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AFCRC-TN-60-196

NEW YORK UNIVLKSITY

INSTITUTE OF MATI IEMATICAL SCIENCES

LIBRARY

25 Waverly P|. lce , New York 3, N Y.

a?\ tt r% NEW YORK UNIVERSITY

m

Institute of Mathematical Sciences

Division of Electromagnetic Research

RESEARCH REPORT No. EM-146

Scattering of a Surface Wave by a Discontinuity

in the Surface Reactance on a Right Angled Wedge

FRANK C. KARAL, JR. and SAMUEL N. KARP

Contract No. AF 19(604)5238

FEBRUARY, 1 960

AFCRC-TN- 60-196

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM-1^6

SCATTERING OF A SURFACE WAVE BY A DISCONTINUITY IN THE

SURFACE REACTANCE ON A RIGHT ANGLED WEDGE

Frank C. Karal, Jr.

and

Samuel N. Karp

Frank C. Karal, Jr.

â– k,

frA/o

Samuel N. Karp

~hu

/^ Y/juwW. fa**

Morris Kline Dr. Werner Gerbes

Project Director Contract Monitor

The research reported in this document has been sponsored

by the Electronics Research Directorate of the Air Force

Cambridge Research Center, Air Research and Development

Command, under Contract No. AF 19(60^)5238.

Requests for additional copies by agencies of tha Department

of Defense, their contractors, and other Government agencies

should be directed to the:

ARMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

Department of Defense contractors must be established for ASTIA

services or have their 'need-to-know' certified by the cognizant

military agency of their project or contract.

All other persons and organizations should apply to the:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

- i r

Abstract

We consider the electromagnetic field that arises when a surface wave

travels along the front face of a right angled wedge and is scattered by the

tip. An impedance type boundary condition is prescribed on the front face of

the wedge and a simple boundary condition (that for perfect reflection) is

prescribed on the other. The impedance boundary condition is such that sur-

face waves are generated. We present an exact mathematical solution in

elementary form for this problem and find the amplitude of the reflected

surface wave. We also give a simple representation for the radiated far field

amplitude. The important special cases of small and large reactance are dis-

cussed. For large reactance the wave is closely bound to the surface, which may

then be regarded as a surf ace- wave -guide. In this limit there is complete

reflection of the surface wave, with a change in phase of it/3, for our geometry.

The question is raised as to what change of phase ought to be assigned, in

general, so as to describe reflection at an 'open end' of a surface-wave- guide.

Table of Contents

1 - â– - c '

1

I. Introduction

II. Solution 5

III. Limiting cases of the reflected surface wave 13

IV. Radiated far field l6

V. Compilation of results 19

References

I. Introduction.

The purpose of this paper is to study the electromagnetic field that arises

when a surface wave travels along the front face of a right angled wedge and is

scattered by the tip. An impedance type boundary condition is prescribed on the

front face of the wedge and a simple boundary condition (that for perfect reflec-

tion) is prescribed on the other. The impedance boundary condition is

such that surface waves are generated. We present an exact mathematical

solution for this problem and find the amplitude of the reflected surface

wave. We also give a simple exact formula for the radiated far field amplitude.

The important special cases of small and large reactance are discussed.

The boundary conditions prescribed in this paper are given by

|jp = , y = 0, x S (1-1)

^-\u=0, x = 0, y Â« (1-2)

where u is the z-component of the magnetic vector, x and y are the usual

cartesian rectangular coordinates and X is a constant characteristic of

the wedge surface. The value of X is given by

X = iwâ‚¬Z = iwe(R - iX) (1-3)

where e is the permittivity of free space, w is the angular frequency and

Z, R and X are the impedance, resistance and reactance of the surface,

respectively. The impedance boundary condition given by (l-2) is an ideal-

ization, but is a good approximation to several important physical config-

urations. For example, the plane interface separating two homogeneous

media, one of which possesses a very high (but not infinite) conductivity,

can be approximated by an impedance boundary condition of the type

given by (1-2). Here R and X are positive in sign, small in magnitude

and approximately equal. For illustrations of this boundary condition see

GrunbergL J, Bazer and Karp *- ^ , Fernando and Barlow L -I and Felsen^-I.

Another example is a corrugated perfect conductor with rectangular grooves

provided the groove spacing is small compared to the wavelength. Still

another example is a dielectric-coated ground plane with a large dielectric

constant. For corrugated or dielectric coated surfaces R and X are positive

in sign, R is much smaller than X and hence Z is almost purely reactive.

See Cullen'-^ Felsen '-'-', Barlow and Karbowiak L J and Kay"- '-I. We note

that when X is positive, equation (1-3) implies that

Re X > (1-M

This condition is employed in our mathematical work. The condition R Â« X

is of course the most important special case insofar as surface waves

are concerned since there is negligible attenuation along the surface.

The problem treated in this paper is not separable because of the

mixed boundary condition. This difficulty can be overcome however by

introducing an auxiliary function which is a linear combination of the

magnetic field and its cartesian derivatives. The auxiliary function is

chosen in such a way that it satisfies the wave equation and simple

homogeneous boundary conditions on both wedge surfaces. Once the auxiliary

function is found, the original field can be determined by solving an

auxiliary partial differential equation. This idea is due to Stoker L

and Levy'- -I who studied problems in water wave theory. It has since been

employed by the authors L 1:L J " L l6 J for solving problems in diffraction theory.

Once the boundary conditions have been simplified by using this idea, the

development in this paper differs considerably from that occurring in water

wave theory since the conditions at infinity and at the edge of the wedge are

quite different.

Section II contains an exact solution of the problem state in the first

paragraph. In Section III we obtain simplified formulas for the magnitudes and

phases of the reflected surface waves for the special cases of small and large

reactance. In Section IV we use a special method to obtain the magnitude of the

radiated far field. The same special limiting cases are also discussed. When

the reactance is small, the ratio of the magnitude of the reflected surface wave

to the incident surface wave is Q./5 n/^Kv - ) > where X = weX. The ratio of the

magnitude of the radiated far field to that of the incident surface wave is

x/2/itkr [cos i e/v/5 cos Â©] [l + (x/k cos 0) 2 ] 1 ' 2 . In the limit of zero

3

reactance the reflected surface wave is zero, while the radiated far field

acquires a shadow line. This is to be expected since the incident excitation

for this limiting case becomes an incident plane wave. Hence for the case of

a surface with a small reactance, most of the energy is radiated and very little

energy is reflected in the form of a surface wave. On the other hand, the solution

behaves quite differently when the reactance is large. Then the ratio of the

magnitude of the reflected surface wave to the incident surface wave is unity.

The ratio of the magnitude of the radiated far field to the incident surface wave

vanishes as \/2/rtkr (2\/k) ' cos - 0. Hence for the case of a surface with a

large reactance, there is very little radiation and most of the energy is closely

bound to the surface in the form of incident and reflected surface waves. Thus,

for large reactance the structure behaves like an open ended wave guide. The

change of phase at the open end is it/3 - The results, and a plot at the

pattern function for several values of the reactance, are compiled in

section V.

I ! 1

In a recent paper Maliuzhinets L J states a solution of our problem

for a plane wave incidence on a wedge of arbitrary angle when the

wedge supports surface waves. His method is entirely different and is

[23]

probably related to the method used by Peters 1 - J in water wave theory. In

Maliuzhinets ' very brief announcement it is pointed out that the complicated

general solution simplifies for certain wedge angles of which our wedge angle

is one. But these special cases are not studied. Also, the limiting cases of

small and large reactance for the reflected surface waves and for the radiated

far field are not considered, radiation patterns are not given and there is no

discussion of the results obtained. One should be able to arrive at our results

and conclusions by starting with his formulas and performing the necessary limit-

ing processes and algebraic manipulations. But the use of the procedure presented

in this paper seems much simpler to use as a basis for our analyses.

5 -

II. Solution.

Consider a right angled wedge defined by the surfaces y = 0, x >

and x = 0, y < 0, as shown in Figure I. In the angular region 019? 3Â«/2

we assume that we have free space. An incident surface wave u g , whose

magnetic vector is linearly polarized in the z-direction, travels in the

positive y direction toward the tip of the wedge where it is scattered.

The form of the incident surface wave is given by

u = exp Xx + i /k 2 + X 2 y x = 0, y < (2-1)

where the magnitude of the surface wave is unity and the time dependence

e is omitted for convenience. It is easily shown that (2-1) satisfies

the wave equation and the impedance boundary condition (1-2) on the front

of the wedge. The boundary conditions, as mentioned in the Introduction,

are given by

y = 0, x > (2-2)

^ " Xu = Â° x = 0, y < (2-3)

We also require that the scattered waves at infinity be outgoing and that

the energy of the elctromagnetic field be finite. We wish to solve Maxwell's

equations subject to the prescribed conditions and obtain the amplitude of

the reflected surface wave.

The time dependent form of Maxwell's equations is

curl H = -iu)â‚¬ E

curl E = i(ju H

(2-4)

- 5a -

Incident

Surface Wave

/////////////////

Perfectly Conducting Surface

/ Impedance Boundary

/ Condition Prescribed

/ on Wedge Surface

'/

Figure 1

- 6 -

where E and H are the electric and magnetic field intensities, and e and u

are the permittivity and magnetic permeability of free space. Because of

the geometry, the field produced is independent of z and hence the field

and

H = H

x y

- bE

E = _ _i_ z

x iue oy

, c>H

y lue ox.

The field component H = u satisfies the equation

(V 2 + k 2 )u =

(2-5)

(2-6)

(2-7]

where v is the rectangular Laplacian and k is the propagation constant of

free space. Therefore the mathematical problem reduces to solving the

homogeneous wave equation (2-7) subject to the mixed boundary conditions (2-2)

and (2-3) and an incident surface wave of the form given by (2-1). In addition

to these requirements, we require that the far field be outgoing and that the

electromagnetic field be finite everywhere.

Let us make the substitution

v = (^ - X)u

Then v satisfies the wave equation

(V 2 + k 2 )v =

subject to the simpler boundary conditions

(2-8)

(2.9)

= y = o, x > o

(2-10)

v = x = 0, y <

and the requirement that the scattered waves at infinity "be outgoing. In

the transformed problem there is no incident surface wave, that is, v =0.

s

If the problem for v can be solved, then we can obtain the solution for the

problem involving u since (2-8) can be integrated. One particular solution

of (2-8) is

u p (x,y) = -e* / e" X| v(Â£,y)dÂ£ (2-11)

x

It is important to point out that the function v does not have to be finite

at the origin. Thus we should first introduce all radiating solutions, no

matter how singular at the origin, that satisfy the appropriate boundary

conditions for v. Out of this class of solutions we should finally select,

only those that yield an everywhere finite value of the elctromagnetic field

in the original physical problem involving u. Now, if we employ the method

of separation of variables and use conditions (2-10), we find that the most

general representation for v is of the form ]jT[ a H^y / (kr) cos 2n+l/3 0.

Since v cannot be too singular, the only admissible function of this class

is

v(r,0) = C-jH^ (kr) cos | (2-12)

In the above expression r and are the usual polar coordinates and

Â§ Â§ 3^/2. The constant c^ has yet to be determined. Equation (2-12)

- 8

obviously satisfies the wave equation (2-9), the boundary conditions (2-10)

and yields outgoing scattered waves at infinity. If we substitute (2-12)

into (2-11) we obtain for the particular solution

u p (x,y) = -(

Xx

f Â°-Â« 4}

x/ > (kr) cos i d|

(2-13)

Note that in (2-11) and (2-13) we integrate into the wedge when y is negative.

This is possible because the wave function v vanishes on the y axis and hence

can be continued analytically to positive values of x. The function u however

does not satisfy all the conditions of the problem. This is because it is not

a wave function in regions including the negative x axis since the y derivative

by (2-1) is not a wave function either since it is obviously discontinuous across

the line y = 0, x Â§ 0. Furthermore the sum of u and u is still not a wave

s p

function. In order to obtain a wave function it is necessary to add another

solution of the wave equation as follows:

c 2 exp[_Xx - i /k 2 + X 2 yj , y <

u = u + u + /

Q Tl \

y >

(2-1*0

xx r -x? Tr (i) ,. v 1 _ ,,

u = -c e e H iA ( kr ) cos "5 e d Â£

x

exp Xx + i \/k 2 + ^ y + c 2 exp Xx - i \/k 2 + X 2 y , y < C

(2-15)

, y >

The introduction of the additional term with the unknown constant c now

2

makes it possible to satisfy the continuity conditions involving u and

the y-derivative of u. We note that the form of the additional term is that

of a reflected surface wave.

We must now determine the unknown constants c and c . We do this

by requiring that

(i) u be continuous across the line y = 0, x <

(II) du/oy be continuous across the line y = 0, x <

Expressing the above as jump conditions we have

(I) [u] = u(x,0 + )-u(x,0") =0, x <

| (x,0 + )- I (x,0-) -0, x (k|||) cosÂ±(*)dg - Cl e Xx f e" X Â«H^(k | ? | )cos |(0)d| J

X o

o Â°Â°

|- C;L e Xx f e-^H^(kU|)cosi(n)d| - c^ f e" X ^(k |g | )cosi(2*)dÂ§

X . o

)â–

Xx Xx ,

e + c_ e ^ =

| =1 I l/5 â™¦ c a . -1 (2-2D

where

I

(k,X) = ^ e" X| H^ (kU|)d|. (2-22)

o

We next impose the second jump condition (il) and obtain after some cancellation

00

joj- l-c^X J e -^H^(kU|)sin|idi (2-23)

o

-c^e^ e Xx f e- X ^H^(k|||)sin^ d 6 + i ^? e* X - i /lS? c 2 e Xx j = (

11

s

2rt

c, < -XI, ,_ + ke 5 I,

^rp c i " A1 i/5 * Ke V 3

Solving (2-21) and (2-21+) for c and c we obtain

3 ^

A câ€ž = â€¢Â£ I. ,, + â€” â€”

V3 2 ^? ^

.2jt

-X L /7 + ke ; I

2/3

where

v/3 f iâ€”

A = * V " ~ "^T 2 ""^ + te ' ' 2 /5

The integral expression given by (2-22) can be evaluated explicitly

and has the value

( \A 2 + k 2 - d v

i A 2 i 2 ,\ 2v n

( y/X +k + X)

: v

[19]

See Magnus and Oberhettinger L J . Further simplification follows if we make

the substitutions

1 / 2 2

sin = + - \/X +k

s k v

cos = + i

Equation (2-28) then reduces to the compact form

2 sin v6

[ â€ž 2jb Â§

v k sin vit sin 0â€ž

- 12

If we now substitute for I , and I / in equations (2-25) - (2-27) and

simplify we obtain

sin

+i*

and

&3 TTf- < 2 ^>

cos - (e s -n)

cos i (6 -hr)

â€” ^â€” (2-53)

cos - (e g -n)

-13-

III. Limiting cases of the reflected surface wave.

It is of interest to determine the values of

cases of small X and large X. In the Introduction we noted that X = iu>eZ =

Itde(R-iX) where R > and X > 0. For corrugated or dielectric coated surfaces

R Â« X and hence X a# -KJeX. This is the most favorable case for surface waves.

In our treatment here we assume that X is real and treat the special cases

when X -* and X =* t] traveling in the positive y-direction. Under such an excita-

tion there is a scattered wave. In fact we shall find the form of the

scattered far field in the next section for this limiting case. It reduces,

f2ll

as it must, to the results given by Reiche L - 1 for the diffraction of a

plane incident wave by a perfectly conducting right angled wedge. Hence

for the case of a surface with a small reactance, most of the energy is

radiated and very little energy is reflected in the form of a surface wave.

The energy, therefore, is loosely bound to the surface and most of it is

scattered in the form of radiation.

When X/k is large we find that

rt , , 2X

P - i log â€”

Substituting into (2-32) and (2-33) we obtain

'1 2 v k

*-^

.2*

Thus the magnitude of the reflected surface wave approaches the value of

the incident surface wave as (2X/k) approaches infinity. The value of c.

- 15-

approaches infinity as (2l/k) approaches infinity. This does not mean,

however, that the scattered field approaches infinity in this limit. In

fact in the next section we shall show that the scattered far field approaches

zero as (2l/k) approaches infinity. Hence for the case of a surface with

a large reactance, most of the energy is reflected in the form of a surface

wave and the radiated energy is small. The energy is confined close to the

surface and there is very little radiation.

It is of interest to compare the magnitude of the reflected surface

wave for a right angled wedge with that for a plane surface with an

impedance boundary condition prescribed on one half of its surface and

a perfectly reflecting boundary condition prescribed on the other. This

problem has been solved by KayL -1 . In our notation we find that the

magnitude of the reflected surface wave for a plane surface with small

reactance is (l/8)(2X/k) . When the reactance is large the magnitude of the

reflected surface wave approaches the magnitude of the incident surface

wave. Thus in the case of a surface with a large reactance, the amplitudes

of the reflected surface waves are the same. We conjecture that this is

true for any wedge regardless of the angle. In the case of a surface with

small reactance, the amplitude of the reflected surface wave for a wedge

approaches zero as the first power of (2X/k), while the reflected amplitude

for a discontinuity of reactance on a plane surface approaches zero as the

square of (2X/k). Thus the reflected amplitude of the surface wave for

a wedge is larger by an order of magnitude.

- 16 -

[V. Radiated far field.

We now determine the radiated far field that arises when an incident

surface wave is scattered by the tip of a right angled wedge. From (2-8) and

[2-12) we have

v - (S " X)Â« (4-1)

V = c

1 h[^ (kr) cos i 6 (4-2)

rhen kr is large (4-2) becomes

v = c i v^f? ex P&( kr " if)] cos \ Q (^-5)

ie expect the far field of the z-component of the magnetic vector u to have

l similar form. Hence

itel ikr {k . k)

fhere t(0) is unknown. Now

5 " -=0, 8 I - i Bin â€¢ Â£ (K-5)

-1/2

Substituting into (4-1) and keeping terms of order r ' , we obtain

-iff cos i 8

rherefore

- 17 _

cos â€” e

U(r ' 9) = \/2$te ^ exp[i(kr +{)"] (k-8)

V2 * kr cos |(e s - Â«) [cos e + cos ej L 4 J

where we have used (2-30) and (2-32).

When X/k is small we find that the radiated far field becomes

u(r,0) = ,/^ ^ [l + â€” ^ -1 exp|i(kr + Â£ )~| (4-9)

VÂ«kr yj cos e L ik cos ej *|_ V ^J

Note that in the limit as x/k approaches zero the radiated far field approaches

a definite limit. This is to be expected since the incident excitation for

this limiting case becomes an incident plane wave of the form exp [iky - iurtTj

traveling in the positive y-direction. Under such an excitation there is a

fell

scattered far field. Reiche L - 1 has solved the problem of an incident plane

wave diffracted by a perfectly conducting wedge. When the direction of the

incident plane wave is parallel to and in the same direction as our incident

surface wave, the results are the same in the limit as X/k approached zero.

When X/k is large we find that the radiated far field becomes

u(r,6) = J^ (f^)" l/5 cos i 9 exp[i(kr - ^)] (4-10)

In the limit as 2X/k approaches infinity the radiated far field approaches

zero. In the last Section we showed that the magnitude of the reflected

surface wave approaches the value of the incident surface wave as (2X/k)

approaches infinity. Hence when a surface possesses a very large surface

impedance there is very little radiation and all energy is closely bound to

the surface in the form of incident and reflected surface waves.

r

It is of interest to compare the magnitude of the radiated far field

for a right angled wedge with that for a plane surface with an impedance

boundary condition prescribed on one half of its surface and a perfectly-

reflecting boundary condition prescribed on the other. This problem has

rifi

been solved by Kay L u . In our notation we find that the magnitude of the

radiated far field for a plane surface with a small discontinuity in

reactance is given by \/2/jrkr(lA)(2X/k) [l-cos 6] . When the reactance

is large the radiated far field is given by v/2/itkr (2X/k) _1 sineQl - cose]" 1 ' 2 .

We note that when the reactance approaches zero, the radiated far field approaches

zero. This is to be expected since the plane surface no longer has a dis-

continuity in its properties. The boundary condition on the plane surface

is the same from - â– Â» to +

NEW YORK UNIVLKSITY

INSTITUTE OF MATI IEMATICAL SCIENCES

LIBRARY

25 Waverly P|. lce , New York 3, N Y.

a?\ tt r% NEW YORK UNIVERSITY

m

Institute of Mathematical Sciences

Division of Electromagnetic Research

RESEARCH REPORT No. EM-146

Scattering of a Surface Wave by a Discontinuity

in the Surface Reactance on a Right Angled Wedge

FRANK C. KARAL, JR. and SAMUEL N. KARP

Contract No. AF 19(604)5238

FEBRUARY, 1 960

AFCRC-TN- 60-196

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. EM-1^6

SCATTERING OF A SURFACE WAVE BY A DISCONTINUITY IN THE

SURFACE REACTANCE ON A RIGHT ANGLED WEDGE

Frank C. Karal, Jr.

and

Samuel N. Karp

Frank C. Karal, Jr.

â– k,

frA/o

Samuel N. Karp

~hu

/^ Y/juwW. fa**

Morris Kline Dr. Werner Gerbes

Project Director Contract Monitor

The research reported in this document has been sponsored

by the Electronics Research Directorate of the Air Force

Cambridge Research Center, Air Research and Development

Command, under Contract No. AF 19(60^)5238.

Requests for additional copies by agencies of tha Department

of Defense, their contractors, and other Government agencies

should be directed to the:

ARMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

Department of Defense contractors must be established for ASTIA

services or have their 'need-to-know' certified by the cognizant

military agency of their project or contract.

All other persons and organizations should apply to the:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

- i r

Abstract

We consider the electromagnetic field that arises when a surface wave

travels along the front face of a right angled wedge and is scattered by the

tip. An impedance type boundary condition is prescribed on the front face of

the wedge and a simple boundary condition (that for perfect reflection) is

prescribed on the other. The impedance boundary condition is such that sur-

face waves are generated. We present an exact mathematical solution in

elementary form for this problem and find the amplitude of the reflected

surface wave. We also give a simple representation for the radiated far field

amplitude. The important special cases of small and large reactance are dis-

cussed. For large reactance the wave is closely bound to the surface, which may

then be regarded as a surf ace- wave -guide. In this limit there is complete

reflection of the surface wave, with a change in phase of it/3, for our geometry.

The question is raised as to what change of phase ought to be assigned, in

general, so as to describe reflection at an 'open end' of a surface-wave- guide.

Table of Contents

1 - â– - c '

1

I. Introduction

II. Solution 5

III. Limiting cases of the reflected surface wave 13

IV. Radiated far field l6

V. Compilation of results 19

References

I. Introduction.

The purpose of this paper is to study the electromagnetic field that arises

when a surface wave travels along the front face of a right angled wedge and is

scattered by the tip. An impedance type boundary condition is prescribed on the

front face of the wedge and a simple boundary condition (that for perfect reflec-

tion) is prescribed on the other. The impedance boundary condition is

such that surface waves are generated. We present an exact mathematical

solution for this problem and find the amplitude of the reflected surface

wave. We also give a simple exact formula for the radiated far field amplitude.

The important special cases of small and large reactance are discussed.

The boundary conditions prescribed in this paper are given by

|jp = , y = 0, x S (1-1)

^-\u=0, x = 0, y Â« (1-2)

where u is the z-component of the magnetic vector, x and y are the usual

cartesian rectangular coordinates and X is a constant characteristic of

the wedge surface. The value of X is given by

X = iwâ‚¬Z = iwe(R - iX) (1-3)

where e is the permittivity of free space, w is the angular frequency and

Z, R and X are the impedance, resistance and reactance of the surface,

respectively. The impedance boundary condition given by (l-2) is an ideal-

ization, but is a good approximation to several important physical config-

urations. For example, the plane interface separating two homogeneous

media, one of which possesses a very high (but not infinite) conductivity,

can be approximated by an impedance boundary condition of the type

given by (1-2). Here R and X are positive in sign, small in magnitude

and approximately equal. For illustrations of this boundary condition see

GrunbergL J, Bazer and Karp *- ^ , Fernando and Barlow L -I and Felsen^-I.

Another example is a corrugated perfect conductor with rectangular grooves

provided the groove spacing is small compared to the wavelength. Still

another example is a dielectric-coated ground plane with a large dielectric

constant. For corrugated or dielectric coated surfaces R and X are positive

in sign, R is much smaller than X and hence Z is almost purely reactive.

See Cullen'-^ Felsen '-'-', Barlow and Karbowiak L J and Kay"- '-I. We note

that when X is positive, equation (1-3) implies that

Re X > (1-M

This condition is employed in our mathematical work. The condition R Â« X

is of course the most important special case insofar as surface waves

are concerned since there is negligible attenuation along the surface.

The problem treated in this paper is not separable because of the

mixed boundary condition. This difficulty can be overcome however by

introducing an auxiliary function which is a linear combination of the

magnetic field and its cartesian derivatives. The auxiliary function is

chosen in such a way that it satisfies the wave equation and simple

homogeneous boundary conditions on both wedge surfaces. Once the auxiliary

function is found, the original field can be determined by solving an

auxiliary partial differential equation. This idea is due to Stoker L

and Levy'- -I who studied problems in water wave theory. It has since been

employed by the authors L 1:L J " L l6 J for solving problems in diffraction theory.

Once the boundary conditions have been simplified by using this idea, the

development in this paper differs considerably from that occurring in water

wave theory since the conditions at infinity and at the edge of the wedge are

quite different.

Section II contains an exact solution of the problem state in the first

paragraph. In Section III we obtain simplified formulas for the magnitudes and

phases of the reflected surface waves for the special cases of small and large

reactance. In Section IV we use a special method to obtain the magnitude of the

radiated far field. The same special limiting cases are also discussed. When

the reactance is small, the ratio of the magnitude of the reflected surface wave

to the incident surface wave is Q./5 n/^Kv - ) > where X = weX. The ratio of the

magnitude of the radiated far field to that of the incident surface wave is

x/2/itkr [cos i e/v/5 cos Â©] [l + (x/k cos 0) 2 ] 1 ' 2 . In the limit of zero

3

reactance the reflected surface wave is zero, while the radiated far field

acquires a shadow line. This is to be expected since the incident excitation

for this limiting case becomes an incident plane wave. Hence for the case of

a surface with a small reactance, most of the energy is radiated and very little

energy is reflected in the form of a surface wave. On the other hand, the solution

behaves quite differently when the reactance is large. Then the ratio of the

magnitude of the reflected surface wave to the incident surface wave is unity.

The ratio of the magnitude of the radiated far field to the incident surface wave

vanishes as \/2/rtkr (2\/k) ' cos - 0. Hence for the case of a surface with a

large reactance, there is very little radiation and most of the energy is closely

bound to the surface in the form of incident and reflected surface waves. Thus,

for large reactance the structure behaves like an open ended wave guide. The

change of phase at the open end is it/3 - The results, and a plot at the

pattern function for several values of the reactance, are compiled in

section V.

I ! 1

In a recent paper Maliuzhinets L J states a solution of our problem

for a plane wave incidence on a wedge of arbitrary angle when the

wedge supports surface waves. His method is entirely different and is

[23]

probably related to the method used by Peters 1 - J in water wave theory. In

Maliuzhinets ' very brief announcement it is pointed out that the complicated

general solution simplifies for certain wedge angles of which our wedge angle

is one. But these special cases are not studied. Also, the limiting cases of

small and large reactance for the reflected surface waves and for the radiated

far field are not considered, radiation patterns are not given and there is no

discussion of the results obtained. One should be able to arrive at our results

and conclusions by starting with his formulas and performing the necessary limit-

ing processes and algebraic manipulations. But the use of the procedure presented

in this paper seems much simpler to use as a basis for our analyses.

5 -

II. Solution.

Consider a right angled wedge defined by the surfaces y = 0, x >

and x = 0, y < 0, as shown in Figure I. In the angular region 019? 3Â«/2

we assume that we have free space. An incident surface wave u g , whose

magnetic vector is linearly polarized in the z-direction, travels in the

positive y direction toward the tip of the wedge where it is scattered.

The form of the incident surface wave is given by

u = exp Xx + i /k 2 + X 2 y x = 0, y < (2-1)

where the magnitude of the surface wave is unity and the time dependence

e is omitted for convenience. It is easily shown that (2-1) satisfies

the wave equation and the impedance boundary condition (1-2) on the front

of the wedge. The boundary conditions, as mentioned in the Introduction,

are given by

y = 0, x > (2-2)

^ " Xu = Â° x = 0, y < (2-3)

We also require that the scattered waves at infinity be outgoing and that

the energy of the elctromagnetic field be finite. We wish to solve Maxwell's

equations subject to the prescribed conditions and obtain the amplitude of

the reflected surface wave.

The time dependent form of Maxwell's equations is

curl H = -iu)â‚¬ E

curl E = i(ju H

(2-4)

- 5a -

Incident

Surface Wave

/////////////////

Perfectly Conducting Surface

/ Impedance Boundary

/ Condition Prescribed

/ on Wedge Surface

'/

Figure 1

- 6 -

where E and H are the electric and magnetic field intensities, and e and u

are the permittivity and magnetic permeability of free space. Because of

the geometry, the field produced is independent of z and hence the field

and

H = H

x y

- bE

E = _ _i_ z

x iue oy

, c>H

y lue ox.

The field component H = u satisfies the equation

(V 2 + k 2 )u =

(2-5)

(2-6)

(2-7]

where v is the rectangular Laplacian and k is the propagation constant of

free space. Therefore the mathematical problem reduces to solving the

homogeneous wave equation (2-7) subject to the mixed boundary conditions (2-2)

and (2-3) and an incident surface wave of the form given by (2-1). In addition

to these requirements, we require that the far field be outgoing and that the

electromagnetic field be finite everywhere.

Let us make the substitution

v = (^ - X)u

Then v satisfies the wave equation

(V 2 + k 2 )v =

subject to the simpler boundary conditions

(2-8)

(2.9)

= y = o, x > o

(2-10)

v = x = 0, y <

and the requirement that the scattered waves at infinity "be outgoing. In

the transformed problem there is no incident surface wave, that is, v =0.

s

If the problem for v can be solved, then we can obtain the solution for the

problem involving u since (2-8) can be integrated. One particular solution

of (2-8) is

u p (x,y) = -e* / e" X| v(Â£,y)dÂ£ (2-11)

x

It is important to point out that the function v does not have to be finite

at the origin. Thus we should first introduce all radiating solutions, no

matter how singular at the origin, that satisfy the appropriate boundary

conditions for v. Out of this class of solutions we should finally select,

only those that yield an everywhere finite value of the elctromagnetic field

in the original physical problem involving u. Now, if we employ the method

of separation of variables and use conditions (2-10), we find that the most

general representation for v is of the form ]jT[ a H^y / (kr) cos 2n+l/3 0.

Since v cannot be too singular, the only admissible function of this class

is

v(r,0) = C-jH^ (kr) cos | (2-12)

In the above expression r and are the usual polar coordinates and

Â§ Â§ 3^/2. The constant c^ has yet to be determined. Equation (2-12)

- 8

obviously satisfies the wave equation (2-9), the boundary conditions (2-10)

and yields outgoing scattered waves at infinity. If we substitute (2-12)

into (2-11) we obtain for the particular solution

u p (x,y) = -(

Xx

f Â°-Â« 4}

x/ > (kr) cos i d|

(2-13)

Note that in (2-11) and (2-13) we integrate into the wedge when y is negative.

This is possible because the wave function v vanishes on the y axis and hence

can be continued analytically to positive values of x. The function u however

does not satisfy all the conditions of the problem. This is because it is not

a wave function in regions including the negative x axis since the y derivative

by (2-1) is not a wave function either since it is obviously discontinuous across

the line y = 0, x Â§ 0. Furthermore the sum of u and u is still not a wave

s p

function. In order to obtain a wave function it is necessary to add another

solution of the wave equation as follows:

c 2 exp[_Xx - i /k 2 + X 2 yj , y <

u = u + u + /

Q Tl \

y >

(2-1*0

xx r -x? Tr (i) ,. v 1 _ ,,

u = -c e e H iA ( kr ) cos "5 e d Â£

x

exp Xx + i \/k 2 + ^ y + c 2 exp Xx - i \/k 2 + X 2 y , y < C

(2-15)

, y >

The introduction of the additional term with the unknown constant c now

2

makes it possible to satisfy the continuity conditions involving u and

the y-derivative of u. We note that the form of the additional term is that

of a reflected surface wave.

We must now determine the unknown constants c and c . We do this

by requiring that

(i) u be continuous across the line y = 0, x <

(II) du/oy be continuous across the line y = 0, x <

Expressing the above as jump conditions we have

(I) [u] = u(x,0 + )-u(x,0") =0, x <

| (x,0 + )- I (x,0-) -0, x (k|||) cosÂ±(*)dg - Cl e Xx f e" X Â«H^(k | ? | )cos |(0)d| J

X o

o Â°Â°

|- C;L e Xx f e-^H^(kU|)cosi(n)d| - c^ f e" X ^(k |g | )cosi(2*)dÂ§

X . o

)â–

Xx Xx ,

e + c_ e ^ =

| =1 I l/5 â™¦ c a . -1 (2-2D

where

I

(k,X) = ^ e" X| H^ (kU|)d|. (2-22)

o

We next impose the second jump condition (il) and obtain after some cancellation

00

joj- l-c^X J e -^H^(kU|)sin|idi (2-23)

o

-c^e^ e Xx f e- X ^H^(k|||)sin^ d 6 + i ^? e* X - i /lS? c 2 e Xx j = (

11

s

2rt

c, < -XI, ,_ + ke 5 I,

^rp c i " A1 i/5 * Ke V 3

Solving (2-21) and (2-21+) for c and c we obtain

3 ^

A câ€ž = â€¢Â£ I. ,, + â€” â€”

V3 2 ^? ^

.2jt

-X L /7 + ke ; I

2/3

where

v/3 f iâ€”

A = * V " ~ "^T 2 ""^ + te ' ' 2 /5

The integral expression given by (2-22) can be evaluated explicitly

and has the value

( \A 2 + k 2 - d v

i A 2 i 2 ,\ 2v n

( y/X +k + X)

: v

[19]

See Magnus and Oberhettinger L J . Further simplification follows if we make

the substitutions

1 / 2 2

sin = + - \/X +k

s k v

cos = + i

Equation (2-28) then reduces to the compact form

2 sin v6

[ â€ž 2jb Â§

v k sin vit sin 0â€ž

- 12

If we now substitute for I , and I / in equations (2-25) - (2-27) and

simplify we obtain

sin

+i*

and

&3 TTf- < 2 ^>

cos - (e s -n)

cos i (6 -hr)

â€” ^â€” (2-53)

cos - (e g -n)

-13-

III. Limiting cases of the reflected surface wave.

It is of interest to determine the values of

cases of small X and large X. In the Introduction we noted that X = iu>eZ =

Itde(R-iX) where R > and X > 0. For corrugated or dielectric coated surfaces

R Â« X and hence X a# -KJeX. This is the most favorable case for surface waves.

In our treatment here we assume that X is real and treat the special cases

when X -* and X =* t] traveling in the positive y-direction. Under such an excita-

tion there is a scattered wave. In fact we shall find the form of the

scattered far field in the next section for this limiting case. It reduces,

f2ll

as it must, to the results given by Reiche L - 1 for the diffraction of a

plane incident wave by a perfectly conducting right angled wedge. Hence

for the case of a surface with a small reactance, most of the energy is

radiated and very little energy is reflected in the form of a surface wave.

The energy, therefore, is loosely bound to the surface and most of it is

scattered in the form of radiation.

When X/k is large we find that

rt , , 2X

P - i log â€”

Substituting into (2-32) and (2-33) we obtain

'1 2 v k

*-^

.2*

Thus the magnitude of the reflected surface wave approaches the value of

the incident surface wave as (2X/k) approaches infinity. The value of c.

- 15-

approaches infinity as (2l/k) approaches infinity. This does not mean,

however, that the scattered field approaches infinity in this limit. In

fact in the next section we shall show that the scattered far field approaches

zero as (2l/k) approaches infinity. Hence for the case of a surface with

a large reactance, most of the energy is reflected in the form of a surface

wave and the radiated energy is small. The energy is confined close to the

surface and there is very little radiation.

It is of interest to compare the magnitude of the reflected surface

wave for a right angled wedge with that for a plane surface with an

impedance boundary condition prescribed on one half of its surface and

a perfectly reflecting boundary condition prescribed on the other. This

problem has been solved by KayL -1 . In our notation we find that the

magnitude of the reflected surface wave for a plane surface with small

reactance is (l/8)(2X/k) . When the reactance is large the magnitude of the

reflected surface wave approaches the magnitude of the incident surface

wave. Thus in the case of a surface with a large reactance, the amplitudes

of the reflected surface waves are the same. We conjecture that this is

true for any wedge regardless of the angle. In the case of a surface with

small reactance, the amplitude of the reflected surface wave for a wedge

approaches zero as the first power of (2X/k), while the reflected amplitude

for a discontinuity of reactance on a plane surface approaches zero as the

square of (2X/k). Thus the reflected amplitude of the surface wave for

a wedge is larger by an order of magnitude.

- 16 -

[V. Radiated far field.

We now determine the radiated far field that arises when an incident

surface wave is scattered by the tip of a right angled wedge. From (2-8) and

[2-12) we have

v - (S " X)Â« (4-1)

V = c

1 h[^ (kr) cos i 6 (4-2)

rhen kr is large (4-2) becomes

v = c i v^f? ex P&( kr " if)] cos \ Q (^-5)

ie expect the far field of the z-component of the magnetic vector u to have

l similar form. Hence

itel ikr {k . k)

fhere t(0) is unknown. Now

5 " -=0, 8 I - i Bin â€¢ Â£ (K-5)

-1/2

Substituting into (4-1) and keeping terms of order r ' , we obtain

-iff cos i 8

rherefore

- 17 _

cos â€” e

U(r ' 9) = \/2$te ^ exp[i(kr +{)"] (k-8)

V2 * kr cos |(e s - Â«) [cos e + cos ej L 4 J

where we have used (2-30) and (2-32).

When X/k is small we find that the radiated far field becomes

u(r,0) = ,/^ ^ [l + â€” ^ -1 exp|i(kr + Â£ )~| (4-9)

VÂ«kr yj cos e L ik cos ej *|_ V ^J

Note that in the limit as x/k approaches zero the radiated far field approaches

a definite limit. This is to be expected since the incident excitation for

this limiting case becomes an incident plane wave of the form exp [iky - iurtTj

traveling in the positive y-direction. Under such an excitation there is a

fell

scattered far field. Reiche L - 1 has solved the problem of an incident plane

wave diffracted by a perfectly conducting wedge. When the direction of the

incident plane wave is parallel to and in the same direction as our incident

surface wave, the results are the same in the limit as X/k approached zero.

When X/k is large we find that the radiated far field becomes

u(r,6) = J^ (f^)" l/5 cos i 9 exp[i(kr - ^)] (4-10)

In the limit as 2X/k approaches infinity the radiated far field approaches

zero. In the last Section we showed that the magnitude of the reflected

surface wave approaches the value of the incident surface wave as (2X/k)

approaches infinity. Hence when a surface possesses a very large surface

impedance there is very little radiation and all energy is closely bound to

the surface in the form of incident and reflected surface waves.

r

It is of interest to compare the magnitude of the radiated far field

for a right angled wedge with that for a plane surface with an impedance

boundary condition prescribed on one half of its surface and a perfectly-

reflecting boundary condition prescribed on the other. This problem has

rifi

been solved by Kay L u . In our notation we find that the magnitude of the

radiated far field for a plane surface with a small discontinuity in

reactance is given by \/2/jrkr(lA)(2X/k) [l-cos 6] . When the reactance

is large the radiated far field is given by v/2/itkr (2X/k) _1 sineQl - cose]" 1 ' 2 .

We note that when the reactance approaches zero, the radiated far field approaches

zero. This is to be expected since the plane surface no longer has a dis-

continuity in its properties. The boundary condition on the plane surface

is the same from - â– Â» to +

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