L. B Felsen. # Relation between a class of two-dimensional and three-dimensional diffraction problems online

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INSTITUTE C?

AFCRC-TN.59.130 ^^^^^

ASTIA DOCUMENT NO. AD 210493

15 Waverly Pbcai;, New York 3, H Y,

yC^^ y^ i^ NEW YORK UNIVERSITY WDV 1 S 1^

So I isj Tl ^ Institute of Mathematical Sciences

"^ V-/ X Division of Electromagnetic Research

RESEARCH REPORT NO. EM-120

Relation Between a Class of Two-Dimensional

and Three-Dimensional Diffraction Problems

L. B. FELSEN and S. N. KARP

Contract No. AF 19(604)1717

JANUARY, 1959

AFCRC-TN-59-130

ASTIA Document No. AD-210l

(e) Infinite parallel planes

t

.-J

(f) Array of strips

Fig. 2 - Equivalent configurations.

- k -

to a perfectly reflecting infinite plane. It is evident that any surface with

rotational symmetry about the Z-axis of Fig. l(a), and described by the equa-

tion f(p,Z) = 0, is mapped by this transformation into the two-dimensional

configuration f(y,Z) = in the presence of the infinite plane at y = 0.

Some special structures in this category are listed in Fig. 2. We shall show

how we can generate solutions for such ring source excited three-dimensional

configurations involving am infinite half plane, from the knowledge of the two-

dimens i onal re s\ilt s .

As a further application it will be shown how the above-mentioned

transformation can be employed to construct the electromagnetic field due to

an- arbitrary source distribution in the presence of a perfectly conducting

half -plane. The construction is carried out explicitly in terms of the solu-

tions of the corresponding scalar Dirichlet and Neiomann problems. The prob-

12 3

lem has been solved previously by Heins ;, Senior , and Vandakurov , for

various dipole excitations through the use of methods which differ from each

other and from the present one. The procedure we employ exhibits explicitly

the modifications necessary in order to convert the scalar solutions into

vector solutions. Thus, we start with the scalar wave functions corresponding

to excitations which are Cartesian components of the arbitrarily prescribed

vector excitation and which are assumed to be known. The vector solution is

then constmcted from these scalar solutions.

II. Relation Between Three-and Two-dimensional Problems

A. Scalar Problems

We define the even and odd Green's functions G (r; r', 0') and

G (r: r' , 6') appropriate to the ring source excited half -plane in Fig. l(a)

o

as

p rcos(72)^

G (r; r', 6') = r'sin 9' / J U (r,r')d*', r= {r,e,^), (l)

o ^o Lsin(*72)J o~~

where (Z? (Â£;,Â£') and sC' (]Â£_,Â£_') are the three-dimensional Green's fimctions

satisfying the inhomogeneous wave equation

\ r sm ott> ' o T' sin

V Q = â€”;r -r- r tâ€” + â€” ,^â€” sm 6 -rrr- , (2a)

r0 2 or dr 2 . ^ o0 d0 ^ ^ ^

r r sm

in the domain 0

AFCRC-TN.59.130 ^^^^^

ASTIA DOCUMENT NO. AD 210493

15 Waverly Pbcai;, New York 3, H Y,

yC^^ y^ i^ NEW YORK UNIVERSITY WDV 1 S 1^

So I isj Tl ^ Institute of Mathematical Sciences

"^ V-/ X Division of Electromagnetic Research

RESEARCH REPORT NO. EM-120

Relation Between a Class of Two-Dimensional

and Three-Dimensional Diffraction Problems

L. B. FELSEN and S. N. KARP

Contract No. AF 19(604)1717

JANUARY, 1959

AFCRC-TN-59-130

ASTIA Document No. AD-210l

(e) Infinite parallel planes

t

.-J

(f) Array of strips

Fig. 2 - Equivalent configurations.

- k -

to a perfectly reflecting infinite plane. It is evident that any surface with

rotational symmetry about the Z-axis of Fig. l(a), and described by the equa-

tion f(p,Z) = 0, is mapped by this transformation into the two-dimensional

configuration f(y,Z) = in the presence of the infinite plane at y = 0.

Some special structures in this category are listed in Fig. 2. We shall show

how we can generate solutions for such ring source excited three-dimensional

configurations involving am infinite half plane, from the knowledge of the two-

dimens i onal re s\ilt s .

As a further application it will be shown how the above-mentioned

transformation can be employed to construct the electromagnetic field due to

an- arbitrary source distribution in the presence of a perfectly conducting

half -plane. The construction is carried out explicitly in terms of the solu-

tions of the corresponding scalar Dirichlet and Neiomann problems. The prob-

12 3

lem has been solved previously by Heins ;, Senior , and Vandakurov , for

various dipole excitations through the use of methods which differ from each

other and from the present one. The procedure we employ exhibits explicitly

the modifications necessary in order to convert the scalar solutions into

vector solutions. Thus, we start with the scalar wave functions corresponding

to excitations which are Cartesian components of the arbitrarily prescribed

vector excitation and which are assumed to be known. The vector solution is

then constmcted from these scalar solutions.

II. Relation Between Three-and Two-dimensional Problems

A. Scalar Problems

We define the even and odd Green's functions G (r; r', 0') and

G (r: r' , 6') appropriate to the ring source excited half -plane in Fig. l(a)

o

as

p rcos(72)^

G (r; r', 6') = r'sin 9' / J U (r,r')d*', r= {r,e,^), (l)

o ^o Lsin(*72)J o~~

where (Z? (Â£;,Â£') and sC' (]Â£_,Â£_') are the three-dimensional Green's fimctions

satisfying the inhomogeneous wave equation

\ r sm ott> ' o T' sin

V Q = â€”;r -r- r tâ€” + â€” ,^â€” sm 6 -rrr- , (2a)

r0 2 or dr 2 . ^ o0 d0 ^ ^ ^

r r sm

in the domain 0

Online Library → L. B Felsen → Relation between a class of two-dimensional and three-dimensional diffraction problems → online text (page 1 of 3)