Lewin:
196Y
If
we
limited
are
interested
frequency
time
requirecl
the
line,
a wave
assume
even
noise
the
quencies
up
Eliminating
las
WO by
into
j, –fa
the
the
a
for
the
length
Discontinuity
L
of
Rice~O,ll
Problems
obtains
one-ohm
spectral
density
the
by
a
noise
of
highest
W. during
time
z
the
Nyquist
microwave
and
for
the
and
~
substituting
second
=
mean-square
and
~~~’(f
=
the
first
of these
T
(29)
At
By
eliminating
fluctuation
to the
formu-
as
similarly
mean-square
in
a
uniform
fb—f.
– f(’),
(31)
fluctuation
W“vr(fb
(30)
these
formula
can
formulas,
also
this
be converted
E2/AtAJ.
ACKNOJVLEDGMEtNT
The
“
(32)
— f.).
WO between
form
and
—2
E
AtAf
with
T in bandwidth
energy
characteristic
as A~ gives
~=_
dissipated
fre-
b -- fa).
redesignating
energy
current
Two(jb
UT2 =
fa)
mean
function
and fb (as it does
f.
321
for
resistor
use):
~wo(f~–
=
only
L/c
density
form
the
for
T=
spectral
zoo between
nearly
of Waveguide
values
propagate
mechanical
to
in current
~
mean
the
value
quantum
formula
to
that
zu(~) has a uniform
by
these
f. <f <fb, write
range
for
and
in
Resolution
W.
author
M.
takes
Gottschalk
pleasure
of the
encouragement,
in
thanking
University
suggestions,
Professor
of Delaware
and
helpful
for
his
discussions.
Discontinuity
On the Resolution
of a Class of Waveguide
Problems by the Use of Singular Integral
Equations*
L. LEWIN~
Surnmarp—It
is shown that a considerable number of solutions
of rectangular waveguide problems appearing in the literature are all
special cases of a general treatment focused around the known solution of a singular integral equation. In terms of this a number of
typical results are re-examined.
The method is then applied to four
new conEgurations, and the range of application and the limitations
are examined.
the
\vaveguide
from
This
number
exact
F
solution
shapes,
ideal
of
is
even
geometry
and
involved
has
and
space
the
infinite
the
way
across
the
or some
in
other
the
form
tinuity.
It is then
solved
axis,
on
by
* Received by the PGMTT,
Telecommunication
the
either
the
March
in
discontinuity
uniform
into
the
the integral
side
Wiener-Hopf
8, 1961.
Laboratories,
of
of
half-
the
along
the
such
setting
of
longer
for
obtainable
This
degrees
has been
waveguide
medium
diaphragms,
by
Hopf
of
a
such
of
at
guide
by
the
dis-
insert.
to take
the
variable,
field
and
be set up for
technique
and
a
this
is
no
to various
in some
Schwinger
and
It
those
form
can be solved
of approximation
done
the
or ferrite
can
Wiener-
of
waveguide,
changes
unknown
the equation
limited
be applied.
change
satisfactory
equation
the
to configurations
is a dielectric
The
but
can
takes
applicable
as the
for
majority
strips,
is it
it is more
section
problems.
quasi-static
cases.
Nor
of integral
usuable,
it
the
of the
diaphragms,
cases
type
in
section
propagation
cross
result
discontinuity
e.g., if there
the
class
free
a bifurcation
involve
field
which
different
regions,
radiation
readily
to which
cross
etc.
continuity,
For
the
the
exalnple,
section,
over
a rigorous
however,
which
over
for
cross
made.
gives
successful,
cases
in
of
treatment
a diaphragm
solutions
equivalent
a different
simple
are
the
of guide,
for
very
to exact
two
are
The
taking
~ Standard
England.
into
equation
few
of
approximations
which
length
capable
conductivity
exceptionally,
guide.
of an integral
axis,
wall
space
and,
a
amenable
Examples
of a semi-infinite
to
common
recently
z >0.
waveguide,
up
the
configurations
separated
z <O
problems
limited
when
A class of problems
has
waveguide
being
of configurations
not.
as,
HE
method
number
is
parameters
solution.
variation
I. INTRODUCTION
r
the
particular
co-authorsl
Lewinz,8
for
u n-
guide
equation
the
discon-
technique,
Harlow,
Essex,
‘ N. Marcuvitz,
“lJTaveguide
Handbook,
” M.1.T.
Rad. Lab. Ser.,
McGraw-Hill
Book Co., Inc., New Yorkr N, Y., p. 147; 1951.
2 L. Lewin,
“The impedance
of unsymlnetrical
str]lps in rectangular waveguides,
” PFOC. IEE,
vol.
99, pt. 4, pp. 168-176, Monograph
No. 29; 1952.
3 L. Lewin, “A ferrite boundary value problem in a rectangular
waveguide, ” Proc, IEE, vol. 106, pt. B, pp. 559–563; ~ ovember, 1959.
IRE TRANSACTIONS ON MICROWAVE
322
symmetrical
guide
waveguide
section.
formation
methods
guide
integral
and
diaphragm
direct
its
of
not
for
in this
the
quasi-static
and
medium,
It
is with
paper
is
sporadically
ticular
in
cases
albeit
the
cases
few
new
of
rather
of
the
Fig.
1 shows
in
terms
the
the
various
unknown
Here
of
2
~
F(v)
only
y
s
F(q)
the
parts,
The
continuity
leads
and
E(y)
and,
B,
of
but
amplitudes
of
susceptance,
the
(1)
it
known
The
does
not
complicates
here,
it.
though
reten-
affect
the
They
their
will
inclusion
straightforward.
(1)
with
term
8,, neglected,
of the
infinite
we
series,
add
and
summing
it
cos (Ty/a)
I .
result
‘n
—
-On
differentiation
and
substituting
(2+
2aB/X,)
; log 2 ] cos (7rq/a)
with
respect
an
in
-
to y this
gives
F(v)
gives
where,
as before,
phragm
aperture.
Now
of the
formb
dq
F(q)dV
-J
integral
sim-
cos (q/a)
— —
of
be
the
(1)
s
of
to an
terms
takes
into
in
if these
can
sin (n7ry/a)
variable
C,
the
the
whole
of
X = cos (mq/a),
(3)
,
COS
range
of the
independent
then
cos (nr~/a)d~.
of integration,
which
has
to
is
hold.
is a small
n, and
Y=
(7rq/a)
—
of both
left-hand
y.
If
cos (Ty/a)
and
put
q is the
is some
take
(3)
(fry/a)
y and
side
we
cos
dia-
constant,
new
variables
F(~)dq
= G(X)dX
becomes
s
Here
also
~, rangthe
The
range
side;
otherwise
The
the
de-
and
of Mraveguides,
” Iliffe
and
Sons,
we
c
limits
can
appeal
limits
q. The
for
Y<B.
(4)
form
value
(4)
“Integral
e F. G. Tricomi,
New York, N. Y., pp. 173-188;
in
to
of higher-order
Y to
the
right-hand
of (4) is unaltered.
of
the
integral
is a singular
to the
A convenient
X corresponding
inclusion
a polynomial
the
Hence
new
for
add
principal
solution.
A<
x–Y=–
B are the
aperture
understood.
vanishing
of
.4 and
would
quantity
quantity,
is a measure
A
terms
its
d Marcuvitz,
op. cit., p. 156.
5 L. Lewin,
“Advanced
Theory
London,
England,
p. 47; 1951.
solve
resulting.
terms
the
a cor-
B G(X) dX
for
number,
solution
though
first
(7rq/a)dq
aperture,
which
to
retained,
cos (rz7ry/a) cos (rz2rq/a)
m
z1
the
the
with
be considered
the
As a
are
coefficients
is relatively
The
unknown,
equation
~. = 1 – (1 – k2aA/#@)’lZ
high-mode
not
value.
As a second,
higher-order
point
order
its quasi-static
be neglected.
sets
a train
the
aperture
terms
mode,
and
and
amplitudes,
This
(1 – 6,,) j’F(v)
= E’ (q),
over
in
by
cos
via
from
higher-order
(1)
(A~/a)
over
the
mode
results.
diaphragm
waveguirle.
a rectangu-
diaphragm.
is as yet
over
above,
E(y)
wave,
of the
be expressed.
the
an integration
B sin (~y/a)
—
— –
as
for
by
It
chosen,
z= — ~
expansion,
field
containing
expressed,
plified
can
two
to the
dominant
from
sides
E(y),
magnetic
equation
can,
in
the
a transmitted
a Fourier
modes
tangential
in
on both
aperture,
by
diaphragm
wave
wave,
of it,
equation
a
treatments;
approximation
is incident
modes
in rectangular
DIAPHRAGM
inductive
A
a reflected
field
ing
an
(ry/a)e–Ik’z,
evanescent
be
INDUCTIVE
waveguide.
Ez ==sin
of
In
with
mode
it can
these
subtract
of con-
nth
of solution,
at any
a wide,
be successful.
that
or
therefore
par-
as particu-
earlier
of
method
outlines
finishes
of problems
degrees
has
of the types
the
by
all
one
tion
solved.
II.
up
range
solved
are
paper
and
should
displace
quasi-static
rigorously
lar
method
been
they
of the
responding
this
occurred
above,
an indication
course,
it extends
various
mentioned
the
that
which
diaphragm
approximation
first
The
treatment,
and
to which
not,
treatment
general
examples
figuration
does
the
have
that
l—Inductive
parture
first
of propa-
have
of applicability.
examples
un-
others.
and
a general
of
for
changes
realized
Fig.
solution
necessary
equations
is now
field
known
lar
be
of
a direct
is
and
literature
It
equation.
be catered
of discontinuities
The
the
limited,
first
class
concerned.
ad hoc methods.
that
strips,
latter
of the
b>- the
can
configurations,
reactive
this
is very
some
integral
equation
inductive
form
method
for
configurations
it appears
wave-
obtainable
quasi-static
integral
trans-
the
explicit
last
those
otherwise
such
for
in
This
include
the
all
way;
symmetrical
gation
solution.
results
solution
a ferrite-loaded
conformal
obtaining
configurations,
However,
for
used
expressions
without
equation
powerful,
and
has also
to obtain
parameters
the
strips,
Schwinger4
July
THEORY AND TECHNIQUES
theory
reference
Equations,
1957.
in
(4)
integral
of these
is
equations
is Tricomi,c
” Interscience
to
be
equation,
for
where
Publishers,
Lewin:
Y967
we find
the
solution
of Waveguide
of
Discontinuity
equation
can be put
(5)
~J7
X+
C is a constant,
known
current
the
this
result
to
integration
by
of
(4)
it
from
taking
an
integration
necessary
is
(.1 , B)
new
constant.
to
to
To
change
the
( — 1, 1). This
l($)d&
apply
range
is easily
a similar
2(X
which
form
for
–
K
(6)
(7)
terms
of
Y. This
transformation
Schwiuger’s
is the
as (3) have
is pre-
‘ttrick”
been
by
means
treated
of
change
(5)
ward
integration,
enter
J
in
constant
of (4),
the
the
G(-Y)dX
=
term
present
tious
at
are
the
The
can
(4)
case,
if
of
~(Y)
un-
x and
:summatiou,
(7r:/b)
= Y,
side
so that
change
of
(4)
is
any
SOIU-
ad hoc guessing
that
is
key
to
of
the
do
the
case
It
integral
point
not
concern
us
realization
that
(3)
of
(5)
is the
means
constant
equation
extension
by
the
The
which
setting
the
edge.
solved
as the
through
solution
the
an
eventual
[y
use
infinite
summed,
to (9).
integrations,
than
in
tensor
a
are
The
de-
given
in
consists
and
The
reason
waves
is that
distribution
the
simple
ferrite
and
gives
plain
a mag-
guide.
on both
paper.
rise
and
account
of the
Hence
sides
conditions.
Heim’s
for
for
reflected
on
in the
is of the
c
ferrite
supports
differs,
is needed,
derived
waveguide
arrangement
than
continuity
Sharpe
C=
this
that
of modes
equation
com-
constant
magnetized
\vhich
from
to satisfy
in
dielectric
that
involved
series
of the
in a. more
of a rectangular
of
more
boundary,
reappearance
albeit
WAV~GUID~
transversely
an infinite
given
(9),
(4).
medium
permeability,
integral
is the
in
FERRITE-LOADED
anything
field
here
equation
arrangement
a
The
~ In
of the
details
Lewin3
the
form
(lo)
MF(Y)+~
ii-
where
forms
method
to
a
the
C, 31 and
field
This
configurations.
UNSYMMETRICAL
by
strip
was
final
awk-
constants,
assuming
This
to the
s:
FY”y’
reappear-
in various
of
of
1($).
to be noted
form
sort
111, THE
the
plicated
netic
is re-
higher-order
expressions
of variable.
singular
result
with
integral
are
a special
same
transmitted
right-hand
was
this
rather
reference.
to
solu-
the
determined
transformation,
to the
filled
with
— ~ < z <O,
Corresponding
reduces
from
at the
(8)
for
singular
is a
apart
zero
series
O <z << ~.
at once.
thing
to
of the
range
vanishes,
since
without
these
ance
wider
Cos
Integration
paper’
IV.
‘–
the
important
the
the
= x,
are
field
original
thus
forms.
be reduced
simple
limits;
Tx >
obtained
details
The
and
polynomial
readily
(6)
both
p~~
follows
a finite
necessary
here.
E(v).
at
in
Y–X
obtained
by
=
vanishes
additional
to
are
placed
to effect
of (7) at once
problem,
together
The
F(q)dv
this
s –1
tions
to
s
in the
solution
the
of both
(2)
follows.
Schwinger’s
tails,
reduces
1 ~1–
.
the
to
range
(7rx/b)
solution,
electric
leading
d(<Y)dX
the
this
Fourier
11
reasons
K = O and
using
cos
of variable
and
tangential
of
case
physical
on
putting
form
the
For
the
= F( Y)d Y becomes
In
n- J –1
in the
and
strip.
The
hitherto.
is
.
which,
is proportional
strip,
A)
in
cos O which
equations
in
Y’
to
= C+S
such
Now
(8)
of
–1
equivalent
cos (ry/a)
l(f)o?~,
done
B–J
cisely
cos (w&b)
variables
XI=.
with
on
1($)
in the
equation,
and
nature
formz
sin (mm/b)
where
This
the
323
s
1
by
is of
in the
C=25
& is over
K
Problems
$+)(Y)
‘
J’,~>-
.f(-~)= +
is given
Resolution
C.\ P.> CITIVE
at the
are
is a singular
from
guesswork;
STRIP
K
(5).
but
constant,
junction
integral
1n the
it
across
is
and
the
F(X)
ferrite
equation,
reference
it
a particular
is related
but
was
c)f a different
solved
case
to
face.
of
partly
by
Carleman’s
equation”
The
setting
similar
route
current
on
L]uknown
up
to
the
to
of
that
strip
be
of
when
the
is used
in the
diaphragm
the
rather
evaluated.
results
the
current
integral
equation
previous
case
thal~
(~
rather
cases.
the
similar
than
) The
follows
except
aperture
type
the
that
field
of
the
{1(I)+(X) — i
is the
‘ 4(Y)
J
--—–- dy =
j(x),
(11)
,y—.r
expression
aperture
quasi-static
a
field
integral
lem
T C. B. Sharpe aud
in a rectangular
l%~o~~
D. S. Heim, “.4 ferrite boundary-value
probwaveguide,
” IRE
TRANS. ON MICROWJiV~
1958.
AND TIZCHNIQUIZS,vol. MT~-6, pp. 42-46; January,
IRE TRANSACTIONS ON MICROWAVE
324
with
the
If dl and
solution
The
1
a’(.r)+k’~’+~az(x)
“[s
-*
of a and
~e7(l)
a(x)f(x)
@(v) =
dy
+
—+
y –
x%’
J1 –
x
,
x1
Hence
All
the
Eq.
integrals
(12)
(10)
case
change
in the
Although
partially
form
(11)
paper
a(x)=
ill,
C by
its
is
have,
Some
the
Section
much
tive
limits
(1,
values
(~dl/a).
– 1) with
dx’
=.
by
tribution
of
transforms
the
ydy
Lewin3
we
which,
together
1? — y’
into
(14)
for
f’ = ~(~ —a)z
factor
on
unaltered
is that
changing
accordingly
get
the
(10),
jX
and
becomes
from
variables.
the
From
con-
(33)
of
equation
mode
with
no
past,
been
of
with
(5)
and
around
extensions
.x=–+
obtained,
the
(6)
gives,
eventually,
lr~
{
as a
this
of existing
1 +KX
1
1+
explicit
which
– C=
with
f’Lz
+
m’(f’
–
1)
(15)
‘
where
K+M
con-
K–M
T
more
W~VEGUIDE
of the
difficult
(see (35)
to
it
it
is
with
is shown
at
the
ferrite
of Lewin3).
(m-d/2a)
aperture
waveguide
task,
combine
located
sin
ferrite-loaded
arrangement
being
WITH
DIAPHRAGM
a formidable
The
diaphragm
the
cos
L=~log
is quite
diaphragm.
the
the
feature
new
solution
IV
new
and
= O. Eq.
polynomial,
(12),
INDUCTIVE
fact,
.
higher-order
formulation
FERRITE-LOADED
Although
gives
change
follow.
V.
of
equation
eventual
solution.
the
central
~
a(t)
case a(x)
C:
in
solution
the
is written.
figurations
~(x)=
a simple
guesswork,
and
(7)
then
(~dZ/a)
values.
special
of the
solutions
case,
principal
as the
character
by
special
are
(5)
replace
O(t) = tan–l
>
involved
the
only
multiplied
~t
—$
contains
is the
solutions
e(t)
—
s _lt
inserts,
– cos
x—y
the
T
by
transformation
(12)
where
‘
diaphragm
given
ax
~–r(?df(y)
1
dz are the
fi are
+A27r2
<a’(y)
T(x) = —
July
THEORY AND TECHNIQUES
not,
in
an
induc-
in
Fig.
2,
of
its
1 (no
center.
f=
face.
The
and
Eq.
can
where
reduces
inserts)
O (diaphragm
(35)
gives
completely
RECTANGULAR
in the
— dz) is the
to
and
be put
d = a — dl — dj
yO = ~ (a +dl
(15)
diaphragm
VI.
factorf
(~yo,ia)
opening
~=
for
sin
of
form
is
Lewin3
when
X = O, as it
across
WAV~GUIDE
the
coordinate
the
must,
guide).
BIJ?CTRCATION
(17-PLANE)
Fig.
3 shows
waveguide,
into
Fig.
2—Inductive
diaphragm
in ferrite-loaded
an ~-plane
bifurcation
in which
two
guide
of width
a single
guide
guides
Yi
t
process
repeated,
at
the
with
the
boundary,
sioned
by
of the
to
up
of
is seen
alteration
and
equation,
equation
setting
addition
it
the
integration,
gral
of
the
integral
equation
the
metallic
diaphragm
that
the
only
is a reduction
independent
the
new
changes
of
the
variable
aperture.
Hence,
the
Fig.
of
3—H-plane
new
is
This
arrangement
by
the
of treating
C=
MF(y)+j:
d.r,
(13)
IT J .x—y
method
— C = 1 +KX
and
the
normalized
reactance,
X,
is
2~
(14)
yF(y)dy.
jx=—
-T f c
These
must
forms
are
taken
from
!9
satisfy
F(y)dy
s a
Lewin.
a Moreover,
yield
F(y)
it
only
to
the
junction,
can
be incorporated
the
waveguide.
These
which,
plicability,
e.g.,
quasi-static
latter
method
material
can
so happens
there,
variants
be used
arrangement
has
oP.
two
of Fig.
O.
8 Marcuvitz,
or both
are foreign
with
the
cit.,
p. 383,
equation
possible
Thus,
on
either
of diaphragm
bifurcation
that
integral
are
of these
to the
its
3.
which
a difside
of
or strips
can
be used
Wiener-Hopf
own
unequal
approaches
rigor-
justification
of attack,
nevertheless,
to
is solvable
The
of variants
or an arrangement
approach,
simple
by
form
techniques
a number
the
together.
in it silnplest
here
dielectric
just
=
I
in rectangular
Wiener-Hopf
is that
ferent
\vllere
bifurcation
inteously
D F(r)
—
I
2.0
l’
1
,
occa-
range
in the
t
x
j
I
is
at z = O
!
\
the
a join
2a.
a
If
of a rectangular
of width
field
of ap-
guides.
overlap
It
in the
1961
Lewin:
Two
basic
metrical
and
the
two
modes
two.
is
the
just
guide,
from
sidered,
For
z <O
replace
into
at
in
the
~.
mairrder
term,
variable
to 0. The
broad
being
some
pressing
junction-the
attention
to
to the
be con-
upper
325
be substituted
1’2~+1 by
mode
the
needs
can
and
and
write
=@
of
by
via
equation
ex-
field
over
the
to
(19)
for
equating
(17)
(2),
-s
COS
E’(O)
%’
sin
The
magnetic
there
(@/2)d0
—
Re$~”)
as a conplernentary
terms.
.4 sin @ + B cos (@/2)
+
We
a re-
part
<a.
(e-’~”
(19).
and
O<y.:a.
have
Ez =
and
forms
are affected
of tangential
obtained
(17)
dominant
7ry/a
simplification
is
in
their
summations
continuity
aperture
mode
our
values
Problems
a sum
reflection.
we confine
we
field
symmetrical
O <y
These
feeding
antisymmetrical
mode
Discontinuity
3, sym-
of
be resolved
the
of
without
figure,
can
of Waveguide
Fig.
method
second-order
the
and
of the
left
is no change
only
by
Any
since
natural
propagates
Hence
the
lVIoreover,
there
field
be supported
antkiymrnetrical.
guides
of these
can
Resolution
(7ry/a)
- +
o 2 [sin (0/2)
Whel”c
–
(23)
s,
sin (@/2)]
s COS
.
S
r
Z,HU = (e-fk” – Re’~”)
s
+ j
(k’/k)
sin (ry/a)
.1 =
– +jk’u(l
– R)
Zo = 1207r,
~n~/a
mode
R~(y,,/k)e’nZ
for
large
For
27r/A,
k’=
n. The
amplitudes
R.
(17)
sin (mry/a),
and
reflection
have
z> O, bearing
in
yet
V.=
v’n%’/a2–
=
Tle–~Ic’z
.
+ ~
coefficient
R and
mind
the
S is a remainder
symmetrical
T,m+,e-rzm+”
– j ~
of
feeding,
the
term
(27rz+ 1 ry/2a)
cos
quasi-static
TZm+l(I’Zm+Jk)
= 2ir/A,
(18)
S=g
1
(
with
A, =A/tii–
as yet
then
in
from
T
,?3’(0) sin (2m
–1
cm
+
1 7ry/2a).
(k/4a)’
= (2m + 1)~/2a
is the
)s
2aI’2m+1
7r(27n +3
the
terms
for
unknown
various
of it
(19)
“~o solve
and
large
1 19/2)
77
)S
–1
E’(O)
(23)
field
mode
in
to the
change
(4/2)
quasi-static
variables
(23)
by
A
= ~(1 +y).
in
(24)
sin (rz@) cos (rzO)dO.
o
common
which
approximation,
putting
sin
factor
becones,
neglect
(0/2)=
cos
on
~(1 +x),
(@/2)
putting
can
be
E’(0)dO
= F(x)dx,
m.
the
amplitudes
can
s
1
aperture
F($)daf
——
_lx
—y
A(l+y)+ll=
be
.
(25)
as follows:
This
..2
E(rr~,/a)
1+R=5
+
o
a-f.
cancelled
expressed
constants
values.
x(
2
e–rz~+”
sin
(O< y <a),
of the differences
. cos (Zm +–i $/2)d0
S and
E(~y/a)
effect
attenuation
cos (7ry/2a)
cos (2m
If
the
mode
- n7r
K’
giving
higher-order
—
Here
sin (0/2)d0.
cos (my/2a)
Tl(K’/k)e-’K”
=
E’(O)
the
to be determined.
1
ZOH,
9d9,
o
k’
their
E.
E’ (0)
o
B = (1 – j2aK’/7r)
where
–
sin
is of the
form
(5),
giving
as solution,
(7rT/a)dn
a. J
F(.Y)
2’
—
--s To
In
on putting
xq/a
grated
part
ishing
of
and
vanishes
at
the
)
boundaries
R.
=0
=
tangential
integrating
both
by
limits
electric
parts.
because
field
(The
=2ti&
[C+
T*(A
+
B +- AX)].
(26)
of the
at
the
order
determine
C, we
note
that
vanmetal
J
(21)
E’ (~) COS Jztkio
to
inte-
Similarly,
2’
—
=
(20)
E’ (d) COSOdd
since
Jo
–1
E vanishes
at
the
limits,
Finally,
therefore,
7rlL s o
–4
T2,.hl
=
7r(2?r2 +
1)
s
m
E’(O)
0
sin (2m
+
1 O/2)d0.
(22)
F(x)
= &
[.4x’+
(A
+
B).V -- ~.~].
(27)
IRE TRANSACTIONS ON MICROWAVE
326
From
(20),
and
B,
together
we
get
be eliminated.
the
with
three
These
normalized
the
two
relations
equations
from
give
the
quasi-static
A
following
A
The
effect
of
B can
wherein
the
limits
defining
which
and
expression
impedance
at
the
for
V.
The
z=—
k’a
357r + j2K’a
~
137r + j30K’a
.j—
We
“
further
here,
cal
If only
one waveguide
is fed from
antisymmetrical
the
wave
in
amplitude
tion
mode
the
other
is now
coefficient
early
terms
viate
appreciably
tain
it,
with
on
(28)
series
Hence
zero
we have
the
given
Thus
the
is the
term
reflec-
(28).
by
first
term
in r~.
to de-
If we
re-
1
F(x)
s
to
—
dt,
(29)
–lx—y
2aI’8
r
E’(O)
sin (30/2)
–
AZ=—
d@,
appropriate
F(x)
solution
7r<l
of (29)
sions
{c*’+
defining
(A
A+
A,
from
l+R
of
tive
It
media
have
unity
k’
C)}.
the
into
latter
the
case
+ j2K’a
–
the
d(l19~
solution.
The
that
the
constants.
(2m/h)
or
to
expresan
the
small
order
cated
~e—
analysis
(31)
of
de-
much
order
to
half
further
if the
to reduce
function
There
is
no
in
the
and
hold
(17)
and
is otherwise
the
with
the
the
obstacle
combina-
a central
strip.
as the
capaci-
known
result2
by
the
for
an
method
of
in
of
4.
note
aperture
It
is apparent
a triple
description
problem
we
Fig.
and
metallic
being
to one involving
that,
because
distribution
ob-
permissible.
of
a single
the
function
sym-
suffices,
~.d2 . ..J
y.dl ---~=~ ---
4—Inductive
same
tions
symmetry
mirror
new
different
case
at
the
image
either,
guides.
of
and
values
right
the
diaphragm
accept
strip in rectangular
waveguide.
we
could
either
side
sary
each
this
aperture.
point
allow
for
of the
discontinuity,
complication
field
=
Hence
of view.
different
which
to
the
left
but
will
+
.
+ ~
Relk’z)
Rzn+le~’’+’%
set
up
the
previous
dielectric
obstacle
equa-
section,
materials
this
on
is an unneces-
be omitted
of the
(e-j”%
we
As in the
here.
can
be written
cos (~y/a)
1 7ry/a)
cos (2tz +
(32)
+1’
h
R~.+17zn+le’2”+”
cos (2n +
1 7i-y/a),
(33)
becomes
of
can
in
the
In
the
upper.
arbitrary
;L.
with
l’z~+l
the
bifur-
j unction.
in
and
value
solvable
of
the
that
to
for
from
except
similarly
is the
diaphragms
I
diaphragm
on the
unaltered,
on the
is
z
+
di-
the
y.
the
dielectric
which
complication,
the
an
of
with
aperture
the
a single
E.
guides
of
(~/2a)z
trouble
with
maintain
is
examine
strip,
is shown
method
In order
=
To
waveguide
dielectrics
either
. (31)
different
e, instead
are valid
configuration
lower
from
is a double
unknown
varied.
without
V is typi-
STRIP
be examined,
capacitive
arrangement
– j14K’a)
include
Thus,
(16)
Similarly,
Another
.+ND
shall
case will
is obtainable
there
stacle,
+ j18K’a)
d(135~
constant
assumed,
(28)
+
matter
— k~c. The
results
any
images.
The
in the
becomes
the
as
matter
(30)
eliminated,
in which
indicates
13~ +j30K’a
far
4~-#/az
also
appears,
+ 28A,)
a dielectric
so
that
be
(28)
is a straightforward
left
re-
proceeds
the
consisting
J
(* A+
and
C, and
to
35r
m
electric
the
then
in Section
diaphragm
inductive
unsymmetrical
metry,
1.
–
1–R
with
obtained.
we
a symmetrical
the
(28)
Ha
.j—
given
waveguide
2C)X’
+
(20),
B and
d ~3A$/(64
parture
(20),
Section
(7)
solution
DIAPHRAGM
example
I
B–+C)X–
in
analogous
quantity
example
INDUCTIVE
a further
Only
Fig.
is substituted
equation
but
of
pursue
of solution
a rectangular
the
this
as the
is
— X2
+(
If
however,
!
1
—
=
(25),
variable
the
in
as in
37
0
The
As
in
tion
retaining
where
s
not,
method
VI 1.
mode
relative
by
of the
cancel
incident
be obtained
A(l+y)+B+c(y2+2y)=
C=A,
to
of
– 1, 1 and
to
shall
directly
change
appears
become
to add
to
case the
R still
can
is augmented
left
amplitude
in this
(24).
from
the
unit
guide.
is *R
in the
(25)
of
2 so that
Improvements
The
range
before.
first
(28)
I–R
an
the
diaphragm
of integration
leads
limits.
stores
l+R
the
solution
altered
junction,
July
THEORY AND TECHNIQUES
~(2~
the
first
+
right
term
nitude
coming
boundary.
versed,
Only
mations
ment.
and
from
this
form
Ez is (1 +R)
The
the
account
the
the
l)m/a
is obtained,
of E.
of z in the
modes
of
(2?z +
for large
except
that
the
mag-
e–]~’z cos (my/a),
equality
sign
changes
odd-order
on
— k2 N
a similar
for
of the
H,.
l)z7r2/U2
sign
of j
appear
symmetry
in
on the
two
exponential
in
the
of
the
series
various
the
sides
is refor
sum-
arrange-
7967
Lewin:
If
the
field
in
then
Fourier
E(~y/a),
the
upper
Resolution
aperture
l+R=~
by
Problems
Discontinuity
The
solution
of the
327
integral
equation
F(y)
—-=4=
=
E(7ry/a)dy
dl
In
order
ing
to determine
Of E(6)
at
the
s
Tdl
—
C we
note
viously,
cause
the
variable
the
integrated
of the
metal
and
integrating
part
physical
for
the
at
As
the
pre-
limit
on E at the
of
mode
.R2.+1
=
~
equation
magnetic
27z +
1
resulting
fields
on
from
the
equating
two
sides
of
the
the
ak.’R cos #J =
4j ~
~ 7r(21L+ 1) s
–
x=; {
I.
are
the
integration
is over
and
the
s
can
From
(34)
at the
and
On
&O
in the
for
large
lower
(3.5)
o,
(W)
we
and
the
this
can
hence
relation
is given
now
calculate
the
normalized
discontinuity.
the
In terms
in
reflec-
reactance
of C we have
x=-:-Agl+pc”
range
of
~dl/a
half
the
<~,
(as
to give
first
and
the
guide.
the
an improved
term
of
simplifying,
the
(35)
(7rdj/a)
E
s
E’(0)
—
Cos
rdlln
in
(7rdl/a)
modulus
various
]K
forms.
Perhaps
the
‘
}
of the
complete
elliptic
functions
K-
by
solution
few
to
may
we
side,
compared
to
Lewin,b
when
dz = $a.
VII
~TNSYMMETRICAL
I.
(36)
2+ — Cos20
5 shows
(relative
guide
be written
sin O cos cjdO
—>
be
cosecz (7rd~/a).
(p.
62)
11- PL.iN~
(42)
to
which
it
ST~P
terms
Fig.
each
<1 – sinz (7rdl/a)
k =
solution).
can
sin’
is given
Now
first
series
–
the
and
for
~djla
Acosl$=
[cos’
reduces
quasi-static
previously,
adding
be put
conditions
1 — &?,+l,
=
can
is
where
variable
O <~d%/a.
correct
of the
+
This
be retained
the
relation
simplest
(41)
2 sin’ (rdz/a)li
problem
n. l-or
8 completely
summing
—
K
aperture
of the
7r(2?2+ 1)
could
C from
(37)
aperture,
X representing
This
i @)dO,
a~z.+l
ignore
1 NY)~Y , hence
,8
–l+—
the
upper
symmetry
maintained
where
vanish-
dy
of
E’(O) sin (2n + 1 @
9.
the
–
=
determination
Appendix
tion
tangential
bc)undary
sin (2,, +
From
the
,(K
a~z.+l
@ = ~y~a
=
y-l-c
written
where
from
–1 sin 6’
J
The
(2H + 1 tl)dO.
sin
E’(0)d8
‘
coefficients
E’(9)
that,
-ldl–y’~1–a–py
——
——
c’).
limits,
Villle
be-
edges
rdzliz
s.dl/a
higher
41
be
vanishes
requirements
parts.
inserts.
Similarly
The
by
(39)
(y+
yg
?rd2/fJ
() =
on changing
is
gives
cos (7y/a)
a
Waveguide
is represented
dg
s
analysis
of
a waveguide
values)
side
z >0.
from
y =a
is stepped
This
wave
with
from
the
region
electric
left,
filled
,s = –
field
with
where
is
k’ =
to
with
to
give
filled
EC=
the
~
At
of width
Za
e2, pz.
(ry/a)
cousta
X’/4a’)
cl, pl
z = O the
medium
e–~k’z sin
–
0.
a guide
with
propagation
k(elpl
a medium
z=
A
is incident
nt
given
by
(43)
1/2.
rd2[a
A = jak’R/4
E’(O)
+
J
To solve
and
and
E’(O)
(36)
this
equation,
sin
put
Odd =~F(y)dy.
sin
(37)
OdO.
~dlla
cos 2$ =a+~x,
The
s
a + 8 = cos (2Tdl/a)
which
values
of e~ and
same,
at
cos
guide,
the
factor
@ cancels
the
1 F(y)dy
—
–ly
—z
modes
cos 20 =a+~y
becomes
A=
The
form,
least
propagate
pz. In
two
dominant
for
particular,
z >0
if both
mocles
can
propagate
mode
and
the
second.
sin
(7ry/2a)
and
e–j%’
that
and a — @= cos (2Tdz/a).
with
(38)
depend
media
i n the
They
respectively,
e–’~”
~ provided
can
K’
=
k(qyz
–
Az/16a2)’12
sin
(7ry/a),
on the
are
the
wicler
are
of
July
IRE TRANSACTIONS ON MICRO WAVE THEORY AND TECHNIQUES
328
,=0
=,
Fig.
5—Change
H-plane
Y.o
PI
where
W2
S2
of cross section and of medium
of rectangular
waveguide.
in
17n =
In
and
putting
of the
K2
=
k(e2p2
–
(44)
x~/4a~)l/~.
(;z%~/4az
fully
When
the
media
In order
these
tion
to treat
two
second
at
in either
most
z >0
mode.
l?l’
of the
to the
Y,
general
case,
must
be
general
R2’,
z = O, we can
=
(1 –
will
If
+
as k’.
the
sence
loading
to
If
the
is no reflecbe reflections
dependent
these
define
R,’)/(l
the
there
phases
loads.
same
considered.
there
with
reflecting
plane
is the
matched,
In
and
KZ
same,
is completely
of amplitude
positioning
the
the
modes
guide
ferred
are
loads
At
effect
k2c~pJ ll? N
equations
have
of the
loading
the
of mismatch
in
z = O the
metal
O for
walls.
termined.
It
the
field
are
ables
to O and
admittances
m7/a
coefficients
terms
l?,’)
can
of the
be
unknown
coefficients
guide.
Thus,
in the
for
is zero
of E is as yet
a dummy
also
varia-
to change
vari-
the
by
analysis
Fourier
For
s
at
unde-
to O. Then
E.
ab-
be E(ry/a)
a, it
to take
determined
as yet
to
y=O,
of y, and
~y)a
(51)
mode.
form
be convenient
~ instead
the
u.
so as to exhibit
second
At
the
large
coefficient
is taken
a<y<2a.
ble of integration
from
form
dominant
on the
re-
for
written
in the
Otherwise
will
this
been
transmission
electric
and
rim/2a
in
modes
is simply
various
in
example,
a
and
l+R=~
Yz =
which
determine
mittances
The
the
Rz’)/(1
effects
relative
guide,
respective
(1 –
the
are
second
now
two
O<y<a,
the
the
first
T, alone
–
to
reflections.
These
wave-admittances
admittance
E(m-q/a)
aO
(45)
R,’),
of the
the
relevant
+
being
in
that
adthe
of the
If we integrate
by
because
vanishing
from
of the
parts,
taking
integrated
of E at the
v to 9 as explained
above,
mode.
for
up.
the
dq.
part
limits,
and
zero
change
we get
.
equations
be set
sin (mq/a)
For
the
electric
z <0
and
magnetic
fields
l+R=~
can
E’(8)
(52)
COSOdO.
To s
we have
Similarly
Ez =
(e-~k”
+
.
.ReJh’z) sin (~y/a)
R.=?
E’(%)
(53)
COS nodo
7rJbs u
+
~lkZOHU
=
~
2
k’(e–i~”
R~e’”z
–
(46)
sin (wry/a)
Rejh”)
and
sin (~y/a)
?rn
(47)
Inserting
field,
these
putting
aperture,
.
s
E’(8)
Tn=~
values
COS
(54)
~fiOdO.
0
in the
z = O, and
equations
equating
the
for
two
the
magnetic
fields
over
the
gives
where
~n =
For
(n2#/a2
— k?elpJ1/2
~
}?m/a for large
n.
(48)
~mpzk’(1
— R) sin @
/J2(71J?t)
s
.
+
s
.
E’(6)
+ j 5
z>O,
cos MdO. sin ?z~
o
T
+
E= =
1 G:
sin (~y/2a)
=
“-’K’”
+ ~~
+
R1’ej~’Z)
~
;J
3
Y1 sin ~~
E’(o)
COS ~~d~
f o
(e-iK22 +
T.e-rwz
plK’
RJe’K”)
sin (mry/2a)
sin (~y/a)
~NIKz
(49)
– j S
3
Yz sin ~
~l(rn/n)
E’(e)
Cos
ode
(O”E(~)cos+~~dO.sin+n@.
Jo
(55)
196?
Resolution
Lewin:
In
order
replace
to
~.
number
this
order
mode,
the
n~/a
and
of early
form:
in
mode.
The
the
done
in
series
and
If
are
in
the
The
here
to
Eq.
summed
1
—
T
sin (m)
=
* sin u~(cos
introduce
two
constants
2K”’a YJ
cos v).
(58)
1 +
[k’a(l
–
s1‘(x)
[(1
.lz
(55)
in terms
with
–
additional
(56)
~Od6
COS
the
+
Y2 + jmpZ/Kl) ].
K2a
Eq.
(57)
after
of their
expressing
half
the
trigonometrical
. [Cos ;0 +
Cos +4(1
A sin ~~ +
+
2#2/AJ]de
function.
ously
having
into
the
center
is
fed
O<
obtained
into
–0
order
the
(58)
@<T.
term,
can
field
<r,
also
be
the
and
at
with
–0
for
limits
is
a metal
wall
–
y’)1/2jd~
yz)llz
E’(o)do
–1
= E(r)
the
<y
<l,
(60)
– E(o)
= o.
(61)
integral
If the
fact
the
known
0, it
takes
still
–m.
equation,
only
also
in
seem
However,
a
range
of
the
final
(61)
sight
it would
un-
absent
which
arbi-
solvable
particular
in Appendix
The
case
can
II.
solution
it
equa-
two
be not
equations,
results.
be
as
were
to
the
first
a simple
the
other
outlined
satisfies
absent
(1 —y2)l/2,
would
a technique
quote
F(x)
be
We
will
to
(60)
is
1)1
=
;;:
J-YP(:%-
1)+X-6(”+-
the
valid
for
—~
‘@z
B cosec q?
—
X19
2(1 +
a’)
~+33_
l–x
[(
)
for
of
E(O).
simplify
the
term
form
Hence,
here
general
appear,
at
(1 —x2)11i!F(.v)
in (1 –Xz)l)z
by
than
it
to transform
time
the
The
quite
by
with
term
techniques.
and
which
which
putting
second
obvi-
along
properties
same
if the
involved
which
in a? were
on dividing
functions
(60),
term
equation
Similarly,
F(x).
more
equation
wave-
(58),
is seen
if in
O and
considerably
become,
converted
Thus
by
equation
see this,
0 = O. In
and
features,
is in
symmetry
that,
mode,
seen
the
at
to be
a symmetrical
is null.
using
note
linearly
of
O <~
0, when
we write
but
zero
placing
electric
for
be considered
second-order
by
This
we
left,
side
one
on
equation,
the
only
for
to
to
problem
the
<O.
itself,
In
goes
by
present
—z <~
of O can
antisymrnetrical
where
@ and
It
alternative
is
to be.
in
by
B sin ~~ cos ?#
Now E(6), qua function
step,
az(l
f 0
(60)
solved
guide
(59)
r
a straightforward
trary
the
–
=
singular
appears
angles,
tion
odd
media),
requirement
F(x)d2
known.
fact,
X2)112 +
1
would
an
equal
as follows:
E’(@
R) (jr
becomes,
=
from
—y
simple
terms
repeatedly
– R)M/Ml
~(1 +
“rhen
occurs
az
2PZ/P1( = 3 for
J o
(-–j8/7r)
by
becomes
o
B =
which
be denoted
= Ay + By(l
A=(j4/7r’)
(jr+ sr
We
u -
1 + 2pJp1
329
a
formula
~ cos (nu)
1
quantity
on will
a2 =
be
down
and
Problems
exact
extended
terms,
Discontinuity
second-
problem,
then
subtracting
now
a finite
their
with
this
we
desired,
be retained
been
happens,
adding
equation
n~/2a.
could
fact,
which
to n = 1 by
quasi-static
17. by
terms
has,
propagating
using
obtain
by
of Waveguide
only,
on
of the
the
left
Cos @ + Cos +0(1 + 2LL2/kl)
—
sin ~e — sin+$
do
where
first
x = (1 – z)/(1+ “c)
hand
becomes
and
1’
—
E’(o)
7r J —r
which
obviously
Finally
and
we
has
y = sin *I$. (This
‘be confused
stead
E’(6)d0
of
the
change
with
E’(0)
= F(x)dx.
we
symmetry
variable
use of x and
their
earlier
introduce
properties
again,
putting
y in this
use
7
stated.
section
F(x)
A
x== sin
will
as coordinates.)
a function
6 = 1 tan-l
T
such
+0
relation
be obtained
(a)
between
by
the
insertion
( = 1/3
for
constants
into
equal
A
(63)
media).
and
B
can
now
~
(64)
(56),
not
Inthat
4B
~(1
– p)(l
–
26)(1
– j2K’a
A=x
1 –
(1 –
2@)2(l
– j2K’al’l/7r)
YJ7r)
—
IRE TRANSACTIONS ON MICROWAVE
330
Finally,
if these
an equation
tance
results
results
relative
are
for
to the
inserted
(1 —R)/(1
wave
into
+R),
(52)
the
admittance
and
input
of the
(57),
admit-
first
wave-
THEORY AND TECHNIQUES
the
obstacle
the
media
be
more
The
guide
is capacitive
can
be met.
involved,
method
July
a change
But
the
apparently,
can
be used
in
may
the
for
permeability
analysis
any
other
finite
of
sometimes
way
round.
number
of feed-
(65)
In
the
case
circuit,
P9>>,uI this
gives
~~~
and
Yi.~0,
an
open
Eq.
(62)
field
can
across
the
be
integrated
aperture,
to
give
the
since
=‘F(I)A.
s
SE’(o)do
=
modes,
o
u
or
tion
l+a’
Herein
[
we must
Now,
for
X.
Hence,
electric
now
the
near
field
we have
smallest
the
is seen
be expressed
‘o
constant
to vary
in terms
X = tan’
as X~ or
of the
P2>>M,
O-m,
the
(m — (?)2$. This
can
coordinate
y across
into
(40)
Ptlt
and
the
hence
exponent
in terms
in
of Ml and
terms
of
a
P2, we
get
through
lrj2
o=
o
(63),
Nlultiply
=2/3/(
2@ =
For
tions
equal
of
fel
media
this
formably
~
r
of
the
problem
and
consideration
surface
other
at
In
mal
at
and
media.
with
(68)
guide
(68)
is ~. In
been
Schwartz-Christof-
exponent
has
of
corner.
which
2P=
waveguide
the
Eq.
varies
on
to
to
is
1, corresponding
be
zero
but
be
the
first
success
will
the
special
arise.
I
the
nor-
linearly
do.
by
v’1
‘/2
(
C
–
s
2/k’)
– (1 – k2 sin’ @2/k’
E–K
—
~Kk2
mod
1 +
– a+~
and
put
k2
<1
–
~0
k2 sin2 O
where
C=l+
On
simplifying
(42)
of the
the
various
{26/(1
terms
ATPENDIX
Following
erator
–
this
a +
leads
3.
to
(41)
and
text.
new.
open-cir-
to the
to
the
of electrical
guide
going
confrom
according
least
with
that
cases
to
~1–a+fl-2@sin’O
through
boundary
(68)
appears
first
mode
to
arisen
in terms
result
the
usual
at
dealing
in
needs
C–l+2sin20
o
calcula-
has
exponent
This
prior
the
geometry
p~>>~l,
gives
sinusoidal
the
the
a value
the
particular,
cuited,
the
re-entrant
with
in
it
by
the
exhibits
parameters
changes
of
the
hand,
2~2/I.J1)l/2.
exponent
boundary
transformation,
a
(1 +
that
equations
in
are
1-a+p).
o=
this
sort
map
tan–l
nor
obstacle
removed,
of
applied
dimensions
related,
of an
methods
retained.
y = – COS 20.
s
Expressing
are
an
elimina-
successfully
of different
APPENDIX
(67)
y~a.
introduces
whose
modes
is believed
integral
mode
solution,
be eventually
new
of singular
In
as
on
compli-
new
the
aperture,
E~(a–y)2~
may
depend
relevant
more
integrally
It
num-
of the
been
extension
account.
limitation
types
where
axial
solu-
of a finite
solution
many
guides
coeffi-
appreciably
the
yet
electric
quasi-static
retention
the
if too
the
transmission
each
not
in which
an
the
into
has
except
which
taken
exponents
where
involved,
in
(~–O/4).
for
various
corner,
(66)
“
cases
calculate
The
not
However,
is complicated
those
1
f?= $, whilst
of the
sharp
XB
_
1+X
x = sin ~0 giving
media
@= ~. D is always
of
(1+X)’
take
equal
X1–B
X1–5 – X1+6
by
being
method
to
conversion.
modes,
equation
The
B cosec T(I
mode
thereby.
tion
used
coefficient,
of higher-order
integral
cated
be
be augmented
additional
——
can
reflection
the
can
ber
and
the
cient,
electric
o
E(0)
ing
fields,
as is to be expected.
TricomiO
TU operating
we
II
introduce
on a function
the
transform
op-
@(z)
wall.
(69)
IX.
The
only
present
to
RANGE
method
rectangular
parallel
plate
ductive
a change
the
boundary
OF APPLICABILITY
is applicable,
so far
waveguides,
including
arrangements.
can
of dielectric
be
When
constant
accommodated.
the
The
as is known,
obstacle
on either
Similarly,
infinite
is inside
principal
Where
of
when
shall
of the
value
no confusion
write
this
of the
integral
of the
variables
simply
T(o).
Eq.
is to be understood.
is likely
(60)
to exist
is a particular
we
case
equation
aT(F)
+
T(bF)
= f,
(70)
1961
Lewin:
Resolution
of Wcweguide
with
Discontinuity
with
a =
a~(l
— y2)l/2,
f =
~Y
b =
+
By(l
(1 –
–
C as yet
Then
.#)’/’,
(76)
subject
(61)
for
addition
to
O =fl-l
F(x)
even.
of
inductive
renormalization,
replaced
the
by
ever,
of
to the
solution
theorem
solution
a large
number
—xz)
a number
with
of
(70),
Eq.
(70)
analysis
o!T(o,)]
solution
of
T(d)
where
eliminating
(1 -
y’)-”{
=
=4
Choose
+
edk/c
C=X–
C,
=
T(cF)
d =(1
+
(e’d/c)
–.v2)1fa,
so
in-
But
we
that
(79)
cd/c
which
“-s
lx–y+y–c
1
ii-
leads
=K.
N“ow
~~(.~)~~
x—y
—1
1
(y –
=
A convolu-
T(o,)
– odv.
(72)
.P)’/’#(x)]
+
c}
,
K=
gives
C) T(F)
since
s –1
I/a,
and
F(x)d.r
substituting
== O by (61).
these
results
(79)
in
finally
a2(l
-
T(dF).
how-
cases,
is
-T.[(1
untrans-
latter
can,
results.
T(OJ
the
F,
in hand.
of useful
(78)
edF.
In
Putting
+
@=
on
–
is
T[@]T(@
The
2“(cF)
(78),
in
T(h)
radicals
others.
and
i-erms
after
this
special
(75)
formed
=
the
give,
the
to
T(h)
of
throughout.
many
on the
problem
but
with
of
and
only
of the
gives
but
unsolved.
concentrate
Tricomi
tion
[1 — (c+dy)~]llz
in
form
would
equation,
a = b, ab = C(l
here
problem,
form
remains
be solved
shall
similar
alternative
diaphragms,
a general
so far
cluding
The
an
a like
the
absence
problem
F(x)dx,
:331
arbitrary.
simplifies
(71)
y’)’l’
From
and
Problems
— y~)’12T(F)
+
T[(l
(73)
— X2)l/~F]
—
“(;: :1’2[T(h)
-+ ;,/a].
(80)
(81)
C is a constant.
The
solution
of Carleman’s
Lo@
If
equation
– I“z[dy)l
=
now
we
choose
T(h)
id~)
then
is
(80)
From
+
h such
h/a
=
becomes
(78)
that
a-~(y
–
C)(I
equivalent
with
k given
to
by
–
y’)-~
j~(y),
(70)
and
(71).
(81)
a(y)g(y)
4(Y)
=
1+
T[(x
–
C)F]
–
TU
(y)
{[
[1 +
= ;[T(k)
1++:},
e-’(’)g(x)
+ .4
a’(x)]”
If
we. put
and
These
now
what
=
formulas
we
circumstances,
example
e’t’)
of
if
[1 +
at
Carleman’s
uz(x)]-’/’
return
all,
to
–
it
can
equation,
and
be
ask
reduced
whose
to
solution
us start
with
the
eT(dF)
~a–~(y
–
= ~H(y)
+~k(y),
a-’(y
C)(l
for
from
(81)
C)(I
–
then
L and
H,
of
ya)–1/2~(y).
II
is given
and
In
y’)-]
h only
by
the
Carleman’s
F(y)
(82)
l~(y),
is given
sign
type,
of
can
in terms
of
by
writing
down
integration
trary
the
–
c)
it
“
becomes
at x = 1 unless
C is g~ven
the
h(y)
(83)
solution
diverges
constant
determines
+
=
2(y
we
–
for
both
an
(75)
h.,
–
H(y)
the
=
+
equations,
F(y)
equation
CF +
=
differing
two
under
know.
Let
~/a
be solved
them
(70)
h,/a]
(y – C) F(y)
an equation
A (x)
–
by
a.
these
C) F/a
(74)
T(H)
Using
(y –
a’(y)
the
constant,
unity.
value
and
apparent
the
This
solution
the
that
so far
arbi-
feature
takes
the
form
where
c, d, e and
Operating
with
i n other
terms,
h are
T, and
we
all
functional
using
(72)
at
our
to express
disposal.
T[eT(dF)
1
1
]
F(y)
yl–d
=
27r(l
get
–
y)(l
+
cl?) {f
[et
=
7’(cF)
LIS choose
–
l’(e)
edF +
T(e) T(dF)
= AZ, a constant,
–
so
T[dFT(e)].
that,
from
(76)
+.
y-d—- 1,
C)(I
–
%’)–1/’,
xl/2-p
s –,y
J(.r)dx
(84)
—x
(73)
where
e=~(~..—
1 /:!
y–x
—1
L
T(h)
Xp—-j(t)dx
(77)
x.
The
Y=
(1 — y)/(1
constant
+y)
and
P is related
X
is a similar
to a by
function
D = (,1/m)
tan-l
of
a.
IRE TRANSACTIONS ON MICROWAVE
332
In
view
at
of the
y = 1 no
indicated
The
requirement
complementary
in (74)
in this
in
(84)
are
variable.
yn
along
The
F(y)
the
lines
when
straightforward
If
X,(1
=
of
and
integrations
1 +R
yl = —T
Y. ‘YA(fi
cosec
we
x.
– 2 –P)/(n
can
the
XO and
relation
recurrence
– 1) is readily
The
obtained.
in-
o
on taking
~ =X/1’
as new
and
p is suitably
(71)
It is readily
when
Y%
–
variable.
j(x)
These
is of
the
of any
that
other
structure
sectorial
sidered
section
in terms
Practical
propagation
mental
of its
curves
V-1ine,
and
conducting
hybrid
medkm
plates,
modes
structure
surface-
binding
are con-
of propagation.
are emphasized.
to determine
various
significance
is discussed.
Experi-
is described.
INTRODUCTION
F? ANALYSIS
A
the
electric
cylinders
bounded
by
usual
higher-order
terest,
“V-Iine,
insulated
hybrid
modes
the
excitation
* Received
the
eliminated
plates,
Io\v-order
cases
in
whereupon
and
consideration
by the PGilITT,
This
in
and
of
A
permits
the
plates
transverse
of
it
the
will
yj
of
more
to
of x.
is found
on collection
arbitrary
been
determined
in this
general
are required.
find
the
complicated
of a diaphragm,
follows.
is elementary.
completely
tools
o
of yz and
value
E(O)
yet
funcby
case
this
more
In particular,
solution
form
except
when
the
appropriate
in the
very
to
special
a=l(,uz=O).
Structure:
biasing
With
the
provide
tion
characteristics.
Although
between
convenient
the
angle
unrestricted,
be an
aliquot
where
the
For
such
the
V-line
A. VIGANTS~
the
of ferroelectric
may
principle,
AND
potentials
use
the V+Line*
angles,
may
electronic
by
the
the
that
propagate
on
cylinders
as well.
The
latter
extensive
analysis
with
respect
i.e.,
order
may
full
waveguide
fields
propaga-
plates
it will
of a semicircle,
modes
bias
of
the
simplicity
n designates
plates.
such
control
included
for
portion
integer
conducting
cylinders,
is,
r/n
of
circular
has
to
radians,
the
mode.
be supported
to its dominant
in
be taken
by
dielectric
undergone
modes.l–c
the
1 R. E. Beam, etd.,‘[Dielectric
Tube W7aveguides, ” NorthwestUniversity,
Evanston,
111., Report
A.T. I. 94929, ch. 5; 1950.
2 S. A. SchelkunotY,
“Electromagnetic
Waves, ” D, Iran INostrand
Co., Inc., New York, -..
V V-., n
194.3
r. 4?7.
-3 A. Sommerfeld,
“‘Electrodynamics,
” Academic
Press, Inc., New
York, N. Y., pp. 177-193;
1952.
4 H. Wegener,
“Ausbreitungsgeschwindigkeit,
Wellenwiderstand,
und Dampfung
electro.magnetischer
Wel]en
an dielektrischen
Zy linderu, ” Forschungberlcht
Nr. 2018, Deutsch
Luftfahrtforschung.
lrierjahresplanInst.
fur Schwingungsforschu
ng, Berlin,
Germany;
.htgust
26, 1944. (CADO
Wright-Patterson
AF Base, Dayton,
Ohio,
Document
No. ZWB/FB/Re/2018.
)
5 C. H. Chandler,
‘(Au investigation
of dielectric
rod as waveguide, ” -T. Ap@L F’hy.r., vol. 20, pp. 1188–1 192; December,
1949.
‘ W. M. Elsasser.
“Attenuation
in a dielectric
circular
rod. ” -7.
Appl.
Phys.,
vol. 20, pp. 1193-1 196; December,
1949.
ern
.!:----
high-order
V-
facilitates
application
16, 1960; revised
script received,
April 26, 1961. This work is based
taken pursuant
to Contract
AF 19(604 J3879 with
T Dept. of Elec. Engrg.,
Columbia
University,
desig-
the
modification
F(x)
that
possible
terms
the
=
in-
here
of
not
1)$%,
and
practical
particular,
August
section
1, leads
modes
that
apex,
modes
cross
structure,
be modified
versatility;
of the
In
on di-
as in Fig.
transverse
prototype
is required.
its
propagation
sectorial
modes.
will
at
are
enhances
of
the
”
of
conducting
hybrid
modes
Iine
set
however,
nated
be
of surface-wave
with
has
mathematical
presence
of
of
for
(70)
the
in
to give
to
MJ3MBJ31Z,
1=,
are presented
their
of the theory
a wedge-shaped
text
b(x)
been
(W –
whence
it is possible
takes
–
expressed
of the
and
not
case
2pxn_,
integration
and
has
the
confirming
function.
dielectric
higher-order
of the ideal
parameters
it
S. P. Schlesinger,
by two
and equations
verification
in
Surface~Wave
a cylindrical
supported
modhications
Design
a(x)
+
readily
(62)
powerful
suffice
specified
is zero,
complementary
of the
comprising
cross
results
form
fl_lF(x)dz
P. DIAMENT~,
Summary—Properties
of
o
paragraph,
solution
tions
radical
A Dielectric
wave
cot @
chosen.
verified
absence
to
are
Its
method,
(84)
the
xl
The
=
x–v
integrate
determination
to satisfy
1)%
previous
Finally,
XP
J
to
the
of terms,
dX
s
be shown
(?’2+
and
—
the
x)-n~jy,
tegral
WI
in
integrals
‘X.(1 – jyp(l + X-”wix
=
write
which
+
rp,
the
X
o
then
involved
involve
problem.
cc
s
as a new
integrability
functions
appear
integrations
is taken
of the
July
THEORY AND TECHNIQUES
nmuuon studies
underthe AF Res. Div.
New York,
N. Y.