; and the normal component of B is determined by its

» initial value and the tangential component of E.
3. Derivation of Integral Equations for the Fields

To derive integral equations for E and B, we use the appropriate form

of Green's theorem from Appendix B, apply it to the potentials in a Lorentz

gauge, and select a Green function for the scalar wave equation, as discussed

in Appendix C.

We start with eq. (BUT) and change the variables of integration

to x' and t', collectively designated "by the four-vector x'. We choose

t+e, where e is an arbitrary positive number, as the upper limit of

integration t]_, and substitute

u(x') = A(x'), (19)

v(x') = aG

R

(x,x'), (20)

where A is the vector potential, G^ a retarded Green function that also

> depends on a four-vector x, and a an arbitrary constant vector. Since

-» A obeys the wave equation (A20 ) and Gr satisfies eq. (C8), we obtain

f , -[ ,v +, ,, 9G

R

(X>X,)

,-> 9i(x) -> f MJ i afc')

9G

R

(x » x '>

-{Vxi(x')}xV , G

R (x,x')-{v , 4(x')}v*G

R

(x,x')

o .(a^lMrvv^-^f

10
C 3Gr> (x,x') r „-*, ,. -^dS'x{Vx!(x ')}—

+j dP- Mix_Iv

, GR(x>x .) ,

r 3G (x,x') 1

+4>dS'-^T V'-A(x') , (21)

where the causality condition (C2) implies that there is no contribution to

time integrals for t' between t and t+e, and that the integrand of the

volume integral on the right-hand side vanishes for t'=t+e. Similarly,

we substitute

u(x') = $(x'), (22)

v(x') = G

R

(x,x'), (23)

where $ is the scalar potential, into eq. (B5l) to obtain

V*(x)+--\ dt'V dV'p(x , )V'G

R

(x,x')

o -f «•(

e o3t h o

-, C T 3G„(x,x') „^, , N

"I

, N 3Gfx,x') +£

df[jd* |(V.Hx'))-VOR

(x, X ')- ^

^

R

3"

O

-£dS*.{(V$(x'))V'G

R (x,x')+(V , G

R

(x,x'))V'$(x')} . (24)

We obtain 3A/9t from eq. (2l), using the property (Bl6) to eliminate the

11
arbitrary vector a, and V$ from eq. [2k). We combine them according to

eq. (AlU) to find the field

E(x) =

ft r f

. 3G (x,x')

\ dt'l dV y j(x')—^r-i + p-p(x')V'G (x,x')

Jt Jv o

" _ o

*-

f f ^

3G (x,x') "I -V dV'^ECx')—^p B(x')xV'G

R

(x,x»)J

o

+ \ dt* 4> dS'E(x , )-V l G

R

(x,x')-i dS' ECx^V'G^x.x')

r r 3G (x,x')~|

-<J> dS ' • { V'G

R

(x,x' ) }E(x' )-<b dS • xB(x' )~J

jp , (25)

where we have introduced B from eq. (A15) and used the Lorentz condition (Al8).

We now substitute eqs. (l°) and (20) into eq. (B5T) to find VxA; we obtain

y V dt'l dV'J(x')-{v*G

R

(x,x')}xa - a»Vxl(x)

o

-. r I"*"?-/ in 8GT1 (x,x l ) I = - \z) dVpf^2 »{v'Gr(x> x')}x1- R

9t , ;.V'xftx ')|

f

I dt'todS'

' a£, M 3G (x,x')

^T^-1 "^-^ +{v'xI(x')}x{(VG

R

(x,x'))xl}

{Vx^x^Ja-V'G (x,x , )-{v*4(x')}{VG (x,x')}xa R

(26)

We use the Lorentz condition (Al8), integration by parts and eq. (B2l) to

show that

12
f dV'lv'SCx'^xV'G-Cx.x')

Jv

R

t'=t

dS'x

3G (x,x')

{V$(x')}—^p +c 2 {v'GR(x,x')}v'.A(x')

o

dS'xjV'G^Cx^^l^Cx') +<t dS'x{v'<Kx')}G (x,x' R

t'=t Js R

t'=t

+pfj

dS'x Jv'1|^1|gr

(x,x') + {v'Gr (x,x')}^^

= t dS'xV'{G

R (x,x')$(x')}

i + ( dt'£ dS'xV' G

R (x,x')^ = 0. (27)

o o

We find the field B from eq. (26) according to eq. (A15) ,

(x) = V \ dt'f dV'T(x')xV'G

R

(x,x')

- kS dV E(x')xV'G

R

(x,x , )+B(x')

3G

R

(x,x')

3t ! t'=t

ft r r 9G (x,x') r +\ dt' -^bdS'xE(x')-^p 4>dS'-{v'G

R

(x,x')}B(x')

o L

+t dS'B(x')-V'G

R

(x,x , )-<t dS'-B(x , )V'GR

(x,x') . (28)

Equations (2 5) and (28) give the fields E and B in terms of the sources,

the initial values of the fields, and boundary values. These equations

involve a retarded Green function with unspecified boundary conditions, and

13
they are essentially the integral equations we seek. They are not solutions

of the set of partial differential equations because both fields appear in

the surface integrals, and we have shown in our discussion of the uniqueness

of the solutions that only the tangential component of one of the fields needs

to be specified. To express the surface integrals in terms of the normal

and tangential components of the fields we replace the vector surface

-> element dS' by its magnitude of dS' times the normal n' , as in eq. (B39)»

and rewrite eqs. (2 5) and (28) in the form

c

t

( r »

a G

R

(x> x ')

i (x) = \ dt'V dVy(x')^ + ^-p(x')V'G

R

(x,x')

-H8G

R

(x,x ? ) ^

-^rE(x') —, -B(x')xV'G

R

(x,x')

t'=t

t

dt '£ds ' {n*xE(x')}xV'G

R

(x,x')

3G (x,x')

+n'-E(x')V , G

R

(x,x')+fi'xB(x , )-^p (29)

B(x) u( dt'f dV'jV)xV'G

R

(x,x')

E(x , )xV'G

R

(x,x , )+B(x')-

9t ,

-\ dt'<t dS* {n , xB(x , )}xV , G

R

(x,x')

SGfx.x')

3G (x,x')

+fi'-B(x , )V'G

R

(x,x')- izfl'xECx')—^p (30)

Ik
To see whether we can find actual solutions to the problem we turn to Green

functions such as G4 and G-k , which satisfy homogeneous boundary conditions

on S given by eqs. (ClU) and (C15). We examine the solution of the scalar

wave equation,

DiKx) = a(x), (31)

where the d' Alembertian is defined in eq. (A22) and a is a given source.

From eq. (BU2) we find

i(x) = ( dt'( dV'a(x')G

R

(x,x')

H

3G (x,x') , ,.

i

dt i

3G_.(x,x') ... ,.

o

(32)

In this case, we need to know the source a(x), the initial values of i[>

and 3ijj/3t at time t Q , and either i|> or the normal derivative 3^/3n on the

surface S. If ty is given on S, we choose a Green function GA that vanishes

on S and the surface integral has a known integrand. If 3\J>/3n is given,

we choose G

R

; , whose normal derivative vanishes on S, and again we know the

integrand. There are more complicated boundary conditions that can be

satisfied by \\> and Gtj, but we will not discuss them here. For eqs. (29)

and (30), we find that neither GX ' nor GA ' will eliminate the unknown terms.

*

(1) For instance, if the tangential component of E is given, we might try G^ ' in

15
ecq. (29) and Gp ' in Eq. (30); the surface term contributions then reduce tc

E

g (x) = -/* dt , /sdS , f(x , )3G^1) (x,x, )/3n , > (33)

B

g (x) = -/' dt'^dS'Hn'xBtx'HxVV^^x')

' .» \t7ir>(2) /,, ,,i\_ LftivP/„i\ir(^) +n'-B(x')V , G; y (x,x')- ^n'xE(x')3G; ' (x,x» )/8t ' ] . (34)

s R C2 R

We can find n*B by means of eq. (l8), but we do not know n*E in eq. (33)

or n x B in eq. (3^+). It is not clear at this point whether further

transformations of the surface integrals or other relations derived from

Maxwell's equations would allow a determination of the fields through Green's

functions for the scalar wave equation by integration alone. The lack of

symmetry between source point and field point for the gradient of a Green

function, as seen from eqs. (C22) and (C23), limits the transformations we

can try.

The scalar Green function is used extensively in the case of monochromatic

waves (Helmholtz equation) by Muller [ 5l 5 where uniqueness is also shown

under more general conditions.

Also for the Helmholtz equation, the reduction of a solution to

integrations is accomplished by Tai [6] with the help of dyadic Green

functions. We examine the use of dyadic Green functions for transient

electromagnetic fields in Appendix D.

16
k. Scattering by Perfect Conductors

We now restrict our problem to the scattering of an incident

electromagnetic pulse by one or more perfect conductors.

We use the integral equations (29) and (30) to derive simpler integral

equations for the surface charge and current densities.

The initial time t Qis chosen so that the incident pulse has not

reached any conductors. The volume V is the exterior of the conductors,

and it is bounded by a surface S that has one or more parts just outside

the conductors and is closed by a surface at infinity. The incident fields

have to be defined so that no contributions to the integrals come from the

surface at infinity, and the scattered fields do not reach this surface at a

finite time because they propagate with the speed of light.

We assume that there are no sources in V after the initial time, and

we separate the fields into incident and scattered parts,

E(x) = E in(x)+E

SC

(x), (35)

B(x) = B in

(x)+B

SC

(x). (36)

The incident fields, by definition, propagate as free fields, and are

given by

(°)/„ „tw^i *ln, tWn ,„(o)

E

in

(x) = -/vdV'[(l/c 2

)E'ln (x')3G^o; (x,x')/8t'-Bin

(x , )xV'G^ '

) (x,x')]

t

, =t , (37)

B

in

(x) = -l/c 2 /v

dV'rE

in (x')xV'G^o) (x,x')+B in (x')3G^o) (x,x')/9t']

t

, =t , (38)

IT
from eqs. (25) and (28) ; the fields are expressed in terms of the initial

values and the free-space Green function g4° •

We introduce the surface charge and current densities on the conductors,

P s and Js

, and relate them to the fields through the boundary conditions

fixE(x)|^

eS = 0, (39)

ft - E(x)

lx- e S = p s (x)/ V < 40>

ftxS(x)

'x%S

= Vs(x) ' (41)

A-B(x)|^

s = 0, (42)

where h is now the outward normal of the conductors, that is,

if n' is the outward normal as seen from V,

ft = -ft'. (43)

Charge conservation on the surface reduces to

V «J +3p /dt = 0, (44)

s s s

a consequence of eqs. (12), (HO) and (Ul) when there is no normal

-> ->• component of the current density j and when we use eq. (B35) for B.

-y The integral equations for p s and J s are obtained from the

boundary conditions (39) to (H2) by expressing the fields on the surface

18
in terms of the sources through the integral expressions for the equations.

Since the sources and the initial scattered fields vanish, the integral

equations (29) and (30) reduce to

E(x) = -/vdV[(l/c2 )E

in

(x')3G^0) /3t»-B

in (x')xVG^0)

] tI=t

o +H dt'f.dS'Kl/e )p (x')V'G^

o)

+y J (x')3G^

o) /3t'], (45)

l o OS KOS K

B(x) = -(l/c 2 )/.,dV , [J

ln

(x , )xV , G^° ) +B

in

(x , )3G^o) /3f ] ._

V K is. t — t o

+j\ dt^

s dS'yoJs (x')x V 'G^

o)

. (46)

We recognize the volume integrals in eqs. (U5) and (U6) as E

in (x) and

B

in (x) from eqs. (37) and (38), so that the scattered fields are simply

E SC

(x) = ll dt'^dS'Kl/eJp Q (x')V'G<o)

+y J (x' ) 3G^

o) /3t' ] , (47)

C o OS KOS K

O

B SC

(x) = !l dt'^dS'y J (x')x V 'G^

o)

J

. (48)

t

J b o s R

o

We use the explicit form (C34) of the free-space retarded Green

function and the properties of the delta function to derive

8G

R

(X,X,)

_ 6'(t-t'-R/c)

3t' 4ttR (49)

nt r (o)/, m - 6'(t-t'-R/c)R/c+6(t-t'-R/c): ,, n

. V G

R

(x,x ) - ^7 R, (50)

19
3p (x 1 )

\t dt^

s dS'p

s (x')V'G^(x,x') -^ds|f^ -^- + ^8

( X .) (51)

£

dt iis ' 3G<

o) (x,x') 8J (x')

1 idS' 1 S t'=T

(52)

£

dt

'i

ds '

o

J (x')xV'G< o) (x,x') =

S K 4ttJ

Rx

i

8 Vx,) iRe 3t*

J t'=T

(53)

where

R = (x-x')/ x-x* , (54)

x = t-R/c, (55)

and the integrands vanish for times before t Q

. The expressions (1+7) and

(U8) for the scattered fields thus become

r

c (x) = i( h Tr 9P

s (x,)

, R

4Ttds \T*E at' +

•'S 7?

L o o

y 3J (x')

• T N O S P

S (X } " R- _9V~Jt'=T

(56)

f

SC

(x) = -His* Rx

y 3J (x') y

Re" 3? + ^J

s (x } (57)

We now substitute eqs. (35) and (36) into the boundary conditions (39/

-> to (U2), and we use eqs. (56) and (57) for the scattered fields. When x

•* ->- is on the surface S, the integrands become singular as x' approaches x,

and we have to determine the contribution to the integral from the region

about this singularity. We follow Van Bladel [7] and let x approach the

20
surface field point xQ along the normal. (Poggio and Miller [8] change

> the surface about x , but they fail to take into account all the different

forms that the integrand can take. )

We separate a small part S-j_ of the surface around xQ and we call the

-> -> rest of the surface S. We assume that p s

, 8p s /9t, Js, and 3J s /3t are slowly

varying functions on S]_, and we approximate them by their values at (xQ

,t).

Furthermore, we choose S-]_ small enough so that it can be considered

->- flat, and shaped symmetrically about xQ

. If we write

x-x = hft, h>0, (58)

o

we take the limit h+0 to let x approach the surface.

We decompose the vector x-x 1 into its parts normal and tangential to

the surface,

R = R-nn+nx(Rxft), (59)

and we compute some integrals of singular functions of R. The first one

involves the definition of the solid angle,

/ (R«M/R2 )dS ? = ft/ dfi = fin, (60)

where ft is the solid angle subtended by Sj_ at x. The tangential component of

R does not contribute, since

21
/_ [nx(Rxft)/ R2] dS ' = 0, (61)

S

by the symmetry that makes contributions cancel from elements in S]_ that

are symmetric about x . Thus,

h+0 Sl

A similar decomposition shows that

lim/ (R/R2 )dS' = 2iTft. (62)

lim/Q (R/R)dS' = 0, (63)

h+0 Sl

as would be expected because the integrand is less singular than the previous

one. The integral of the tangential part vanishes again by symmetry, and

the integral of the normal part is bounded by that over a circular region

S2 of radius _a enclosing S-i ; we compute

( ^dS' = fdTdP —2 ^~ = 2irh(a-g§k = irh log 5^i, (64)

k, R

Jo JO JP3F vb^h^ J0P +h e h

which vanishes when h tends to zero. On the other hand, the integral in

L = limL (l/R)dS' (65)

h+0 Sl

22
has a finite limit that depends on the size and shape of S-j_. For a circular

region of radius a,

f ^- = lX -2&- = 2^ (v4r^a^"-h) , (66)

Js 2 R )o/h^p^

W

lim/_ (l/R)dS* = 2wa « 6.28a, (67) S2

while for a square of side 2a we find

•tt/4 ra sec<j) rTr/4

^f-=8\ d A -====8\ d*(/ha+a 2 sec 2 «D-h), (68)

s 3 R

Jo Jo /h^F Jo

lim / (l/R)dS' = 8ajT/4dcj>sec4> = 8a log (vf+l) - 7.05a. (69)

h^O

b3 u e

If a region S-j_ is completely inside a circle of radius a-|_ and if a circle

of radius a2 is inside S-j_, the value of the integral over S]_ is bounded by

2i\a 1 > L > 2TTa 2. (70)

We can thus evaluate the integrals over S]_ for the scattered fields in eqs.

(56) and (57) when the field point lies also on the surface S.

From the boundary condition (Ul), we obtain

23
o s nx^Cx)

x"eS 4 " Ds

Rx

1 "s (x,)

1 +

'

7TZ ~ TT» + ^TJ

S (x 1 ) R

z c 3t'

9J (x)

xeS

(71)

and substitute the integrals in eqs. (62) and (63) to find the integral equation

J

s (x) ^nxfln

(x ) M o X«S

2?

J;

dS'Rx

R^c

xtS

(72)

Some authors [l,8] define principal value integrals by considering the

limiting case of a region S when the excluded part S]_ tends to zero in

area, presumably in a symmetric manner. The integral in eq. (62) is

independent of Si , but that in eq. (65) tends to zero, as shown by the

inequality (TO). Thus eq. (72) retains the same form when the integral

over S is replaced by a "principal value" integral, and in this context the

integral over a small region S-j_ about the observation point vanishes.

Since the integral in eq. (72) involves values of J s and 9Js /3t at times

earlier than t, a judicious choice of the time steps (small enough not to

allow the electromagnetic fields to propagate from the center of one patch

to the neighboring ones ) allows us to compute the current density by a

time-stepping procedure that involves no matrix inversion [l,8]. The

integral equation (72) is derived from the expression of the scattered

magnetic field, and it is known as a "Magnetic Field Integral Equation"

(MFIE).

Another equation, the "Electric Field Integral Equation" (EFIE), can be

derived from the boundary condition (39) and the expression for the scattered

2U
electric field (56). We have

= fixE ln

(x)

: f S+ ^1)s

dS

To

-^P (x') R

z c 3t' e R

or s o %

8 J (x*)~

s R at'

p.W r..,t

3J

s (x »fds-j

xeS

(73)

The contribution from the first integral over Sj_ vanishes by eq. (63). The

second one is given by eq. (62), and the vector product vanishes. The

third one is expressed by eq. (65), and it vanishes when a principal value

integral is considered. When a small finite region around the observation

point is excluded, the integral equation takes the form

8J (x)

s9t —r-nx (n*E (x)

y oL

xeS'

. 1 fixIdS 1 1 3J (x')

1 s3t' Jt*=T x«S

(74)

which does not have precisely the form given by Mittra [l].

The surface charge density can be obtained from the divergence of the

surface current density from eq. (UU), which expresses conservation of

charge. Also an integral equation can be obtained from eq. (Uo), and,

proceeding as above, we find

25
P s (x) 2e ft-E in

(x)

o xeS

T-A-V dS'

t

8p

s (x,) r

z c at' ,L ; , i

3 VX,) + Fp

s (x )_ R^ 3t'

xeS

(75)

The remaining boundary condition, eq. (U2), leads tc

= fi.B

in

(x) -f^-fds' Rx

y 3J (x 1 ) y + R^J

S (X } R^c 9t'

xeS

(76)

which does not show a contribution from the region S]_. There is no obvious

contradiction, as would have been the case for the equation derived from

the boundary condition (39) 3 because B

in should be tangential to the surface

when the incident pulse reaches for the first time a region on the surface,

so that neither term in eq. (76) differs from zero.

Which equations are best suited to carry out a numerical calculation

depends to some extent on the shape of the scatterer. Others could also be

used as a check on a solution obtained numerically.

26
5. Conclusions

In this note we have presented a possibly original derivation of integral

equations that relate time-dependent electromagnetic fields to their sources,

initial values, and boundary values. The integral equations (25) and (28)

involve an arbitrary retarded Green function for the scalar wave equation and

its derivatives, but no derivatives of the fields appear. Integrals over unknown

boundary values do not disappear completely if the scalar Green function is made

to vanish or to have a zero normal derivative on the boundary of the spatial

region. To obtain a formal solution of Maxwell's equations, we use dyadic

Green functions in a formulation shown in an appendix.

We applied eqs. (25) and (28) to the problem of a pulse scattered by a

perfect conductor. We obtained simpler integral equations for the surface change

and current densities, mainly the MFIE, eq. (72), and the EFIE, eq. ilk). The

derivation shows in detail what are the contributions to the fields from

sources on a small region of the surface about the observation point. The

EFIE has a form different from what is found in the literature. Once the

induced sources are found, the scattered fields can be obtained by

integration.

Although time-dependent fields can be obtained from monochromatic fields by

a Fourier integral, the integral equations in the space-time domain can be

advantageous in some cases because causality allows for a numerical solution by

a time-step procedure that does not require the inversion of big matrices.

27
6. Acknowledgements

I thank Drs. George Lieberman and Daniel Maystre for many helpful

discussions.

28
References

Mittra, Raj , Integral Equation Methods for Transient Scattering, in

Transient Electromagnetic Fields, edited by L. B. Felsen (Springer-Verlag,

Berlin, 1976).

Yaghjian, Arthur D. , A Direct Approach to the Derivation of Electric

Dyadic Green's Functions, NBS Technical Note 1000 (December 1977); Electric

Dyadic Green's Functions in the Source Region, Proceedings of the IEEE 68 ,

2U8 (1980).

Marx, Egon, Covariant Formulation of Faraday's Law, Journal of the

Franklin Institute _300, 353 (197 5).

Stratton, Julius A., Electromagnetic Theory, p. h86 (McGraw-Hill,

New York, I9U1).

Muller, Claus, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).

Tai, Chen-To, Dyadic Green's Functions in Electromagnetic Theory (intext,

Sc rant on, 1971).

Van Bladel, J., Electromagnetic Fields, p. 392 (McGraw-Hill, New York,

196U).

Poggio, A. J. and Miller, E. D. , Solutions of Three-Dimensional Scattering

Problems, in Computer Techniques for Electromagnetics, edited by Raj Mittra,

p. 162 (Pergamon, Oxford, 1973).

29
Appendix A

The Electromagnetic Field Tensor and Potential Vector

In this appendix we recall the basic equations of electromagnetic theory.

We emphasize the underlying unity of space and time, as well as that of

the electric and magnetic fields, "brought about by the special theory of

relativity. We also recall how potentials are introduced and some of the

properties related to gauge invariance.

The discovery that Maxwell's equations were invariant not under Galilean

transformations but under Lorentz transformations led to new concepts

that are basic to the special theory of relativity. Space and time variables

are combined into a single four-dimensional space with an indefinite metric.

A point in that space is an event , and the coordinates form a four-vector

x = (x ) = (ct,x), y = 0,1,2,3, (Al)

where c is the speed of light in vacuum. The Lorentz metric is

(gyv

) = (g^) = diag(l, -1,-1,-1), (A2)

although the opposite signs can also be chosen. This metric tensor can be used

to raise or lower indices of vectors and other tensors; for vectors, this

operation changes the sign of the spatial components. Derivatives of a field

f (x) are indicated by

(3 f) = (f ) = (3f/3xy

) = (c

1

3f/3t,-Vf). (A3)

y > y

30
We use the modified summation convention for repeated lower Greek indices

a b = a b -a«b. (A4)

y y o o

The electric field E and the magnetic flux density B can be combined

into a second rank antisymmetric tensor

, Ei E 2 E 3

-Ei cB 3 -cB 2 <V =

\ -E2 -cB 3 cBi J> (A5)

-E 3 cB 2 -cBi

and the charge and current densities form the four-vector

(j y

) = (cp,I). (A6)

Maxwell's equations can then be written in the form

E = y cj , (A7)

yv,v o y

F

D =0, (A8)

yv,v

where, in terms of the completely antisymmetric Levi-Civita tensor, the

tensor dual to F,, vis given by

31
F

yv 2

£ yvaB

F

a3 (A9)

(the result of this operation is to replace E by cB and cB by -E )

Since the product of a symmetric tensor and an antisymmetric tensor vanishes

identically, eq. (AT) leads to

d 3 F = u cj =0, (A10)

y v yv o

J y,y v/

which translates into conservation of charge in its differential form,

3p/3t+V«J = 0. (All)

The homogeneous Maxwell equations (A8) imply that the fields can be

derived from potentials Ay, that is,

F = c(A -A ). (A12)

yv y,v v,y

In terms of the usual scalar and vector potentials.

(A ) = (§/c,A~), (A13)

E = -V$-3A/3t, (A14)

B = V*A. (A15)

32
These potentials are to some extent arbitrary, and they can be changed by

gauge transformations

A'(x) = A (x)+A (x), (A16)

where A is an arbitary function of x and t, without changing the fields Fuv .

The relations (A12) imply that the homogeneous Maxwell equations (A8)

are automatically satisfied, and we substitute the fields in (AT) to obtain

the equations for the potentials,

A -A = u j . (A17)

y,vv v,vy o y

Given a set of potentials, it is possible to choose A so that certain

additional conditions are satisfied. These conditions can also be imposed

from the beginning to simplify the equations that have to be solved. A

Lorentz gauge is characterized by the covariant Lorentz condition,

A = V-A+(l/c2 )8$/3t = 0, (A18)

Jl.V

and eqs. (AIT) reduce to

3 2A -jij , (A19)

]i o

J y'

or

33
DA = y Qj\ (A20)

0$ = p/e , (A21)

where the d' Alembertian operator is

D = 3 2 = (l/c 2 )3 2 /3t 2 -V 2

. (A22)

In Cartesian coordinates, all components of A„ obey the scalar wave

equation, which permits us to find a solution by means of the Green function

for that equation.

Another gauge that is widely used is the Coulomb gauge, characterized by

V«A = 0. (A23)

In a Coulomb gauge, the scalar potential obeys Poisson's equation,

V

2 $ = -p/e , (A24)

o

and the equation for the vector potential is

DA = y J-(l/c 2 )V9<I>/3t. (A25)

When the current density is defined for all space, it can be decomposed

into longitudinal (ir rotational) and transverse (solenoidal) parts by setting

3U
j\(x,t) = -v/dV'lV'-T(x' ,t)]/(4Tr|x-x' |), (A26)

Jt (x,t) = VxfVx/dV , [J(x , ,t)/(47T|x-x r |)J. (A27)

From the solution of eq. (A21.

(x,t) = (l/4ire )/dV*p(x\t)/|x-x'|, (A28)

and conservation of charge (All) we show that

V9$/9t = T£ /e

Q

, (A29)

so that eq. (A2 5) becomes

DA = U

oJt

. (A30)

The Coulomb gauge has the advantage that there are essentially only two

components of the potential that obey an equation of motion. The longitudinal

part of the vector potential and part of the scalar potential change under

gauge transformations; they vanish in the Coulomb gauge, leaving only

gauge independent quantities.

A disadvantage of the Coulomb gauge is the nonlocal nature of the

expressions (A2l) and (A22). Thus, if we have a surface current, the

transverse part of that current is no longer confined to the surface.

Similarly, the transverse part of the current density is not obtainable

from eq. (A22) when the current is not known everywhere in space.

35
Appendix B

Vector Identities and Green's Theorems

In this appendix we collect or derive vector identities and the

generalized versions of Green's theorem used in this note.

We have

V(u+v) = Vu+Vv, (Bl)

V(uv) = vVu+uVv, (B2)

V'(u+v) = V'u+V'v, (B3)

Vx(u+v) = Vxu+Vxv, (B4)

-»- V'(uv) = vVu+uV'v, (B5)

Vx(uv) = (Vu)xv+uVxv, (B6)

V(u-v) = (Vu)«v+(Vv)-u = vVu+u«Vv+vx(Vxu)+ux(Vxv) , (B7)

V* (uv) = (V'u)v+u«Vv, (B8)

V'(uxv) = (Vxu)«v-u« (Vxv)

,

(B9)

V(uxv) = (Vu)xv-(Vv)xu, (BIO)

Vx(uxv) = uV • v+v • Vu-vV • u-u • Vv

,

K&1.L)

VxVu = 0, (B12)

VVxu = 0, (B13)

Vx(Vxu) = VV«u-V 2u. (B14)

36
There is a mild controversy about this last identity: some authors consider

it the definition of the Laplacian of a vector field. When the Laplacian of

a vector function is defined as the divergence of the gradient dyadic, eq.

(BlU) is indeed an identity. The forms of the Laplacians of scalar and

vector fields differ in curvilinear coordinates.

Gauss's divergence theorem states that

/ dVV-u(x) =

f dS-u(x), (B15)

where the volume V is bounded by the closed surface S and the vector surface

element points out of V. The conditions under which this theorem is valid

depend to some extent on the mathematical context and the definitions of

integral and differential operators, and we do not specify them here.

When a is a constant but arbitrary vector, we have

a»u = => u = 0. (B16)

If we set successively

u(x) = av(x), (B17)

u(x) = axv(x)

(B18)

u(x) = a«v 1 (x)v2 (x)+a'V2 (x)v 1 (x) (B19)

in eq. (B15), then the property (Bl6) allows us to derive

37
/v

dVVv(x) =

f dSv(x), (B20)

7 dVVxv(x) ,dSxv(x), (B21)

/..dVV-[v 1 (x)v2 (x)+v2 (x)v 1 (x)] = |dS-[v1 (x)v2 (x)+v2 (x)v 1 (x)]. (B22)

Sometimes scalar or vector, fields are defined only on a surface S. A

surface gradient operator V s can be defined for such a field. If the surface

is given by parametric equations

x = x(c,n), (B23)

the surface gradient is given by

V =

->3 3

(B24)

where, in terms of the unit normal n,

-> 3x a a = — xn

3n

9x _3x

3£

X 3n

-1

(B25)

t - 3x b- nx- 3x 3x

3?

X 3n

(B26)

This operator can be used to define the surface gradient, divergence, and

rotation or curl. Some useful relations are

38
ft'V u = 0, (B27)

V xfi = 0, (B28)

where we use the definition (B2U) and

fi 2 = 1, (B29)

n«3n/3C = fi-3n/3n = 0, (B30)

n«3x/35 = n«3x/3n = 0, (B31)

3x m dh _ _3x

m _3fi _ 3 zx „ rTn'n

35 3n 3n " 3? " 3£3n

* n ' { 5

'

to prove such relations. Eqs. (Bl) through (BIO) are valid when the operator V

is replaced by V

s

.

When the fields are also defined off the surface, we combine the surface

gradient with the normal derivative in

V = V +ft3/3n, (B33)

s

and we have

fixVu = fixV u = -V x(ftu), (B34)

s s

n-Vxu = ft.y xu = -V • (ftxu)

,

(B35)

s s

n-V x(v u) = 0. (B36)

s s

39
Green's theorem is derived from the divergence theorem (B15) when we let

u(x) = v 1 (x)Vv2 (x)-v2 (x)Vv 1 (x); (B37)

we find

/vdV[v 1 (x)V 2v2 (x)-v 2 (x)V 2 v 1 (x)] =^ s dS[v 1 (x)3v2 (x)/3n-v2 (x)3v 1 (x)/3n], (B38)

where we have set

dS = fidS, (B39)

3v./3n = n-Vv., i = 1,2. (B40)

This theorem is used to obtain solutions of the Laplace equation by

means of Green's functions. Other partial differential equations can be

solved in a similar manner with the appropriate Green functions and

generalizations of Green's theorem.

In what follows we assume that the fields u, v, u, and v are also

functions of the time t, that is, they are functions of the four-vector x.

We derive the forms of Green's theorem that give integral equations for the

fields when applied to the potentials in a Lorentz gauge. We start from

expressions in which the d' Alembertian operator is applied to scalars or

vectors, and use the identities (Bl) through (BlU) to obtain terms that can

be partially integrated. The simplest generalization of Green's theorem to

wave equations is obtained from

1*0
udv-vdu = (l/c 2 )(3/3t)(u3v/3t-v3u/3t)-V« (uVv-vVu). (B41)

We integrate each side of this equation over a time interval and over a

volume V bounded by a surface S, which may be multiply connected. We carry

out an integration over t and use Gauss's theorem (B15) to obtain

£dtj«(.TM) - i^d»[«|= -^-j^dS (^ -v|H). (B42,

o 00

More complicated forms of Green's theorem required in the text are obtained

from

/

~*"\ Jvby ,», A+v 3uOU 1X 3d //3udU 3v\ dV \ ,„•)-*-. ,,-,9-K 3vd /r,,-,?-*\,?^\ 3udU /tw->\ (Ou). - + (Ov). — ..^—f— ' d-(V2u)-^-(vM- ^, (B43)

V* [ux(Vxv)+uV«v] = (Vxu)'Vxv+(V«u)V«v+u«V2v, (B44)

(3/3t)[(Vxu)'Vxv+(V«u)V«vJ = V* [ (3u/3t)x (Vxv)+(3u/3t) V«v]

-(3u/3t)«V 2v+V-I(3v/3t)x(Vxu)+(3v/3t)V-u]-(3v/3t)-V2u, (B45)

where the last equation is obtained by using (BUU) once with u replaced by

3u/3t and once with v replaced by 3v/3t. Replacing the last two terms in eq.

(B43) by an expression obtained from (BU5), we find

kl
(Qu). ^+(Dv). — _3_

3t

-V1 3u 3v

c

z 3t 3t

+(Vxu)«Vxv+(V« ^ -1 u)V«v

|| x( Vxv> || x(vx^) + ||v4+ ||v4 dt dt dt 3t (B46)

pdtj dv (ou>[|f + (ov> |f] = j dvP^lf • ll+cvx^.vx^cv.^v.^]'

1

o o

-(^dttdsT]! x(Vx^)+|| x( V

K 3u -* 3v -*• H x(Vxu)+ ^V-v+ -^V-u

dt dt (B47)

Similarly, we have

(Du)Vv+(Dv)Vu = ^r |I(|^Vv+ |^Vu) - ^v(|^ |^)- (V 2u) Vv-(V2v) Vu, (B48)

V- [(Vu)Vv+(Vv)VuJ-V[(Vu)«Vv] = (V 2 u) Vv+(V2v)Vu, (B49)

(Ou)Vv+(Ov)Vu = \|r(|7Vv+ |^Vu

C dt \dt dt (Vu)-Vv- ^z|^||]-V-[(Vu)Vv+(Vv)Vu] (B50)

\ dtl dV[(Du)Vv+(Dv)Vu] = ^M dV |^Vv+ jk

o o

+ 1 dt li dS J(Vu)-Vv- ^z -jp y-rV dS«{(Vu)Vv+(Vv)Vu} (B51)

where we have used eqs. (B20) and (B22 ) to obtain the surface integrals. Also

(qu) • Vxv- (qv) • Vxu = (l/c 2 )3/3t(3u/3t«Vxv-3v/3t«Vxu)

-K >

-(l/c 2 )(3u/3t«Vx3v/3t-3v/3fVx3u/3t)-(V2u)-Vxv+(V2v)-Vxu , (B52)

\2
V- (3u/3tx9v/9t) = 3v/3f Vx3u/3t-3u/3f Vx3v/3t, (B53)

-v . -K V» [(Vxu)x(Vxv)] = [Vx(Vxu )]'Vxv-(Vxu)«Vx(Vxv), (B54)

V* [(V-u)Vxv-(V'v)Vxu] = (Vxv)'VV«u-(Vxu)'VV«v, (B55)

-K -*•„ -K -> (au)-Vxv-(Dv)'Vxu = (l/c 2 )3/3t(3u/3f Vxv-3v/3f Vxu)

+V; [(l/c2 )3u/3tx3v/3t+(Vxu)x(Vxv)+(Vxu)V-v-(Vxv)V-u] (B56)

\ dtl dV[(au)-Vxv-(Dv)-Vxu] = K\ dV^

-Vxv- p- • Vxu

Jt h c Jv L

9t 9t J t o o

I dtl dS- \^z 1^- + x |X +(Vxu)x(Vxv)+(Vxu)V-v-(Vxv)V«u . (B57)

Jt o

Another form of Green's theorem can be derived by starting from

?* !?» - £& !? £ + TM• "] -'• (If'- 1?'-) • < B58 >

whence

C

d idv (^°v+ £°u

) = J,

dv

fc *

* +<vu H[

l -\yidH^v+ »• (b59)

o o

We add some relationships that involve a slightly different form of the

vector wave equation, which can be used with dyadic Green functions. We

recall that, from eq. (B9),

U3
whence

V«[ux(V*v)] = (Vxu)'Vxv-U'Vx(Vxv), (B60)

V* [ux(Vxv)-vx(Vxu )] = v«Vx(Vxu)-U'Vx(Vxv)

,

(B61)

[ (1/c 2

)

2 u/9t 2+Vx(Vxu) ] -v-[ (1/c 2

)

2v/9t 2+Vx(Vxv) ] -u

= (1/c 2

) (3/3t) (v* 3u/3t-u» 3v/3t)+V« [ux (Vxv)-vx (Vxu) ]

,

(B62)

pdtf dvfW |§ +Vx( V

x^)J 4-K ||| +Vx(Vx^)U

o

dS« [ux(Vxv)-vx(Vxu)] (B63)

kh
Appendix C

Green's Functions for the Scalar Wave Equation

The Green functions for the scalar wave equation are functions of two

four-vectors, x and x' , that satisfy

OG(x,x') = 6 (A) (x-x'), (CI)

where the source is a Dirac delta function in four variables. To specify

the solution further, appropriate boundary and initial conditions have to

be added. Two widely used classes of Green's functions are the retarded

and advanced ones, which satisfy

G

R

(x,x') = 0, t<t', (C2)

G (x,x*) = 0, t>t*. (C3)

If the region of space is bounded by a surface S, the most common

boundary conditions on G are that either G itself or its normal derivative

vanishes when the field point is on S. Other initial and boundary conditions

are also possible, but we restrict ourselves to the ones above.

In the form (B^2) of Green's theorem, we use x", t" as variables of

integration and set

u(x") = G

R

(x",x), (C4)

v(xM

) = G

A

(x",x'). (C5)

U5
We let tQ*-00 and obtain

i>idv, i

i>ids"

G

R

(x",x)6 (4) (x"-x')-GA

(x",x')6 (4) (x"-x)

9G ,(x",x') 9G (x",x)

dV Gr(x.., x)-A_ R___^

a(x,, x

, ) t"=-oo

9G (x",x') 9G (x",x)

G

R (x",x)-A^n - V G

A

(,",x') (C6)

We choose t-]_ greater than both t and t' ; the volume integral on the

right-hand side vanishes because the integrand is zero at the upper limit

since G^ satisfies eq. (C3), and also at the lower limit since Gr satisfies

eq. (C2). Furthermore, the surface integral vanishes because either G^ and

G^ vanish when x" is on S , or 9GR

/9n" and 9GA/9n" vanish on the surface.

Thus, we find that

G

R (x\x) = GA

(x,x T ), (C7)

and, consequently,

D'G(x,x') = 6 (4) (x-x') (C8)

If we replace GA

in eq. (C5) by Gr(x" ,-t" ;x' ,-t ' ) , we prove in the same

way the reciprocity relation for Gr (or GA ),

k6
G

R

(x',t';x,t) = G

R (x,-t;x',-t') (C9)

When we proceed in a similar manner with eq. (B59) , we obtain

J-oa J\

.->.. .. "> ,-».. .. ->. dV"

3G (x",t";x,t) 3G (x",-t";x',-t') ,,.

' (4) (x"-x')+

—^

—, 6 (4) (x"-x)

3t" 3t"

f,

dV

T 3G

R

(x",t";x,t) 3G

R (x",-t";x\-t')

3t M ~ 3t M

+{v"G

R (x",t";x,t)}.V"GR

(x",-t";x',-t')

t"-ti

t"=-co

-n

dt

"i

as

" 3GD (x",t";x,t) 3GD (x",-t";x',-t')

3t" 3n'

+

3G

R

(x",t";x,t) 3G

R (x",-t";x , ,-t')

3nM (CIO)

and the same arguments show that the right-hand side vanishes. We then have

3Gp(x' ,t';x,t) 3G (x,-t;x"' ,-t') + - = 0, at' 3t (Cll)

whence, from the reciprocity relation,

3GR

(x,t;x\t')

St

3G

R

(x,t;x*,t')

3V (C12)

To look for a symmetry for the spatial derivative, we start from eq,

(B51) and find

47
( V'f dV" [v"G

R

(x",t";x,t)6 (4) (x,, -x

, )+V"GR

(x,, ,-t,I ;x',-t')6 (4)

( x

"-x)]

'"GB (x",t";x,t)—

9Gfx",-t";x',-t')

9GR (x",t";x,t) +

K

3t „ V"GR (x",-t";x\-t')

t"-ti

t" =-oo

( V'f dS" n" (v^G

R

(x", t ";x,t)).V^G

R

(x",-t";x',-t')

9G

R

(x",t";x,t) 9G

R

(x",-t";x',-t')

± 9GR

(x", t";x,t) 9G

R (x",-t";x\-t'

)

9n" 9n"

"^

9t" 9t"

9GR (x*\t";x,t) ^ ^

9G_.(x",-t";x\-t')

9n" ?R(x"'- t

" ;X ''- t,) --i

9^ V'G

R

(x",t";x,t) • (C13)

We now distinguish between GA , which satisfies

G

R

1} (x,x')

xeS

= 0. (C14)

and GA ' , for which

9G

R

2) (x,x')/9n

x«S

= 0. (C15)

Eq. (ClU) also implies that

V.eJ" («.«•)

xeS

0, (C16)

U8
ac^x.xM/at = o. (C17)

xtS

The reciprocity relation (C9) implies that similar relations are satisfied

vhen the source point x' instead of the field point x is on the surface.

We have

G

R

(x,x ) x't S

- o, (C18)

dG^2) (x,x')/dn' = 0,

x'eS

(C19)

VjG^Cx,*')

x'eS

= 0, (C20)

SG^Cx.x')/^' = 0. x'eS

(C21)

When we substitute eqs. (ClM , (Cl6), and (CI?) or eq. (C15) in eq.

(C13 ) , we obtain

V'G^1) (x',t';x,t)+VG^1) (x,-t;x',-t')

dt-4 ds"-^ -J>i 3n" 3n"

(C22)

h9
V'g£2) (x» ,t' ;x,t)+VG^2) (x,-t;x' ,-tV)

C

dc"£ds" {v"G^ 2) (x",t";x,t)}-V'G^ 2) (x",-t";x',-t')

± 8G^2) (x",t";x,t) 3G^2) (x",-t";x',-t')'

(C23)

3t" 3t"

and no symmetry can be shown in either case.

The free-space Green function Gp° has translational invariance, and

consequently it is a function of x-x' alone. In that case,

Vg£0) (x,x') = -V'G^o) (x,x*). (C24)

This property also means that it is sufficient to evaluate the free

space Green function for x'=0. If we express that function in terms of its

Fourier transform,

G(x,t) = (2^)" 2 /d 3 kdwe" i(wt" k * x)

G(£,03), (C25)

eq. (Cl) leads to

G(k,co) = - -jTrrj ——

(C26)

To perform the integration over u, we have to specify the behavior of the

path at the poles at ±|k|c, which is linked to causality. When t<0, the

contour can be closed around the upper half-plane with no contribution from

50
the semicircle of infinite radius. For the retarded Green function, the

condition (C2) implies that we have to deform the contour so that it lies

above the poles. For t>0, we close the contour around the lower half-plane,

and compute the residues at the poles to obtain

_o -it -* GB (x,t) = c(2tt)

J /d 3 k[sin(kct)/k]e1 X

, (C27)

R

where k is the magnitude of k.

To do the angular integrations in momentum space, we choose a spherical

coordinate system with the polar axis along x. For an arbitrary function

of k, we find

_-> ->- /d 3 kf(k)e±ik ' X

= (Wr)Dcdkf(k)sin(kr), (C28)

where

r = Ixl. (C29)

In particular.

G^Cx.t) = (c/27r 2 r)/^dksin(kr)sin(kct). (C30)

R U

The integrand is an even function of k, so that we can extend the range of

integration to negative values of k and take one half of the result. In

terms of exponential functions, we have

51
G-(tt) = -(c/16Tr2 r) /^dk{ei(ct+r)k

-e i(ct- r)k

-ei(ct -r)k+e- i(ct+r)k

}, (C31) R

where the integrals on the right represent Dirac delta functions. Since

t is positive and r is non-negative, ct+r cannot be zero and only two terms

contribute to

G

R

(x,t) = (l/4Trr)6(t-r/c), (C32)

which is also valid for t<0 as the delta function vanishes. If we set

R = x-x\ R = Ix-x 1 I, (C33)

the retarded Green function for the scalar wave equation in free space is

G<°>(x,x-) - {(t^-R/c). (C34)

Similarly, we obtain the advanced Green function if we deform the contour

to pass below the poles. The result is

G

(o) (x x .) = 6(t-t'+R/c)

( 35)

and the symmetry properties follow from those of the delta function.

52
Appendix D

Dyadic Green Functions for Transient Fields

We have seen in the main text of this note that the use of Green's functions

for the scalar wave equations has some limitations when we apply them to finite

regions of space, which give rise to boundary conditions.

The natural operator that relates what are essentially vector fields to

vector sources is a dyadic Green function. These functions have been studied

extensively for monochromatic fields that satisfy the vector Helmholtz equations;

see, for instace, a book by Tai [6] and an article by Yaghjian [2].

In this appendix, we develop the corresponding basic theory for transient

fields that obey Maxwell's equations in free space. We first show the equivalance

of Maxwell's equations and the vector wave equation, we then proceed to derive

some properties of the dyadic Green functions and we use them to find the solutions

for the fields in a finite region of space. Finally, we use the method presented

in Appendix C for the scalar Green function to find the free-space dyadic Green

function.

Our original problem is to solve Maxwell's equations with given sources,

V-E = p/e , (Dl)

o

VxE = -3B/3t, (D2)

V-B = 0, (D3)

VxB = jj j+y e 9E/3t. (D4)

o

J o o

53
+ -* Also given are the initial values of E and B subject to the constraints

>

(Dl) and (D3), and either the tangential component of E or the tangential

-> component of B on the boundary S.

Taking the curl of each side of eq. (D2) and using eq. ( DU )

, we obtain

-» a differential equation for E alone,

(l/c2 )3 2 E/3t 2+Vx(VxE) = -y 3j/3t. (D5)

Thus, any solution of Maxwell's equation is a solution of eq. (D5). To

> solve eq. (D5), we must know the initial value of 9E/9t in addition to

> that of E; we find it by using eq. (DU) to determine

3E(x,t )/3t = c

2 VxB(x,t )-j(x,t )/e (D6)

o o o o

-* -* in terms of the initial values of B and j , both assumed to be known. If the

> tangential component of B is given, eq. (D2) shows that we know the tangential

component of Vxg.

-> Furthermore, once the field

->-»

E is known, the field B can

be found from its initial value and its time derivative defined by eq. (D2).

->• -> To show that the field E that satisfies eq. (D5) and the field B defined

by eq. (D2) satisfy eq. (DU) not only initially by eq. (D6) but at all

times, it is sufficient to show that the time derivative of the combination

of these terms also vanishes. We have

(3/3t)(3E/3t-c 2 VxB+j'/e ) = 3 2 E/3t 2 -c 2 Vx3B/3t+(l/e

o

) 3j"/3t

= c

2 [(l/c2 )3 2 E/3t 2+Vx(VxE)+y

o

3J"/3t] = 0, (D7)

5k
where we used eq. (D2), now part of the definition of B, and the equation

of motion (D5). The constraints (Dl) and (D3) are then also satisfied for

all times, as shown in Section 2. Equation (D5) is thus equivalent to the

set of Maxwell's equations when eq. (D3) is used to define B, and eq. (D6)

>

is used to obtain the initial value of 3E/3t.

We demand that the dyadic Green function obey

(i/c2 )3 2'G(x,x , )/3t 2+vx[vx<S'(x,x*)] = 6 (4) (x-x')T, (D8)

where I is the unit 3

X 3 dyadic, and we further define retarded and advanced

Green functions by setting

G^(x,x') = 0, t<t', (D9)

G (x,x') = 0, t>t'

.

(D10)

We first demonstrate some symmetry properties of these Green functions.

In the form (B63) of Green's theorem, we use x" and t" as variables of

integration and set

u(x") = ?

A

(x",x).a, (Dll)

v(x") = (LCx'V^b, (D12)

where a and b are two arbitrary constant vectors. We proceed as in Appendix

55
C, and find

= ±z[ dV"

c Jv

+[ dtn 4 ds"

J-oo Js

dV"[{G1,(x",x , )-b}46 (4) (x"-x)-{G A (x",x)-a}.^6 (4) (x"-x')]

K A

9G .(x",x)l (3G_(x",x f ) {GR

(x",x').£}- ]-~^ -a -{G

A

(x",x).l}. -^

b

[{GA (x",x)4}x{V"xG

R

(x",x , )}-b

t"=ti

t"=-oo

-{GR(xM ,x

, )-b

V}x{V"xfA

(x»,x)}-l] (D13)

The volume integrals vanish because the integrands vanish at both limits

due to eqs. (D9) and (DIO), since t]_ is greater than both t' and t". The

vanishing of the surface integral depends on the boundary conditions imposed

on the Green functions. Two of the possible choices are [6]

n-G <1) (x,x')

n-VxG (2) (x,x*)

2«S

= °'

xeS 0,

(D14)

(D15)

and in either case the surface integral vanishes. Thus, the left-hand side

of eq. (D13) is also zero, that is,

a-G

R

(x,x')-b = b-G

A

(x',x)-a, (D16)

and, since both a and b are arbitrary,

56
#R

(x,x') = ^A

(x',x) (D17)

where the tilde indicates the transpose of the dyadic GA

. Thus, if GA

(x',x)

satisfies eq. (D8) with respect to the variables x/, , we have

(l/c 2 )3 2^(x,x , )/3t ,2+V , x[v'xG D (x,x')] = 6 (4) (x-x' )f. (D18)

Similarly, we set

u(x") = ^(x",-t";x,-t).l (D19)

instead of the field given in eq. (Dll), and we show the reciprocity relation

G

R

(x,t;x',t') = G

R (x',-t';x,-t) (D20)

This relation and eqs. (DlM and (D15) then imply that, when the source

point is on the surface

,

ft'.f^Cx.x')

X «S

(D21)

fi'-V'xf^Cx.x')

x e S (D22)

We now use the x„ as variables of integration in eq. (B63), and set

57
u(x') = E(x'), (D23)

v(x') = ^(x,x')-a, (D24)

and, since E and Gr satisfy eqs. (D5) and (Dl8) respectively, we obtain

-vJl dt'/v

dV'3j(x')/3t'.G^(x,x , )4 - a-E(x)

o

= -(l/c2 )/vdV'[{G^(x,x')-a}-8E(x')/3t-E(x').3f

R

(x,x , )/9t'-a]

t

, =t

o +H dt'/c dS , .[E(x , )x{V , x^(x ,x

, )-l}-{^D (x,x , ).a}x{V , to xf(x ')}]. (D25) K K

O

The initial value of 3E/3t is given by eq. (D6), and we have assumed that

>

eq. (D2) holds for all t as part of the definition of B; hence, eliminating

-* the arbitrary vector a,

E(x) = -v Jl dt'LdV'a_(x,x')-3j(x , )/3t'

O L V IV O

+ (l/c2 )/vdV'[G

R

(x,x , )-{c2 V'xg(x ')4(x ')/ eo }-9^(x ,x')/3t'.E(x')]

t

, :=t o

-II dt , /c dS'fi , -[E(x , )x{V , xS(x,x, )}-8B

>(x')/3t , xS> (x,x')]. (D26)

t b K K

* "(1) When the tangential component of E is given, we choose a Green function G-^ ;

in that case, the first term in the surface integral is known and eq. (D2l)

makes the second term vanish. If, on the other hand, the tangential

* * "(2)

component of B is given, we can compute 3B/3t on S and a choice of G^ allows

58
us to express the surface integral in terms of known functions. Thus, eq.

(D26) represents a solution of Maxwell's equations in terms of integrals

over known functions if the Green functions are available; in practice,

these Green functions are seldom known.

We now derive an expression for the dyadic Green function for free

space, G-rj (x,x')» We have translational invariance, so that Gp° is a

function of x-x' alone. We proceed as in Appendix C, where the scalar

Green function was derived, and we take a four-dimensional Fourier

transform. We write

Gp (x,t) = (2u)-

2 /d 3 kda)ei(tot - k * x)

o:(k, W ), (D27)

and eq. (D8) is transformed into

>-£• ^ ,7>- i& ,„ s~2-*> -(a) 2 /c 2 )Q-tx(^xQ>

) = (2tt) Zf. (D28)

To solve for Q, we first multiply from the left by k to obtain

-(o) 2 /c2 )S-0T= (2tt) 2 k, (D29)

-+ «• and, expanding the triple vector product in (D28) and substituting for k»Q

from eq. (D29) we find

->-> -*-o-«-»- -(u) 2 /c 2 )0+(c 2

/4TT 2

a) 2 )kk+k2 Q = (l/4w2 )I, (D30)

59
Q =

" 4^

i f-(c 2

/a) 2 )££

)2/c 2 -£2 (D31)

We perform first the integration over oj, and we notice that Q has simple

poles at ± |k|c and a double pole at the origin. To obtain a retarded Green

function we deform the contour in the complex co-plane to pass above the

poles. Then, for t>0, we close the contour around the lower half-plane and

find the value of the integral from the residues at the poles. We find that

Jdwe

r. Q(k,uj) = -2 A c T-kk/k2

r ikct -ikct^ , itc 2

kk* ^[8^ k~( e " e J + 4^k2-J : (D32)

and

VS' c) - uyd

3keiS-5[(:-$ kk\sin(kct) , ctkk

P7 k + -&~ (D33)

t

R a,t) = -^md 3k-^^^

ik«x

(D34)

Although the integrands of the last two terms are singular at the origin,

we perform the angular integrations in spherical coordinates and use eq.

(C28) to obtain

60
(L(x,t) =

-7T7 -I\ dk sin(kr)sin(kct)

In

If

d ksin(kr)sin(kct) _ ctf dk

sin(kr)

r )ok r )o

k

(D35)

where all integrands are even functions of k and finite at the origin. We

extend the ranges of integration to -00 and deform the contour to pass below

the origin before separating the sine functions into exponentials. We then

integrate by finding the appropriate residues, and write

(l/2Tr)/

cdke

iak

= 6(a), (D36)

(l/2^)/

cdke

iak

/k = 10(a), (D37)

(1/2tt)/ dke

iak

/k2 = -a6(a), (D38)

where 9(a) is the unit step function and C is the deformed contour. We

could equally well have chosen the contour to pass above the origin; in

that case, the results in eqs. (D3T) and (D38) would have been different,

but the final result is not changed. We have

G_(x,t) = (c/4Trr)f{-6(ct+r)+6(ct-r)}+(c/2Tr)VV[(l/4r){(ct+r)e(ct+r)

R

-(ct-r)e(ct-r)-(-ct+r)0(-ct+r)+(-ct-r)0(-ct-r)}-(ct/2r){e(r)-9(-r)}], (D39)

and, remembering that r and ct are non-negative, this expression reduces to

61
t(tt) = f-feiS*=Ei _ vv (ct-r)6(ct-r)]

R 4tt |_ r r J (D40)

The gradients of well-behaved scalar functions of Ixl are

Vf (r) - ^&i, (D4l)

TO(r, . rf(r)-f(r) S+ f^, ^

Thus far we have dealt with singular functions in a heuristic manner,

avoiding the mathematical complications of the theory of distributions.

The well-known result

V

2 (l/r) = -4^6 (3) (x) (D43)

suggests that special care has to be exercised in evaluating contributions

at the origin. One way of finding these contributions consists of evaluating

the integral over a vanishingly small sphere of radius e about the origin

using the divergence theorem or related equations such as (B20). We use

^dVVf(r) = ^dSf(r), (D44)

/v

dVVVf(r) = ^dSVf(r), (D45)

and find, using eq. (DUl) for the second relation,

62
/ dVVf (r) = e 2 f(e)/dftn = 0, (D46)

/y

dVVVf(r) = e

2 f'(e)<£d£!nn = (4/3)^e 2 f ' (e)T. (D47)

Thus, at least for functions that do not diverge faster than 1/r, eq.

(D41) remains unchanged, while eq. (D42) becomes

uu*f \ r£"(r)-f ' (r)->->, f ' (rVffr 4 . . , 2 *t , Uj (3)A$> VVf(r) = ^—^-3 ^-^xxi ^-^-1+ ^l^mte^f ' (e) }6 V (x)I, (D48)

In particular.

VVI = Jx^fl _ 4^(3) £,£ (D49)

r r

3 3

and the scalar invariants of these expressions reduce the equation to (D43)

.

We now use

d8(a)/da = 6(a), (D50)

06(a) = 0, (D51)

f (a) 6(0-0') = f (a' ) 6(a-a' )

,

(D52)

tu derive from

f(r) = (ct-r)r

_1

e(ct-r), (D53)

63
f'(r) = -ctr

2 0(ct-r), (D54)

f"(r) = r

1 6(ct-r)+2ctr 3

9(ct-r), (D55)

Wf(r) = [r

3 6(ct-r)+3ctr 5

6 (ct-r) ]xx"-[ctr

3 (ct-r)+(4/3)^ct6 (3)

(x) ]¥, (D56)

CL (x,t) = (l/4ir)[{r 1 6(t-r/c)+c 2 tr

3

9 (t-r/c)+(4/3)7rc 2 tS

(3)

(x) }l K

-{r~ 3 6(t-r/c)+3c 2 tr"

5 9(t-r/c)}xx]. (D57)

We note that Gp vanishes for t<r/c, as expected from the special theory of

relativity. Finally, the free-space retarded Green function is

G^

o) (x,x') = (l/4^)[{R

_1

6(t-t'-R/c)+c 2 (t-t , )R~

3 e(t-t , -R/c) R

+ (4/3)7Tc 2 (t-t' )6(t-t' )6

(3)

(R) }f

-{R"3 6(t-t'-R/c)+3c 2 (t-t , )R"

5 0(t-t , -R/c}RR]. (D58)

where

R = x-x'. (D59)

64
NBS-114A irev. 2-ao

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN 1157

2. Performing Organ. Report NoJ 3. Publication Date

February 1982

4. TITLE AND SUBTITLE

Integral Equations for Transient Electromagnetic Fields

5. AUTHOR(S)

Egon Marx

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D C 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as above.

10. SUPPLEMENTARY NOTES

21 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Integral equations for the electric and magnetic fields in free space are derived

from Maxwell's equations. The fields are expressed in terms of their initial

values, boundary values, and sources with the help of a retarded Green function

for the scalar wave equation. These equations are then used to derive integral

equations for the surface charge and current densities induced by the scattering

of a transient electromagnetic field by perfect conductors. An alternative

solution of Maxwell's equations with the help of dyadic Green functions is also

presented.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words bv semicolons)

dyadic Green functions; electromagnetic scattering; integral equations;

perfect conductors; transient electromagnetic fields; wave equations.

13. AVAILABILITY

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| | For Official Distribution. Do Not Release to NTIS

~\ Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D C

20402.

[jj] Order From National Technical Information Service (NTIS), Springfield, VA 22161

14. NO. OF

PRINTED PAGES

68

15. Price

USCOMM-DC 6043-P80



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