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NBS TECHNICAL NOTE 1000

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

A Direct Approach to the Derivation of

Electric Dyadic Green's Functions
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NATIONAL BtJKEAt; OF STANDARDS

LIBRARY rrn 9 2 1978

A Direct Approach to the Derivation of

^^^'^^

Electric Dyadic Green's Functions

\^cKa". <La\ (OoV^

Arthur D. Yaghjian

Electromagnetic Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

•*•' •' Co f

\

\

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

U'. NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued January 1978
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1000

Nat. Bur. Stand. (U.S.), Tech. Note 1000, 68 pages (December 1977)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

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I
CONTENTS

Page

INTRODUCTION 1

I. A STRAIGHTFORWARD SOLUTION FOR THE MAGNETIC AND ELECTRIC

DYADIC GREEN'S FUNCTIONS IN FREE-SPACE 6

II. THE INADEQUACIES OF DELTA-FUNCTION METHODS 15

III. THE SOLUTION IN THE PRESENCE OF BOUNDARIES 22

IV. SYMMETRY, EVALUATION, AND INTERPRETATION OF L 29

IV. 1 Symmetry of L 29

IV. 2 Evaluation of L for Various Principal Volumes 31

IV. 3 A Mathematical and Physical Interpretation of L 36

V. TWO-DIMENSIONAL PROBLEMS 39

VI. SUMMARY 43

ACKNOWLEDGMENT 47

APPENDIX A. EQUIVALENCE OF MAXWELL'S EQUATIONS AND THE HELMHOLTZ

EQUATIONS FOR E AND B 48

APPENDIX B. A DIRECT PROOF FOR THE SOLUTION TO THE WAVE EQUATION. ... 51

FIGURES AND TABLES 56

REFERENCES 62

111

A DIRECT APPROACH TO THE DERIVATION OF

ELECTRIC DYADIC GREEN'S FUNCTIONS

Arthur D. Yaghjian

A straightforward approach that does not require delta-function

techniques is used to derive a generalized electric dyadic Green's

function which remains valid within the source region. Although the

electric field expressed by the dyadic Green's function proves to be

unique, the exact form of the dyadic itself depends, in the source

regions, upon the geometry of its "principal volume." The dependence

on principal volume is determined explicitly, and the different Green's

dyadics derived by a number of previous authors are shown to emerge

merely through the appropriate choice for the principal volume.

Moreover, delta-function techniques, which by themselves are shown

to be inadequate to extract uniquely the proper electric dyadic Green's

function in the source region, can be supplemented by a very simple

procedure to yield unambiguously the correct Green's function and associated fields.

Key words: Dyadic Green's function; electromagnetic theory; principal

values; source region.

I

INTRODUCTION

The problem we want to solve dates back to Maxwell himself [1] . Given

I TMaxwell's equations in a homogeneous region of space, find the time harmonic

electromagnetic (EM) fields in terms of the applied sources of current. The

ields in an infinite region are assumed to obey the outward radiation condiion, and if boundary surfaces are present, the tangential component of the

lectric or magnetic vector is assumed zero or otherwise specified on these

furfaces.

One might think that the general solution to such a basic problem would

ave been investigated thoroughly and laid to rest long ago. And indeed this is

he case for the solution to the fields outside the source points. However, at

he points where sources are present there exist in the literature a number of

different results to the same problems depending on the method of solution which
is used to find the electric dyadic Green's function.^ Yet uniqueness theorems

([2], Ch. VI) demand that there be just one correct solution for the fields even

at the source points. Thus, it is the main purpose of this paper to first

present a straightforward method for finding the correct solution to the electromagnetic fields inside as well as outside the source region, and second to

explain and reconcile the discrepancies which exist between previous results.

The previous results have been obtained by two basic methods of solution to

arrive at a dyadic Green's function, one, the integral formula method used, for

example, by Stratton ([7], Sees. 3.4, 8.1), Wilcox [8], and Van Bladel [3a]; and

two, the method of distributions with delta generating functions (hereinafter

referred to merely as the delta-funqtion method) used, for example, by Morse and

Feshbach ([9], Ch. 13), Tai [4], Collin [10], Howard [5], and Rahmat-Samii [6].^

A fundamental difference between the two methods involves the question of what

we shall call the principal volume (principal area for two-dimensional problems

discussed in Section V). For the integral formula method the geometry of the

limiting principal volume which excludes the singularity of the Green's function

must be stated explicitly to get the right answer when solving for the electric

dyadic Green's function in the source region and the associated fields computed

•^For instance compare at the source points the free-space electric dyadic Green's

function (and the EM fields computed from it) derived by Van Bladel [3a], Tai [4b]

and Howard [5]; or compare the results of Tai [4c] with that of Rahmat-Samii [6]

for a rectangular cavity. These are but a few of the number of instances which

can be sited from the literature where the EM fields at the source points do not

agree. Also, it should be emphasized that the reasons for the disagreement are much more fundamental than the error addressed and corrected by Tai [4a] of

using only solenoidal wave functions in the expansion of the dyadic delta function.

^ Actually, in some of these and other papers on the subject, an approach is

used which combines the integral formula and delta-function methods (see Section

II). However, in most cases one method predominates, and in all cases the

various steps in the particular approach taken can be classified as belonging to

the integral formula method, the delta-function method, or both. It is appropriate to distinguish between the two methods because each one can be used separately to solve for the EM fields in terms of the current distribution.
from it [3a]. The delta-function method as applied in the past, however, does

not require any mention of limit procedures Involving the geometry of a principal volume. (It should be noted here that limit is used in the generalized

sense which allows the value of the limit to depend on the geometry of the

principal volume. This use is nothing more than the generalization to three

dimensions of the one-dimensional concept of limit from the left and limit from

the right ([11], p. 13).)

In the following sections the dependence of the electric dyadic Green's

function on the geometry of the principal volume is verified and determined

explicitly. It is further shown that delta-function techniques as they have

been applied in the past suppress the specification of a principal volume and

thus will not yield the correct fields at source points from the electric dyadic

Green's function unless the suppressed principal volume is identified. And

although the principal volume cannot be identified from the ordinary deltafunction techniques alone, a surprisingly simple procedure can be used to supplement the delta-function method and properly incorporate the effect of the

principal volume. A number of other conclusions are reached, some of the more

important of which are summarized in the concluding Section VI.

Before entering the main body of the paper, a couple of other introductory

discussions may prove helpful. The first has to do with the terminology "geometry of the principal volume," which is used repeatedly throughout the paper.

Here "geometry" entails the specification of not only the shape of the surface

of the principal volume (area in two dimensions) but also its position and

orientation with respect to the singularity which it excludes. For example, it

is not sufficient in order to specify the geometry of a rectangular principal

rolume to simply state that it is a rectangular box. In addition, the position
I

of the singularity within the box and the orientation of the box with respect to

the singularity in the coordinate system chosen for the problem must be specified. We will find notable exceptions to this rule, such as the spherical or

cubical principal volume with singularity at the center, for which the dependence on principal volume remains independent of their orientation, but in general all three (the shape, position, and orientation of the principal volume)

must be specified.

The second discussion has to do with the physical explanation of why the

principal volume should affect the determination of electric field at source

points. (This subject is treated in greater depth and detail in Section IV. 3.)

Consider the hypothetical scheme fQV measuring the electrostatic field within a

dielectric by removing an infinitesimal volume of dielectric material and inserting a point probe which measures the electric field. It is a wellknown phenomenon [15b] that this measured electric field, often referred to as the "local

field," will depend on the geometry (shape, position with respect to the point

probe, and orientation in the dielectric) of this deleted volume. The unique,

mathematically defined macroscopic electric field is related to the local field

by adding to the local field a polarization or source term which also depends on

the geometry of the deleted volume. That is, both local field and source term

depend on the geometry of the deleted volume, but their sum, which equals the

macroscopic electric field, remains independent, as it should, of the deleted

volume.

Now for time harmonic fields, a similar phenomenon occurs as part of the I

derivation of the electric dyadic Green's function in source regions of current,

even when no polarization is present. At source points the electric dyadic

Green's function separates into two parts, which correspond to the analogous

local field and polarization or source term of electrostatics. In addition, the
principal volume corresponds mathematically to the deleted volume of dielectric

material. Both the local field and source term depend on the geometry of the

principal volume, but their sum remains independent of the principal volume and

equal to the unique, mathematically defined electric field. The analogy to

electrostatics is valid in a physical measurement sense also . For if the principal volume of time harmonic current sources were actually removed physically

and a point electric-field probe were inserted, the measured field would correspond to the local field, whose value again would depend on the geometry of the

deleted volume. Section IV. 3 demonstrates in detail the fundamental correspondence between electrostatic fields in dielectric media and electric dyadic Green's

functions in source regions.
I. A STRAIGHTFORWARD SOLUTION FOR THE MAGNETIC AND ELECTRIC

DYADIC GREEN'S FUNCTIONS IN FREE-SPACE

We begin with Maxwell's time harmonic equations in free-space written in

terms of the electric field E, magnetic induction B, and applied current J:

VxE = iwB

VxB = -iwe y E + y J.

o o o

(la)

(lb)

tl

As usual, e and y are the permittivity and permeability of free-space, and

e time dependence has been suppressed. The SI system of units is used

throughout. The E and B fields are restricted to satisfy the outgoing radiationl|

condition at infinity, and J lies in a volume of finite extent. (In Section V

two-dimensional problems are considered where the current J extends infinitely

far in the ± z-direction. ) By taking Vx (1) we find after a little manipulation

the following vector Helmholtz equations for B and E alone:

(2a)

I

V^B + k^B = -y VxJ

o

E - ^icoe + k^

0-' E - -7^ 10)8

VxVxJ

iwe

(2b)

(k = co/y e ) o o

In free-space (2) also imply (1), and thus (1) and (2) are equivalent.

That is, (2a) and (2b) are equivalent to Maxwell's equations. This familiar but

rather surprising result, which depends strongly on the fields obeying the

outward radiation condition, is proven in Appendix A.

Later in Section III, where we consider problems with boundary surfaces,

the fields must satisfy in addition to the vector Helmholtz equations (2) the

divergence requirements.

V«B = and V- E - ^iwe

= , (3a, b)

!

f
in order for (2) and (1) to be equivalent sets of equations. In the free-space

problem, as Appendix A shows, (3) are derivable from (2) with the help of the

radiation condition, and need not be stipulated as separate requirements.

The free-space solutions to (2) can be written down immediately as

B(r) = T^ lim / [V'xJ(r ' ) ]i|;(r,r' )dV' (4a)

V^V -V

£ J e

E(r) = - , .^^ lim / [V'xV'x7(r')]i|;(7,7')dV' + fV ' <^b)

^^^'"^o V^V -V "-"^^o e J e

The scalar Green's function ^i is defined as

i|j(r,r ) = e ' ' / r-r

and V merely indicates a volume within which all sources (J) are contained.

V is the "principal volume" which excludes the singularity of the function

i|^ at r' = r . The geometry (shape, and position and orientation with respect to

r) of V is specified for each r as the limit of V approaches zero, i.e.,

shrinks down around the point r. The principal volume V may but need not be

the same for each point r. Because the limit of the principal value integrals

in (4) do not depend upon the geometry of the principal volume V , it is customary to omit any explicit reference to V or the principal value sense in which

the integrals are defined. However, if (4) are to be properly differentiated or

transformed by integral formulas, both of which will be done below, it proves

necessary and in some places crucial that the limiting process be retained

explicitly [14]. This fact becomes clear immediately if we try to prove by

direct substitution that the solutions (4) do indeed satisfy (2).
A mathematically rigorous proof that (4) are solutions to the Helmholtz

equations (2) has been given by Kellogg ([12], Sec. VI. 3) for k = and by

Miiller ([2], Th. 7) for arbitrary k > 0. The usual textbook proof, by direct

substitution bringing the V^ operator under the integral and making use of the

fact that

V^ij; + k^ip = -4TT6(r-7'), (5)

is simply not valid since the singular point r' = r is excluded by the principal

volume V . The method of proof, by direct substitution can, however, be made

sufficiently rigorous if it is realized that the principle volume V as defined

by the r' coordinates is a function of the observation point r. In other words,

the limits of integration for the triple integrals in (4) depend on the variable

r. And thus, when we differentiate with respect to r, we must take this dependence into account and not merely interchange integration and differentiation.

Appendix B shows that when done correctly the method of direct substitution

yields the result that (4) satisfies (2)

.

Now (4) are solutions to Maxwell's equations in free-space in terms of the

derivatives of J, in particular, Vxj and VxVxJ. A solution is desired in terms

of J alone. To accomplish this we simply transform (4a) once and (4b) twice

using standard integral formulas which convert a volume integral to a surface

integral. Care must be taken to include the surface integral over V which

excludes the singularity.

Beginning with (4a) let

(V'xJ),j; = v'x(7i|j) - v>xj, (6)
I

and the integral in (4a) transforms as follows:

/

(V'xJ)i|; dV = - / n'xj^; dS' - / V>xJ dV . (7)

V -V S V -V

J e e J e

(n' is the outward normal to V ) . The integral over S vanishes because this

surface has been chosen to lie outside the source region. The integral in (7)

over the surface S of the principal volume also vanishes as V -K) since i|j

behaves as —, but the surface integration behaves as R'^ as R' =|r'-r| -> 0. Thus

(4a) has been transformed to

y ^ _ B(r)=-T7lim / (V x^;!) . j dV , (8)

V

^

V -V

e J e

where I is the unit dyadic and use has been made of the identity

V'iJ;xJ = (V'x,J;I).J . (9)

The free-space magnetic dyadic Green's function (G ) defined by

B(r) = y lim / G°-J dV (10)

o V

^

V iv ^

e J e

thus determined from (8)

G°(r,r') = -^ V'x(^i), rW r . (11)

It is a simple matter to show that because V'lf; behaves as 1/R'^ as R'-^O the

volume integral in (10) remains independent of the geometry of the principal

I
voltune V . Of course, this has to be the case since (10) was derived directly

e from (4a) whose volume integral is also independent of the geometry of V .

The magnetic dyadic Green's function presents no special difficulties. It

has been derived in free-space and for various interior and exterior regions by

both integral formula and delta-function techniques, and the results agree

consistently. Its derivation has been included here chiefly as an initial step

for determining the electric dyadic Green's function from (4b).

Transformation of (4b) by the same technique applied to (4a) yields

^(7)=-^liM / G°.(V.J)dV +^ , (12)

J e

an equation which displays the same properties as (10) . Still (12) does not

give E in terms J alone but includes the derivatives Vxj as well. To recast

(12) in terms of J alone we need merely to transform the integral a second time.

Substitution from the identity.

g"mm. (V'xJ) = -(V'xj) . G = -V'-(JxG ) - J-V'xG , (13) m m

into (12) and converting the divergence integral to a surface integral allows

(12) to be written as

I

E(r) = -1-^ lim

o V ->0 £

/ (V'xG )-J dV + /(G X n')-J dS' + J -' m -' TTi V -V ^

s ^

J e e

(14)

Use has been made of the fact that G and V'xG are antis5mmietric and symm m

metric dyadics, respectively.

10
We shall now prove the very important result that, unlike in (7), the

integration in (14) over the surface S of the principal volume contributes to

the electric field at source points. In addition, the value of this surface

integral contribution depends upon the geometry of the principal volume. The

value of the volume integral also varies with the geometry of the principal

volume and in just the right way to keep the sum of the volume and surface

integral contributions a unique value. That is, both the volume and surface

integrals in (14) have individual values which depend on the geometry of the

principal volume, but the variation in the volume integral is the negative of

the surface integral and together they sum to give a unique E(r) independent of

the geometry of the principal volume.

The surface integral in (14) can be rewritten with the help of (11) as

lim / (G° X n') • J dS' = V

^ S ^

£ £

4lT lim / (I X V'i|;)xn' dS'

V^ S

^ £ e

J (15)

When V'ijj is expressed explicitly as

ikR'

S/'il> = [ik - -,] e^,

R' - R' R' (16)

where e , is the unit vector, e , = R'/R' = (r'-r)/ Ir'-r I , the right side of

(15) reduces to

477 lim

V^ S

£ £

(I X e ,) X n'

/ ^ dS'

R

12

(17)

r The ik term has vanished in the limit as V ->0. Further simplification results

upon substitution of the identity

11
(Ixe^,) X n' = n'e^, - (n'-e^^,)! ,

into (17) to give

A-IT J

(n'e^.)

lim , V ^0 ^^

S' R'2

e e

dS' - I -J ,

(18)

I

(19)

after the solid angle integration is set equal to 4tt, i.e.

n 'e

e R

i2

Rj_ dS' = / d^' = Att . S

(20)

Substitution of (19) into (14) yields the final expression for the electric

field in free-space in terms of J,

E(r) = icoy lim / G° -J dV + ¥^ , ° V -.0 V,-V ^°

e J e

(21a)

where

'R'

Att dS'

S R

e i2

(21b)

and

G° = - i^ V'xG° = ^

VxG° , rV r . eo k^ m k"^ m

(21c)

=o . As a brief review of the notation, G is the magnetic dyadic Green's function

given by (11), n' is the unit normal out of the principal volume V , and e ,

12
is the unit vector pointing from r to r' (see Fig. 1). The limit as V -^0 is

omitted in (21b) because the surface integral depends only on the geometry

(shape, and position and orientation with respect to r) of V , not its size.

The conventional form [3b] of the free-space electric dyadic Green's function emerges as G when G from (11) is put into (21c)

,

^ eo m

C = W^ " ^ ^ ^^^^ =h ^^-^W ^^^'^' rW r . (.22)

Equation (21a) confirms Van Bladel's result [3a] that the conventional

electric dyadic Green's function is not sufficient at source points to determine

the correct value of E(r). The extra dyadic L, which depends only on the geometry of the principal volume, is also required. The fact that the unique electric

field at source points cannot be given by the volume integral alone becomes

obvious when one investigates the volume integration near the singularity of

G and finds that its contribution depends on the geometry of the principal

volume. The dyadic L, which also depends on the geometry of the principal

volume, compensates exactly to produce the unique E(r) regardless of the principal volume shape. This last statement can be proved directly from (21), but

this is unnecessary if we realize that the right side of (21a) and (4b) are

equal. Since (4b) is independent of the geometry of V the same must be true of

(21a)

.

In Section IV. 1 the dyadic L is proven to be a symmetric dyadic with unity

trace and is evaluated for a number of different principal volumes. It is found

that the different electric dyadic Green's functions derived by a number of

authors [3a, 4, 6, 8, 10] emerge merely through the proper choice for the geometry

of the principal volume. Except for Wilcox [8] and Van Bladel [3a], however.

13
the electric dyadic Green's function was derived applying results from delta-function

techniques, for which no mention of the geometry of the principal volume was

made or appeared necessary. This apparent paradox introduces the next section,

where it is shown that a basic assumption used in the ordinary application of

delta-function techniques fails at the source points of the electric dyadic

Green's function. Fortunately, a simple procedure is also found to remedy this

failure.

14
II. THE INADAQUACIES OF DELTA-FUNCTION METHODS

The delta-function method alone can be applied to derive an expression for

the magnetic or electric field from Maxwell's equations in terms of the source

current J. We shall briefly outline and discuss the shortcomings of this

approach for electric dyadic Green's functions. But before doing so, a very

common approach to the problem, which combines both an integral formula method

and delta functions, will be discussed.

This latter approach begins with the dyadic Green's theorem.

/ [(VxVxP).Q - P»(VxVxQ)]dV = / n • [P x (VxQ) + (VxP) x Q]dS . (23)

V S

The vector P is then set equal to the electric field E, which satisfies (2b),

and the dyadic Q is set equal to G , which satisfies the dyadic wave equation

VxVxG - k^G = 5(r'-r)I . (24)

e e

In order to compare the results to those of Section I, assume the free-space

problem; and thus E and G will also satisfy the radiation condition at infinity. After some algebra, theorem (23) yields the familiar equation

E(r) = iuy / G- J dV . (25)

(The surface integrals outside all sources vanish because of the radiation

condition.) To solve for E, given J, (25) indicates we need only solve for G

in (24) under the condition of outward radiation at infinity. There appears no

need for the mention of principal volumes.

15
The discrepancy between (25) and (21a) at source points is obvious. The

answer to what has gone wrong lies in the use of Green's theorem (23). This

identity requires that the vector P and dyadic Q be continuous through their

second derivatives. Sometimes this restriction is a mere formality, but in this

I

case the singularity in G is serious enough to render (23) invalid if Q is

set equal to G . In short, the left and right sides of (23) are not equal (if

the singularity of G is within the volume V), and the resultant equation (25)

becomes invalid at the source points.

The derivation from theorem (23) can be made valid by excluding the singularity of G with a small principal volume. Then instead of (25) emerging, the

correct equation (21a) results (assuming the singularity of G is determined or

known a priori) , eliminating any discrepancies between the two methods of derivations .

Next we consider the linear operator approach of delta-function theory to

derive the Green's function [13]. This method does not explicitly depend upon

Green's theorem (23) nor any similar integral formula. It begins with writing

the equation for the electric field in operator form,

D(E) = iojy J , (26)

where D is a linear differential operator (specifically VxVx-k^) . The inverse

of (26), assuming it exists, can be taken and the following linear equation for

E results:

E = icoy d""*"!!] . (27).

Expressing J(r) in the dyadic delta function form,

J(r) = / 6(7-7') I-J(7')dV' , (28)

16
allows (27) to be written as

E(r) = iooy D

* / 6(r-r') I.J(r')dV' . (29)

IF we can bring the inverse linear operator inside the integral, (29) becomes

E(r) = / D

• [6(r-r')I]'J(r')dV' . (30)

As usual, D [6(r-r')I] is denoted by G and from (26) G satisfies the

equation

D(G ) = 5(r-r')I , (31a)

or

VxVxG - k^G = 6(r-r')I . (31b)

e e

As one can see, the same problem of the discrepancy between the equation

(30) derived purely from the delta-function method, and the previous result

(21a) has reared a second time. Once again the explanation of this discrepancy

Involves the singularity of the Green's function. For the delta-function derivation just outlined, this singularity interferes with the interchange of linear

operators in a subtle yet crucial way. Specifically, in going from (29) to (30)

the volume integral and linear operator D were interchanged with respect to

their order of operation. For most linear operators in circuit theory, for

example, this interchange is perfectly valid. Also, for the linear operators

17
associated with the magnetic field, or with the electric field away from sources,

this interchange is perfectly valid. But for the electric field linear operator

at source points, this interchange becomes invalid, i.e.,

d"^ J

?4 / d"^ . (32)

V V

J J

An easy way to prove (32) is to take the correct inverse linear operator given

in (21a) and demonstrate that the left and right sides of (32) are indeed

unequal, the right side dependent upon the geometry of the principal voliime, the

left side not.

In summary, the interchange of linear operators basic to the ordinary

delta-function method for solving differential equations becomes invalid at the

source points of the electric dyadic Green's function, and this is the fundamental reason for the discrepancies between the various electric dyadic Green's

functions found in the literature .

If the delta-function method is used in spite of this shortcoming, it often

yields the electric dyadic Green's function corresponding to a particular principal volume, depending on what eigenfunctions are used and how they are manipulated.

Unfortunately, the delta-function method does not reveal to what principal

volume, if any, the resulting electric dyadic Green's function corresponds.

That it often (not always) corresponds to a particular principal volume was an

observation made only after evaluating L in (21b) for a number of different

geometries (see Section IV. 2). In those cases where the electric dyadic Green's

function derived by the delta-function method cannot be associated with a particular principal volume, the method has yielded bogus results which will give

erroneous fields in the source region regardless of the principal volume chosen.

13
Fortunately, in all cases an extremely simple way, explained below, exi'^ts to

remedy the delta-function method to include the proper effect of the principal

volume

.

But first, more discussion on the interchange of linear operators involved

with the electric and magnetic Green's functions seems appropriate at this

point. Consider equations (21). The electric dyadic Green's function can be

written symbolically for all r' as

=° = h [VxG°

e k^ m

g! = ^ [VxG„ - 6(r'-r)L] . (33)

It is emphasized that (33) has meaning only in the sense of (21a) , since the

first dyadic on the right side of (33) is not defined at r' = r and the second

dyadic is nonzero only at r' = r. For example, the usual delta-function result.

G^ = ^ [VxG° - 6(r'-r)I] , (34)

is not equivalent to (33). But when (11) and (33) are manifested as (10) and

(21), the resulting fields obey Maxwell's equation, provided the principal

volume and the interchanges of differentiation and integration are handled

properly as explained in Section I and Appendix B. Replacing (33) by (34),

i.e., L by I in (21a), gives an erroneous electric field at the source points,

since it can be shox^^l that L ^ I for any principal volume. (A quick way to

prove that L cannot equal I is to use the results of Section IV. 1 that the trace

of L must equal unity.)

Similarly, if the divergence of (33) is taken without regard to its symbolic nature, the result is

19
V-G° = - ^

V6(r'-r) • L (35)

rather than the familiar delta-function result obtained from (24),

V-G^ = - ^ V6(r'-r) . (36)

However, if the divergence of (21a) is taken, and the interchange of V* and the

integral is performed properly, the correct divergence expression for E is

preserved; i.e..

V-E = J/iwe . (37) o

It should not be concluded from this section that delta-function techniques

should be discarded as a way of finding the electric dyadic Green's function.

On the contrary, the results of Section I summarized in (21) reveal how deltafunction techniques for the electric dyadic Green's function at source points

can be salvaged

.

Away from the source points the delta-function method gives the proper

results and at the source points (21) shows that the conventional electric

Green's function requires but a minor modification. In particular, if we find

the electric dyadic Green's function outside the sources by the delta-function

method (or any other method), at the source points this same function can be

utilized if: 1) the geometry (shape, and position and orientation with respect to

the source point r) of the principal volume is designated; and 2) L for this

principal volume is calculated from (21b). Nothing more is required.

20
T Equations (21) , on which the above discussion and conclusions were largely

based, were derived for the free-space problem. In the following section the

derivation and conclusions are shown to extend to problems with boundary conditions on finite surfaces as well.

I

21
III. THE SOLUTION IN THE PRESENCE OF BOUNDARIES f

Equations (21) show that in the free-space problem the electric field can

be found from the conventional electric dyadic Green's function plus a source

dyadic L which depends only on the geometry of the principal volume used in the

evaluation of the volume integral. It is the purpose of this section to show

that the same is true for problems involving boundary surfaces on which tangential E or B are zero (or otherwise specified) . The excitation of resonant

cavities and the scattering from perfectly conducting bodies are typical examples

of such problems.

Consider the interior problem shown schematically in Fig. 2, and assume for

the moment that tangential E or B is zero on the boundary. The applied current

J lies within the volume V bounded by the perfectly electric or magnetic conducting surface S. Excitation at the resonant frequencies of the cavity is avoided. \

(The corresponding exterior problem, which has no real resonant frequencies, or

problems with tangential E or B specified but nonzero, can be treated by the

same analysis as the interior problem away from resonant frequencies, and thus

need not be considered separately.)

The electric field within S obeys the vector wave equation

VxVxE - k^E = iwy J , (38)

and, as the surface S is approached, the boundary condition for the tangential

fields.

E^ = 0, or B^ = ^

(VxE)^ = . (39a, b) tan tan lo) tan

22
I

The solution to (38), which has been shown to exist uniquely away from resonant

frequencies ([2], Ch. VI), can be divided into a particular solution E and a

homogeneous solution E, , with E = E^ + E . The particular solution is any

solution which satisfies the same equation as (38) but without regard to the

boundary conditions. The homogeneous solution satisfies the homogeneous vector

wave equation

Vxvxl" - k^E = , (40)

and the boundary condition on S

e5! = -E? , or (VxE^) = -(VxE^) . CAio h\ tan tan ' ^ '^tan ^ "^tan K^ia^D)

For the particular solution one may choose the solution given in (21)

,

i.e.

,

:rp,-, . ,. / G° . J dV +^^ . E (r) = X(jj]j lim ^ eo xwe (42) ° V ->0 V-V °

e e

(It is a straightforward matter to show that (42) satisfies (38) as well as

(2b).)

The solution to the homogeneous part E, can be expressed in terms of a

Green's function integrated over the boundary condition at the surface (see

e.g., [4d], Sec. 17),

\(r) = / Gg(r,r') • E^^n^""' ^'^^' * ^^^^

23
(The boundary condition (41a) has been chosen arbitrarily. A similar equation

and subsequent analysis can be written for the boundary condition (41b).) The

surface Integration In (43) does not encounter any singularities of the Green's

function G (r,r') since r remains Inside S where G Is continuous, and thus no

limiting principal values are required.

The field E^ can be found from (42), tan

1

F (r') = Iwy / [G° (r',r")]^ • J(r")dV" . tan (44) o ^ eo tan ^^^^

The principal volume part of (42) Is not contained In (44) since r" never equals

r' If we assume the sources J are within S. Substitution of (44) Into (43)

gives the homogeneous solution In terms of J,

E (r) = Iwu / G (r,r') • J(r')dS' . (45)

V

The homogeneous dyadic Green's function G, Is defined as

G^(r,r') = / Gg(r,r") • [G°^(r",r ' ) ]^^^ dS" , (46)

which, by the standard theorems of advanced calculus ([11], Sec. 4-6), Is a continuous function of r and r' since G„ and G are continuous for r and r'

S eo within V.

When the homogeneous solution (45) and the particular solution (42) are

added and G Is defined as G + G, , an expression for E Identical in form

eo eo h ^ to (21a) emerges.

24
E(r) = iojy lim / G -J dV + k^ . (47a)

° V ^0 V-V ^° ^"^^o

e e

The singularity of G is the same as G , and L has the same definition as in ^ -^ eo eo

(21b),

I

n'e

-47 ! -R^ ^^' • (^^^b)

S

e

The part of the electric dyadic Green's function denoted by G is given by

G = G° + G, , r' ^ r, (47c) eo eo h ^ ^

where G is the free-space dyadic Green's function (22) and G, is the solution

to the homogeneous dyadic vector wave equation.

VxVxG^ - k^G^ = , (48a)

under the boundary condition

nxG, = - nxG° , (48b) n eo

or

so n X (VxG, ) = -n X (VxG ) , (48c)

n eo

depending upon whether tangential E or H is zero on the boundary, respectively.

25
From a delta-function point of view the G required in (47a) is simply the

solution to

VxVxG - k^G = 6(r-r')I , (49a) eo eo *

outside the singular point r = r', and satisfying the Dirichlet or Neumann

boundary conditions,

nxG =0 or n x(VxG ) = . (49b, c) eo eo

Equations (47) were derived for Dirichlet or Neumann boundary conditions

and an interior region away from resonant frequencies. The same analysis and

results apply to exterior regions and to any specified boundary conditions for

which a unique solution exists.

The significance of (47a) lies in the fact that just as in the free-space

equation (21a) the singularity of the dyadic G is excluded in that volume

integration. That is, G need not be specified right at the singularity.

However, to use (47a) the geometry of the principal volume must be designated

and the extra dyadic L determined from (47b). This is a small price to pay for

the elimination of the problem of extracting the proper delta functions at the

singularity (source point) of the electric dyadic Green's function—an effort

which we have shown is futile in the sense that the problem cannot be solved

uniquely by any means unless the geometry of the principal volume is designated.

Another advantage of not having to deal with the singularity of G in

(47a) (except in terms of a principal volume integration) is that G can be

expanded in solenoidal wave functions. That is, since V*G = for r ?^ r',

the G under the volume integral of (47a) involves only solenoidal wave functions. In terms of the familiar M, N, and L notation for the Hansen vector wave

26
functions found in a number of textbooks (e.g., Stratton [7], Morse & Feshbach

[9], and Tai [4d]), this means that the L functions (not to be confused with the

dyadic L) are not needed. Only the solenoidal eigenfunctions M and N

are required in the expansion for G in (47a) . If for a particular problem the

electric dyadic Green's function is determined or found tabulated in the literature in terms of M and N functions alone (e.g., Tai's book [4d]), this is

exactly the right G needed in (47a). At source points the expression for the

electric field is amended simply by specifying the geometry of the principal

volume chosen to evaluate the volume integral and by adding the extra term in

(47a) involving the dyadic L. In the symbolic notation of previous authors the

complete electric dyadic Green's function G can be expressed as

G = G - K 6(r'-r)L . (50) e eo k'' As an example, suppose that the electric dyadic Green's function is wanted

for a circular waveguide, and that the electric field is to be computed numerically from (47a) using a thin pencil shaped principal volume (centered on the

singularity and aligned with the axis of the waveguide) as the patch which

excludes the singularity of G (see Fig. 3). From Section 22, equation (5) of

Tai's book [4d], G for this problem can be written down immediately.

00 00 Meny Meny NenX N^nA

n =^ I I (2-6 ) [ -^ ^ + -^ ^ ] . (51) ^° "^^ nio mil ° y^I k X^ T k,

y y A A (The definitions of the functions and parameters in (51) are unimportant for the

sake of this discussion but can be found, if desired, by referring to [4d].)

The dyadic L for a pencil-shaped principal volume is tabulated below in Section

IV. 2,

I L = 2^

• (52)

27
Equations (51) and (52) substituted into (47a) will then yield, for an applied

current J, the unique electric field everywhere inside the waveguide in the

limit as a pencil-shaped "patch" (centered on the singularity and aligned with

the z-axis) is shrunk down about the singularity.

In the following section, the source dyadic L is proven to be a sjnnmetric

dyadic with unity trace; then it is determined analytically and tabulated for £

number of principal volumes which might be commonly used as the volume "patch"

which excludes the singularity in the numerical evaluation of (47a); and finally,

a general mathematical and physical interpretation of L is provided. In Section

V, where two-dimensional problems are considered, the source dyadic is evaluated

for a number of principal areas as well.

28
IV. SYMMETRY, EVALUATION, AND INTERPRETATION OF L

It has been shown that the dyadic Green's function equation (47a) for the

electric field requires, at source points, the specification of the principal

volume V and the corresponding source dyadic L. Away from the source distribution J, the L term in (47a) vanishes and the remaining volume integral becomes

independent of the geometry of the principal volume since the singularities of

G would lie outside the source volume V,.. At the source points, the two terms

eo J on the right side of (47a) add to give a unique electric field E(r) independent

of the geometry of the principal volume; yet the value of each term separately

does depend upon the geometry of V , and thus it must be specified in order to

compute E(r) from (47a).

Once the geometry of V (shape, and position and orientation with respect to

the excluded singularity) is specified, L is determined from (47b) . The particular principal volume can be chosen arbitrarily according to the preference of

the problem solver and the convenience with which it fits into the geometry of

the problem and the associated computer programs. This section proves that L

is a symmetric dyadic with unity trace for any principal volume, and then evaluates and tabulates L specifically for a sphere, right circular cylinder, rectangular box (includes the cube), and pillbox of arbitrary cross-section. Finally

a general mathematical and physical interpretation is given for L.

IV. 1. Symmetry of L

As a preliminary exercise, it will be proven that the dyadic L is symmetric

for an arbitrary principal volume; and, in addition, the trace of L always

equals unity. In rectangular unit vectors, L takes the form, after dropping the

primes on the variables in (47b),

29
L =

S

1 "'^ "'- "- -- -- -- — [n xe e + n ye e + n ze e + n xe e + n ye e + n ze e ,t-^ >. _3 XXX xxy xxz yyx yyy yyz C53a) + nzzx xe e + nzzy ye e + n ze e ] dS. zzz

Consider the integral.

(n y - n x)

dS = e

R^ n X — dSr3

= e n X ^

dS

R^

(53b)

(I

S+S

e 00

The last equality holds because the integral over the surface at infinity is

zero. Since there is no singularity of the integrand between S and S , the

following vector integral formula applies

n X ^^ dS = r3V X — dV.

r3

s +s

e oo V -V

(53c)

The curl of R/R^ is zero, leaving (53c) zero, and from (53b)

n yX

R^

dS = n X

_Z_

r3

dS, (53d)

Substitution of (53d) and the corresponding results for n z and n z into (53a)

proves the dyadic L is indeed symmetric, i.e..

L = -r^I

47r

ne

R

dS = 7^

R^

417 ^R^

R^

dS = [L] transpose,

(53e)

30
A convenient way to summarize the proof is to say that the "vector of the dyadic

L" has been proven zero and thus the dyadic is symmetric ([14], Sec. 101).

This S3mmietry displayed by L holds for principal volumes of arbitrary

geometry and is confirmed for the specific principal volumes considered below.

The coordinate axes are so positioned and oriented in the following examples

that L takes the form of a diagonal dyadic as well.

Finally, we note from (53a) that the trace of L given by

trace (L) = -;

HIT dS, (54a)

r2

S

e is merely the normalized solid angle integration over a closed surface and thus

is always equal to unity, i.e.,

trace (L) = 1. (54b)

IV . 2 . Evaluation of L for Various Principal Volumes .

1) Sphere

For the sphere with origin (singularity) at the center, n' = e , and (47b) K

becomes, after dropping the primes on the variables,

_ - 27r TT ^ ^ L = -;— f / e_ e„ sin 6 d6 d*

4tt ^ -* R R

o o The angles 9 and (f) are the usual spherical coordinates. Expressing e in

K

rectangular unit vectors,

A ^ /V /V e_, = e sin 6 cos 4 + e sin 9 sin * + e cos 9 , R x . y z allows the integration to be performed. Upon integration all but the diagonal

terms vanish, leaving the familiar results [3a, 8] for a spherical principal

volume.

31
L = \ [e e +ee +eel=|- . (55)

(sphere)

2) Right Circular Cylinder

Consider the right circular cylinder with origin at the center and its axis

aligned with the z-direction. The edge of the cylinder makes an angle with

this axis. The unit vectors e and n expressed in a useful combination of

cylindrical and spherical coordinates are

e^ = e^ sin 9 + e cos 9 R P z

r± e on top and bottom of cylinder

n =

I /

^ e on side of cylinder

P

The element of surface area dS can also be written in terms of R, 9 and cf),

r R^ sin 9 dS = I cos 9 ^

R^ d9 dd) on side

d9 dcj) on top and bottom

Substitution of these relations into (47b) allows the integration to be performed

after lettingtopxy e = e cos d) + e sin d). L for the cylinder emerges as

L = (1 - cos )e e +77 cos 9 I^ , (56a) /.I .- -i-iN ozzz ot

(right circular cylinder)

where I is the unit dyadic transverse to the z direction. In rectangular and

cylindrical •^ coordinates, e.g., °txxyyppc()4) I =ee +ee =ee +ee^

If the circular cylinder is made into a circular pencil-shaped principal

volume (9 = 0) , L becomes

o

i L = 1^/2 . (56b)

(circular pencil)

32

i
Or if the circular cylinder is flattened to a pillbox (0 = 90°),

= ^

^ (56c) L = e e , z z (circular pillbox)

which checks with the results below for a pillbox of arbitrary shape (not necessarily circular).

3) Rectangular Box

Assume that the rectangular box has its origin at the center of an xyz

coordinate system and that the edges of the box lie parallel to these axes. Let

the length of the sides be denoted by a, b, and c, respectively. Then for the

side of the box in the plane z = c/2, we have

z c n e_, = -rr— [xe +ye+-;7e] , R R x y 2 z

and for the opposite side in the plane z = - c/2.

z c ne=-:r— [xe +ye -^e] R R X y 2 z

The contribution to L from these two sides is then

, ^ ^ b/2 a/2 e e y— e e J / — dS = —. fi , - b/2 -a/2

^

where Q is twice the solid angle subtended by the origin at the center of the

box and a side of the box perpendicular to the z-direction.

Since the same analysis applies to the x and y directions, we can write

down L immediately as

33
L =-T-[Qee+nee+nee] , (57a)

, T,x47rxxx yyy zzz

^'

(rectangular box)

where fi , Q , and f2 are the solid angles subtended by the two sides perpendicular to the X, y, and z direction respectively (Q. + Q + Q, = 4tt) . If c is

allowed to approach zero, the rectangular box goes to a rectangular pillbox,

Q = Q, =0,fl =4tt, and L becomes

X y z

L = e e , (57b)

z z (rectangular pillbox)

which checks with (56c) and the general pillbox result below.

If a is set equal to b and both a and b approach zero, then the rectangular

box becomes pencil shaped with square cross-section, 9. =0, Q, = Q, =27r, and

L becomes

(square pencil)

which is identical to the circular pencil result of (56b)

.

If a, b, and c are all set equal to produce a cubical principal volume,

Q = Q. =9. = 4Tr/3, and L becomes

X y z '

L = 1/3. (57d)

(cube)

Note that the source dyadic L for a cube and sphere are identical. This could have

been predicted from the sjmunetry in both cases. Symmetry also predicts the fact

that L = l/3 for any orientation of the cube (and, of course, the sphere) as

long as the origin (singularity) remains at the center.

4) Pillbox Of Arbitrary Cross-section

Finally consider the pillbox shaped principal volume. By pillbox we mean

a disk or flattened right cylinder (not necessarily circular) . In computing

34
L, the contribution from the infinitesimally thin sides of the pillbox is zero.

The contribution from the top and bottom is just

2t7 Tr/2 ^ = 4^ / / ^z^z ^^"^ Q de d<\> = e^e^ . (58)

(pill box) o o

Regardless of the shape of the pillbox principal volume, as long as the top and

bottom approach each other about the singularity (origin) faster than the sides,

L is simply the unit dyad along the axis of the pillbox.

The e e dyad in (58) is a very familiar term. It is the same result that

Tai [4a], Collin [10], and Rahmat-Samii [6] got for the extra source term using

delta-function techniques to find the electric dyadic Green's function in

rectangular waveguides. Although no mention was made to principal volumes in

their work, it is not surprising that they obtained the results corresponding to

a disc-shaped principal volume with axis along the z-direction, since each used

an eigenfunction expansion which preferred or "favored" the z-axis ([9], p.

1850).

If a nonfavoring or symmetric eigenfunction expansion is used to solve a

rectangular problem, one might expect to get the sjmimetric result (L = l/3)

displayed by the cubical principal volume. And indeed this proves true in

Rahmat-Samii 's analysis of the rectangular cavity [6] (assuming he omitted a

factor of 1/3 in the second term of his equation (27)). In Tai's analysis of

the rectangular cavity [4c], he again favors the z-direction in the eigenfunction expansion and again gets the e e result.

All of these comparisons illustrate the important result that the simple

process of tabulating the values of L for different principal volumes allows us

to explain many of the differences between the results of previous authors for

the electric dyadic Green's function at source points. The results of the

35
integral formula method of Wilcox [8] and Van Bladel [3a], who carefully specify

a spherical principal volume, and the results of the delta-function method can

be reconciled through the proper choice of principal volume and corresponding

L.

The values of L for the different principal volumes considered above and

their correspondence to results of previous authors is summarized in Table I.

The values of L for all of these and many other simple geometries could probably

be determined from sjmmietry considerations alone, but this intriguing topic will

not be treated in this paper.

IV . 3 . A Mathematical and Physical Interpretation of L

Finally, before leaving the subject of the source dyadic L it is appropriate to give a mathematical and physical interpretation into the character of

L. From a mathematical point of view, L defined by (47b) has the form of a

generalized or dyadic solid angle normalized by 1/4tt. That is, if a dot product

were placed between n' and e , in (47b), the integration would be that of a

is. solid angle integrated over a closed surface, resulting in a value of 47r steradians, regardless of the geometry of the principal volume. (We noted this fact

above in Eq. (54a) when proving the trace of L is always unity.) With no dot

between n' and e , the integration can still be considered to have units of

steradians, but the result is the symmetric dyadic L whose value does depend on

the geometry of the principal volume. Still, an appropriate name for L might be

the "normalized dyadic solid angle."

The dyadic L has an interesting physical interpretation. Suppose one were

to measure the electric field at a point within a current distribution by removing an infinitesimally small volume of current and inserting an ideal point

probe. The measured field would be that given by (47a) but without the L

i

i 3S
term, since the current J at this point has been removed. This measured or

"local" field would also depend upon the shape of the infinitesimal volume and

its relative position and orientation with respect to the point probe, because

we have shown that each term separately in (47a) (and thus the first term)

displays said dependence. Thus L determines the perturbation in electric field

caused by the hypothetical measurement scheme o£ removing an inf initesimally

small volume of current V £f^ given shape , and position and orientation with

respect to an ideal point probe , which measures the electric field .

The perturbation of electric field caused by the removal of an infinitesimal volume of dielectric material is a familiar phenomenon in electrostatics

[15]. And, in fact, if the electrostatic field is correctly expressed in terms

of polarization P within dielectric material, a source dyadic identical to L

appears. A brief derivation begins with the electrostatic field E given by the

equation

V X V X E = 0, o or

or

V2 E = V(V-E ) = -V(V«P) -V X V X p v2 P

o e o

' = -V X V X (59)

assuming the free charge is zero. Equation (59) is the same as (2b) with the

-i(joP replacing J and a)=o. The analysis of Section I applies to (59), giving the

following result for E ,

E = —^

^^"^ o 4Tr£ V

^ o e G • PdVo (60a)

V -V

P e where

^o -- ^^

r-r

(60b)

and L is the same dyadic defined above for the time harmonic fields-.

37
^|Mj — L • P

The quantity E -\ is a generalization of what is sometimes referred

o to as the "local field" in electrostatics [15]. For a spherical principal

volume with the probe at the center, L = 1/3 and the familiar local field i

E + P/3e within a dielectric emerges. For time harmonic fields the analogous

local field is given by E —-. , which also becomes E H if equivalent _ o _ o polarization current -iwP replaces the free current J.

In the following section, two-dimensional problems are considered and the

source dyadic for different principal areas are evaluated.

38
V. TWO-DIMENSIONAL PROBLEMS

So far only three-dimensional (3-D) problems have been considered, i.e.,

where the current sources J were contained in a volume of finite extent

.

Physically, of course, these are the only problems one encounters. But from a

mathematical point of view it is often convenient to model certain problems by a

two-dimensional (2-D) geometry where the sources extend to ± infinity in a

certain direction (here chosen to be the z-direction)

.

The analysis is completely analogous to the 3-D case. It begins with

Maxwell's equations (1) to derive the Helmholtz equations (2). In free-space a

procedure similar to that given in Appendix A, but using ordinary Bessel functions instead of spherical Bessel functions, proves that (2) are equivalent to

(1), and thus the solution to (2) can be concentrated on without further reference to (1). But just as in the 3-D case, if boundary surfaces are present on

which tangential electric or magnetic fields are given, (2) must include (3) as

separate conditions to be equivalent to Maxwell's equations (1).

Since J is a constant function of z for 2-D problems (although J can still

have a z-component) , the z part of the volume integration in (4) can be performed. Specifically,

0° ik|r-r'| .^

.

_ _

/ ^-^^3 dz' = TTi H^-^Mk|t-t'|) , t ?t t', (61) if I o r-r I _ _

r = t + z e

z

r' = t"' + z'e

z where H is the Hankel function of the first kind. Using (61) in (4) transforms the 3-D solution (4) into the 2-D solution,^

^Even though (61) diverges at t = t', it can be shown that it is permissible to^ elongate the principal volume in (4) to ± infinity in the z-direction. Thus, t never has to equal t' and the principal volume reduces to a principal area.

39
y_,i /-IN B(t) = -J- lim / (v; X J(t') H'^-'^(k|t-t'|)dA' (62a)

e A^-A

J e

E(t) = - T-7- lim / (v; X v; X j)H^l^(k|t-t'|)dA' . (62b)

^"^^o A -^0

, , ^ ^ °

e A -A

J e

In (62) the singularity of H is logarithmic, and the principal volume

converts to a principal area which excludes this singularity. The area integral

extends over a large enough area A to cover all 2-D source current J. The

logarithmic singularity is weak enough so that the value of the integrals in

(62) do not depend upon the geometry of the principal area. But, as in the 3-D

case, the limit process is retained explicitly to enable the desired differentiation and integral transformations to be performed correctly on (62). That

(62) is indeed the free-space 2-D solution to (2) can be verified in a manner

directly analogous to the procedure used in Appendix B.

Equation (62a) can be transformed once and (62b) twice using the 2-D

equivalents of the integral transformations performed in Section I. Such a

procedure yields the following 2-D equations analogous to (10) and (21)

:

B(t) = y lim / g° .J dA' (63a)

A "^0 A A e A -A

J e

£(¥) = icoy lim / g° • J dA' + 4^^ (63b)

A -^0 . . o

e A^-A

J e ^ ^

e

The magnetic integral (63a) is again independent of the geometry of the principal

area and the 2-D magnetic dyadic Green's function is

40
i° = I

V' X (H^l) ! ) . (64)

m 4 t o t

The electric field equation (63b) again has two parts, the conventional integral

part and the source dyadic term. The value of each of these depends on the

geometry (shape, and position and orientation with respect to t) of the principal

area A , but the sum does not, and produces a unique electric field E(t) . The

conventional 2-D electric dyadic Green's function can be written

8° = - fr V' X i° = ^i^ V^ X V x(H^^^l ) = ^(! + h V VJH^-^^ . (65) °eo k^ t m 4k^ t to t 4t k'^tto

The extra 2-D source dyadic term Z defined by (63c) is the obvious analogue

of the three-dimensional dyadic L defined in (21b) or (47b) . The line integral

in (63c) is performed over the curve C bounding the principal area A . The

outward normal to this curve is denoted by n' , and e , is the unit vector from

the singularity point within A to the integrating points on the curve C (see

Fig. 4). Just like L, the 2-D dyadic i can be shown to be a symmetric dyadic

with unity trace.

If a dot product were placed between the n' and the e , in (63c) the result

of the integral would merely be the l-n angle swept out by the closed curve C .

Whereas L could be described as a "normalized dyadic solid angle," I can be

described even more simply as a "normalized dyadic angle." The analogous physical interpretation of I is that it determines the perturbation in electric

field caused by the hypothetical scheme of removing an inf initesimally thin

current volume (extending from z = -H» to -«>) with cross-sectional area A of

given shape, and position and orientation with respect to an ideal point probe,

which measures the electric field. The equations (63) derived for free-space

can be generalized to include boundaries as was done in Section III for 3-D

problems.

41
In fact the 2-D results are so closely related and analogous to the 3-D

results that it becomes unnecessary to discuss the 2-D problems in great detail.

It suffices to say that like the 3-D case the 2-D electric dyadic Green's

function requires the specification of a principal area, and can be conveniently

expressed by the conventional solenoidal Green's dyadic plus an extra source

dyadic I which simply and unambiguously provides the correct fields at the

source points. The normalized dyadic angle I has been determined from (63c) and *'

tabulated in Table II for some common principal areas or 2-D patches which might

be used, for example, in a computer evaluation of (63b).

42
VI. SUMMARY

The intent of this paper was motivated by the conflicting results which

exist in the literature for the electric dyadic Green's function and the fields

computed from it in the source region. Different authors have obtained different results for the same problems depending on the method of solution which was

used. Yet fundamental theorems of EM theory assures the existence of a unique

solution for these problems even at source points. Moreover, the disagreement

is much more fundamental than the error addressed and corrected by Tai [4a] of

using only solenoidal wave functions in the eigenfunction expansion of the

dyadic delta function.

The various methods of solution reduce to just two basically different

approaches, the integral formula approach and the delta-function approach. The

two different approaches or methods of solution often yield different results at

the source points of the electric dyadic Green's function. The concept of

principal volume was introduced, and it was shown that the fundamental difference between the two basic methods involved the specification of this principal

volume. The integral formula method requires the explicit specification of the

geometry of the principal volume while the delta-function method does not.

To shed light on this discrepancy, a simple, straightforward method of

solution, which began with the known solution to the scalar wave equation, was

used to obtain the correct electric dyadic Green's function and associated field

for free-space inside and outside the source region. (Later the derivation was

extended to problems with boundary conditions on finite surfaces as well.) It

was found that this electric dyadic Green's function actually required the

specification of two dyadics, the conventional dyadic outside the source point

and a sjmimetric, unity trace, source dyadic L which depended solely on the

43
geometry of the principal volume. (However, the electric field computed from

these two dyadics remains independent of the geometry of the principal volume -

as it must.) Thus, the results of the integral formula method were confirmed,

and the delta-function method was investigated to determine the reason for its

apparent failure to reveal the principal volume and thus give unique fields at

the source points.

The problem with applying the ordinary delta-function techniques to obtain

the electric dyadic Green's functions was traced to a very basic assumption in

the theory that the order of two linear operators could be interchanged. This

interchange is valid for most linear circuit problems, for the magnetic dyadic

Green's function, and for the electric dyadic Green's function away from source

points. But at the source point of the electric dyadic Green's function the

interchange becomes invalid; i.e., the expressions before and after the interchange are not equal, one depending on the geometry of the principal volume, the

other not.

If these shortcomings of ordinary delta-function techniques are ignored,

the result often (not always) coincides with that of a particular principal

volume, depending on the eigenfunctions which are used to expand the electric

dyadic Green's function. However, the particular principal volume must be

identified explicitly if the Green's function is to be used to compute the

unique electric field from applied current sources. And the ordinary deltafunction techniques do not identify the principal volume to which its solution

corresponds

.

Fortunately, it was shown that the delta-function techniques could be

remedied in a very simple way to assure the proper electric field at source

points as well. Once the electric dyadic Green's function is found outside the

source point (by the delta-function method or by any other method), then the

44
I

proper modification at the source point can be made by

1) designating the geometry (shape, and position and orientation with

respect to the source point) of the principal volume and

2) computing the symmetric source dyadic L (from (47b)) corresponding to

this principal volume.

Nothing more is required.

K Moreover, since the conventional dyadic Green's function is required only

^outside the singularity (source point) when using the modified delta-function

method just described, only the solenoidal vector wave functions (M and N) are

required in our eigenfunction expression of the Green's function. The need and

contribution at source points of the nonsolenoidal vector wave functions (L) is

completely replaced by the contribution of the source dyadic L, which is determined from the geometry of the principal volume alone.

The consequences of this result are significant. If we can find the electric dyadic Green's function valid outside the source region tabulated in the

literature (e.g., in Tai's book [4d]), then the proper modification at source

points is made by simply specifying the principal volume (or "patch" used in the

computer algorithm) and evaluating the corresponding dyadic L. A specific

example for circular waveguide is given in Section III.

The value of L was determined and tabulated for a number of common principal volumes or "patches," including the sphere, circular cylinder, rectangular

box, and pillbox. The results of many previous authors for the electric dyadic

Green's function were shown to correspond to different principal volumes (see

Table I). That is, the apparent discrepancies between the results obtained

by a number of authors using different methods of solution for the electric

dyadic Green's function at source points are explained and reconciled merely

through the proper choice of principal volume and evaluation of the corresponding source dyadic L.

45
An appropriate mathematical interpretation of L was found to be that of a

"normalized dyadic solid angle." An interesting physical interpretation of L

stems from the fact that it determines the perturbation in electric field caused

by the hypothetical measurement scheme of removing an infinitesimal volume of

current of given shape, and position and orientation with respect to an ideal

point probe, which measures the elect ic field. It was shown further that the

same source dyadic L can be used to express the local electrostatic field within

a dielectric.

Finally, in Section V, two-dimensional problems were considered and an

analysis similar to the three-dimensional case was performed. The 2-D results

were completely analogous to th^ 3-D results, with cylindrical Hankel functions

replacing the spherical waves, and principal areas merely replacing principal

volumes

.

46
ACKNOWLEDGMENT

The author is grateful to Drs. E. T. Kornhauser and C. F. Stubenrauch for their helpful reviews of the manuscript. Discussions with Dr. DM Kerns

were instrumental in discovering the physical interpretation of the source dyadic.

47

Hi
APPENDIX A

EQUIVALENCE OF MAXWELL'S EQUATIONS AND THE HELMHOLTZ EQUATIONS FOR E AND B

The vector Helmholtz equations (2) of the main text were derived in the

usual manner from Maxwell's equations (1). Here we shall derive (1) starting

with (2) in free-space, and prove in a very simple fashion, the often assumed

result that the wave equations (2) and Maxwell's equations (1) are equivalent in

the free-space region. In addition, it will be shown that the result extends to

regions with boundary surfaces only if the fields in (2) are made to obey the

supplementary divergence conditions (3)

.

Begin by taking the divergence of (2a) and realizing that the V* and V^

operators commute, to give

V2(VB) + k2(VB) = . (Al)

According to Theorem 13 of Miiller [2], the twice continuously dif ferentiable

solution to the differential equation (Al) can be expanded in a complete set of

spherical harmonics Y (6,({)) and spherical Bessel functions j (kr),

£,m

where the C are coefficients which do not depend upon r, 6, or cj).

In free-space B and thus V*B must satisfy the radiation condition as

r -> 00, i.e.

,

V«B -^ C(e,(}))e^^^/r or faster. (A3)

As r -> 0°, the spherical Bessel functions behave as sin kr/r or cos kr/r . And

since the Y (9,(j)) are orthogonal functions, the only way (A2) can equal (A3) as

48
r -J- °° is for all the C„ and thus V*B to be zero. That is, the radiation con-

£iii dition (A3) in the free-space problem has obviated the need to specify

V-B = , (A4a)

as a separate condition to the Helmholtz equation (2a) . In a similar fashion

(2b) and the radiation condition on E insures

[E - iwe ] = . (A4b)

Still it remains to prove that (2) imply (1). To do this, reT^^rite (2)

utilizing the zero divergences (A4), in vector wave equation form

VxVxB - k'^ B = y VxJ

o (A5a)

VxVxE - k^ E = ia3u J . (A5b)

Take Vx the second equation and equate the result to Vxj in the first equation.

After some rearranging and use of (A4a) we find

v2[VxE - itoB] + k2[VxE - iooB] = . (A6)

Because E and B must satisfy the radiation condition, so also must

VxE - icoB decay as fast or faster than the radiation condition demands. Thus,

the same analysis we used to show from (Al) and (A3) that V«B = implies,

from (A6) and the radiation condition, that

VxE - iojB = .

Substitution of (A7a) into (A5b) gives the second Maxwell equation

(A7a)

VxB + ioje y E - y J = . o o o (A7b)

49
Consequently, in the free-space problem where E and B satisfy the radiation

condition, we have proved that (2) and (1) of the main text are equivalent sets

of equations.

In regions where there are boundary surfaces on which tangential E or B

are specified, the same proof of equivalence fails to apply. The main reason

for the failure is that (Al) no longer implies that V»B = if boundary conditions other than the radiation condition are present. Thus the divergence

conditions (A4) (eqs. (3) of the main text) on the fields must be included as

separate requirements supplementing (2)

.

If the solutions to (2) are made to satisfy the divergence conditions (3),

(A5) are again implied by (2) and the following equations similar to (A6) can be

obtained:

VxVx[Vx'e - iuB] -k2[VxE - iuB] = (A8a)

VxVx[VxB + iwe y E - y J] - k2[VxB + iwe y E - y J] = . (A8b)

If on the boundary surfaces the tangential fields are zero,

B^ = ^

(VxE)^ =0 or E^ = — (VxB)^ =0, , „ ,v tan xw tan tan xwe y tan (A9a,b) V » /

then, as a consequence of the theorems in Ch. VII of Miiller [2], the only solution to (A8) and (A5) for an exterior region or an interior region away from

resonant frequencies is the trivial solution; and once again Maxwell's equations

are implied. That is, (2) plus (3) are equivalent to (1) for problems with

surfaces on which E^ or B^ are zero. Also, by a similar proof, the same tan tan •' r » results can be extended to given tangential boundary conditions other than zero.

50 —- - —"^

~-
APPENDIX B

A DIRECT PROOF FOR THE SOLUTION TO THE WAVE EQUATION

As part of this work on tracing down the inconsistencies among different

methods for finding the electric dyadic Green's function at source points, it

became clear that a simple, yet sufficiently rigorous proof, by direct substitution was desirable to verify the familiar free-space solution to the wave

equation. In short, a straightforward proof is desired to verify that

W^-^ilim / IC^I-e^'^l'^'-ldV (Bl) ' \-*o

v-v I'^'-'^l e is indeed the outgoing wave solution to the wave equation

V2<j) + k2<|) = f (r) . (B2)

Such a proof leads to an understanding of when and why it is or is not permissible to interchange the order of the differentiations and integrations.

Of course, a rigorous proof that (Bl) satisfies (B2) uniquely can be found

in Ch. VI of Kellogg [12] for k = 0, and in Ch. I of Muller [2] for k > and

outgoing radiation, but the proofs are indirect and presented from basically a

mathematician's point of view, which can be somewhat prohibitive.

Also, it should be mentioned that strictly speaking the integral formula

method used in Sections 3.4 and 8.1 of Stratton [7] does not constitute a proof

that (Bl) is the solution to (B2) . Stratton' s analysis says that if a solution

to (B2) exists, then it can be manipulated using integral formulas to prove that

(Bl) satisfies (B2) . Stratton's analysis is a subtle uniqueness proof, but

existence remains to be proven. The books by Kellogg and Miiller are testimonies

to the necessity of such existence theorems.

51
As a trivial example to illustrate this point, suppose we assume initially

2

that a positive real number solution exists to the equation x = -1. We can

then square this equation, find x = 1, and erroneously assume that this is the

unique solution. The error lies in not first proving the existence of a solution. If we do not prove existence initially, the solution obtained through

implication must always be checked by substituting back in the original equation. Exactly the same limitation applies to the analysis in Sections 3.4 and

8.1 of Stratton. It does not constitute a proof that (Bl) satisfies (B2)

.

The usual textbook method of proving (Bl) satisfies (B2) consists in substituting (Bl) into (B2) bringing the V^ operator through the limit and the

integral sign, and letting ^

S/^ + k^lp = -47T 6(P-7)

(B3)

, iklr'-r I , |—, —

I

\l) = e ' '/|r-r|.

This proof is valid away from the source point, i.e., where f(r) = 0, but at

source points it does not recover the function f(r) unless the limiting process

is carelessly ignored. In other words, the limit excludes the singularity of i|/,

and thus the delta function in (B3) is never encountered. Away from the source

points no singularities exist, the limit process becomes superfluous, and the V^

operator and the integration can be interchanged rigorously by the standard

Leibnitz rule of advanced calculus (e.g., [11], Sec. 4-12) which apply to continuously differential integrands.

Actually, one might think that Leibnitz's rule could be applied at source

points as well, since the singularity is excluded by the limit process. Obviously

this is not the case since, as explained above, such a procedure does not recover

the source function f(r).

52
One tempting explanation of the trouble is that the differentiation and limit

process cannot be interchanged; i.e., is it true that

V^ lim / = lim V^ / . (B4)

£ V-V e V-V

e e

The answer is not obvious since such a theorem is not true in general. However,

it can be proven true for the particular type of singular integrals (Bl) involved

here. (The proof will not be included here because we eventually use an approach

which does not require such a theorem.)

Thus, we can write

^2x It- r72 f f(^) iklr'-rl ,„, v2^ = -_lini V^ / —-^e ' IdV . (B5)

V° V-V 1^ -^1

e

>w the essential reason why V^ and the integration cannot be interchanged

immediately in (B5) becomes clear. The limits of integration depend upon the

variable of differentiation r. Since V encloses the point r, with respect to

the integration variable r' the surface equation for V is a function of r. H Fortunately, there is a disarmingly simple approach to circumvent this

problem. Return to the original equation (Bl) and make the change of integrating variable

r" = 1^' - r ' (B6)

converting (Bl) to

(i)(r) = - -.— lim 1 —^^—

n

e dV . v^'/

V ->0 e V-V

53

i.^
No longer is the equation for the surface of V a function of the position

vector r (with respect to the new integration variable r"). That is, the limits

of integration in (B7) are not a function of the r coordinates. Furthermore,

since the value of the integral in (B7) does not depend upon the geometry of the

principal volume, if we choose a spherical principal volume and write dV" in

spherical coordinates,

dV" = (r")^ sin e" de" df dr", (B8)

the limit process is accomplished inherently, and (B7) becomes

(|)(7) = - T^ / f (r" + 7)e^^^" r" sin 6" dG" dcj)" dr" . (B9)

V

Provided the source function f has well defined derivatives, Liebnitz's

rule permits (B9) to be differentiated by bringing the differentiation under

the integral sign. Specifically,

V2(j)(7) = - T^ / V2f(7" + 7)e^^^" r" sin 9" d0" dcj)" dr" . (BIO)

If the volume integral in (BIO) is returned, remembering that the limiting

process must also be reinstated explicitly to exclude the singularity (since

integral formulas which will be used to transform (Bll) below are valid only if

the limit process is retained explicitly) , (BIO) becomes

7M2c {~ti

v72^/-^ It r V'^f (r" + r) ikr" ,.,„ ,^,,. V^(b(r) = - -.— Ixm I ^^—

n

e dV . (Bll)

V ->0 e V-V

e Use has been made of the fact that

Vf(r" + r) = V"f (r" + r) . (Bl2)

54

i .
The rest of the proof follows exactly the procedure used in Section I of

the main text. Equation (Bll) is transformed by the divergence theorem twice,

carefully retaining the surface integral about the principal volume V , to yield

the desired wave equation,

V2(j, = - k2(j, + f (7) . (B13)

In summary then, a simple method by direct substitution can be used to

prove (Bl) satisfies (B2). The method consists in making the change of integration variable (B6) , which allows the V^ operator to be taken inside the

integral sign with impunity and operate only on the source function f . The

resulting integral is transformed twice by the divergence theorem and the solution to the wave equation is verified.

55
Figure 1. Notation associated with a principal volume V used to define the

dyadic L.

56
Fieure 2. Interxor region V with boundary surface S on which E^ or B is

. r . ^ tan tan specified.

57
(p>tz")

^ z CIRCULAR

WAVEGUIDE

PENCIL SHAPED PRINCIPAL VOLUME

1 (r AT CENTER)

(_ ()

D —

Figure 3. Circular waveguide with pencil-shaped principal volume.

y i

58
Figure 4. Notation associated with a principal area A used to define the 2-D

dyadic £.

59
PRINCIPLE VOLUME

(r AT CENTER)

PROBLEM &

AUTHOR(S)

SPHERE

FREE-SPACE

WILCOX (1957)

VANBLADEL (1961)

RIGHT CIRCULAR CYLINDER

^ /N (1-COS 0Q)e^ e^

+ h COS Sq T^

CIRCULAR

e_

OR SQUARE

PENCIL f g

a,b-^0 D-»-0

RECTANGULAR

BOX

^e

y

Tj—(fi ee^ + Slee +fiee)(fi +fi, +fi,

471 ^xxx )l y ^ zzz'^^x y z

fi^, fi , AND fi^ ARE TWICE THE SOLID ANGLE

SUBTENDED BY A SIDEXTO THE x, y, and z, DIRECTION RESPECTIVELY.

= 47r)

CUBE

RECTANGULAR CAVITY

RAHMAT-SAMII (1975)

PILL BOX

^z \

RECTANGULAR WAVEGUIDE

AND CAVITY

TAI (1973, 1976)

WAVEGUIDE ONLY

COLLIN (1973)

RAHMAT-SAMII (1975)

Table I. Tabulation of source dyadic L, and correspondence to previous authors.

60
PRINCIPAL AREA

(t AT CENTER)

CIRCLE

/

,\ '* e RECTANGLE

-e.. o—(6 e e + 6 e e ) 2tt^ XXX y y y

(e^ + e^ = 27t)

SLIT

^e.

d -^

e e

X X

SQUARE AND

EQUILATERAL TRIANGLE

Table II. Tabulation of 2-D source dyadic I.

61
[2]

[3] a)

b)

[4] a)

REFERENCES

[I] Maxwell, James Clerk, A Treatise on Electricity and Magnetism, Vol. II

(Dover Publication, slightly altered republication of the 3rd ed.

published by the Clarendon Press in 1891, Part IV., Ch. XX, New York,

1954).

Miiller, Claus, Foundations of the Mathematical Theory of Electromagnetic Waves ( Springer-Ver lag. New York, 1969).

Van Bladel, J., Some remarks on Green's dyadic for infinite space, IRE

Trans. Antennas Propagation, AP-9 , pp. 563-566 (Nov. 1961).

Electromagnetic Fields (McGraw-Hill, Sec. 7.9., New York, 1964).

Tai, C. T., On the eigenfunction expansion of dyadic Green's functions, IEEE Proceedings, 6j^, pp. 480-481 (April 1973).

b) Eigenfunction expansion of dyadic Green's functions. Math Note 28 , Weapons Systems Laboratory, Kirtland Air Force Base, Albuquerque, NM

(July 1973).

c) Different representations of dyadic Green's functions for a rectangular cavity, IEEE Trans. Microwave Theory Tech., MTT-24 , pp. 597-601

(Sept. 1976).

d) Dyadic Green's Functions in Electromagnetic Theory (Intext, Scranton,

1971).

[5] Howard, A. Q., Jr., On the longitudinal component of the Green's

function dyadic, IEEE Proc, 62_, pp. 1704-1705 (Dec. 1974).

[6] Rahmat-Samii, Y., On the question of computation of the dyadic Green's

function at the source region in waveguides and cavities, IEEE Trans.

Microwave Theory Tech., MTT-23 , pp. 762-765, (Sept. 1975).

[7] Stratton, J. A., Electromagnetic Theory (McGraw-Hill, New York, 1941).

[8] a) Wilcox, C. H., Debye potentials, J. Math, and Mech., 6, pp. 167-201

(March 1957).

b) The diffraction of radiation by inhomogeneous lenses, Lockheed Aircraft Corp., Sunnyvale, California, Rept. LMSD5126 (Aug. 1958).

[9] Morse, P. M., and Feshbach, Herman, Methods of Theoretical Physics

(McGraw-Hill, New York, 1953).

[10] Collin, R. E., On the incompletelness of E and H modes in waveguides.

Can. J. Physics, 5]^, pp. 1135-1140 (June 1973).

[II] Kaplan, Wilfred, Advanced Calculus ( Add i son-Wesley, Reading, Massachusetts, 1952).

62
Kellogg, 0. D., Foundations of Potential Theory (Frederick

Ungar, New York, 1929).

Argence, Emile, and Kahan, Theo, Theory of Waveguides and Cavity

Resonators (Hart, New York, Ch. 6, 1968).

[14] Phillips, H. B., Vector Analysis (Wiley, Sees. 55-57, New York, 1933)

[15] a) Lorrain, P., and Corson, D. R. , Electromagnetic Fields and Waves

(Freeman, 2nd Ed., Ch. 3, San Francisco, 1970).

b) Robinson, F. N. H., Macroscopic Electromagnetism (Pergamon, Ch. 13,

Oxford, 1973).

63
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U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

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4, TITLE AND SUBTITLE

A DIRECT APPROACH TO THE DERIVATION OF ELECTRIC

DYADIC GREEN'S FUNCTIONS

3. Recipient's Accession No.

5. Publication Date January 1978

6. Performing Organization Code

276.05

7. AUTHOR(S) 8. Performing Organ. Report No Arthur D. Yaghjian

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

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15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

A Straightforward approach that does not require delta-function

techniques is used to derive a generalized electric dyadic Green's function

which remains valid within the source region. Although the electric field

expressed by the dyadic Green's function proves to be unique, the exact

form of the dyadic itself depends, in the source regions, upon the geometry

of its "principal volume." The dependence on principal volume is determined

explicitly, and the different Green's dyadics derived by a number of previous

authors are shown to emerge merely through the appropriate choice for the

principal volume. Moreover, delta-function techniques, which by themselves

are shown to be inadequate to extract uniquely the proper electric dyadic

Green's function in the source region, can be supplemented by a very

simple procedure to yield unambiguously the correct Green's function and

associated fields.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) fl "' Dyadic Green's function; electromagnetic theory; principal values; source

region.

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This monthly magazine is published to inform scientists, engineers, businessmen, industry, teachers, students, and consumers of the latest advances in science and technology,

with primary emphasis on the work at NBS. The magazine

highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution

abatement, health and safety, and consumer product per- formance. In addition, it reports the results of Bureau pro- grams in measurement standards and techniques, properties

of matter and materials, engineering standards and services, instrumentation, and automatic data processing.

Annual subscription: Domestic, $12.50; Foreign $15.65.

NONPERIODiCALS

'Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. :Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations, and regulatory bodies.

Special Publications—Include proceedings of conferences

I sponsored by NBS, NBS annual reports, and other special

I publications appropriate to this grouping such as wall charts,

j pocket cards, and bibliographies.

j Applied Mathematics Series—Mathematical tables, mani uals, and studies of special interest to physicists, engineers,

chemists, biologists, mathematicians, computer programmers,

I and others engaged in scientific and technical work.

j National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of

materials, compiled from the world's literature and critically evaluated. Developed under a world-wide program co- ordinated by NBS. Program under authority of National

Standard Data Act (Public Law 90-396).

NOTE: At present the principal publication outlet for these data is the Journal of Physical and Chemical Reference Data (JPCRD) published quarterly for NBS by the American Chemical Society (ACS) and the American Institute of

Physics (AIP). Subscriptions, reprints, and supplements

available from ACS, 1155 Sixteenth St. N.W., Wash., D.C.

20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research

results, test methods, and performance criteria related to the

structural and environmental functions and the durability

and safety characteristics of building elements and systems. Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject

Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often

serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The purpose

of the standards is to establish nationally recognized require- ments for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS administers this program as a supple- ment to the activities of the private sector standardizing

organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest

to the consumer. Easily understandable language and

illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, D.C.

20402. Order following NBS publications—NBSIR's and FIPS from

the National Technical Information Services, Springfield, Va. 22161.

Federal Information Processing Standards Publications (TIPS PUB)—Publicjiions in this series collectively constitute the Federal InicrTTion Processing Standards Register.

Register serves as the oificial source of information in the Federal Government regarding standards issued by NBS

pursuant to the Federal Property and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717

(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulation?).

NBS Interagency Reports (NBSIRl—A special series of

interim or final reports on work performed by NBS for outside sponsors (both government and non-government).

In general, initial distribution is handled by the sponsor;

public distribution is by the National Technical Information

Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey bibli- ographies are issued periodically by (he Bureau:

Cryogenic Data Center Current Awareness Service. A litera- ture survey issued biweekly. Annual subscription: Domestic, $25.00; Foreign, $30.00.

Liquified Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey

issued quarterly. Annual subscription: $30.00. Send subscription orders and remittances for the preceding bibliographic

services to National Bureau of Standards, Cryogenic Data

Center (275.02) Boulder, Colorado 80302.
U.S. OEPARTMEMT OF COMMERCE

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Washington, D.C. S0S34

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