Download The Solutions of Wave Equation in Cylindrical Coordinates The

EMT 93
The Solutions of Wave Equation in Cylindrical Coordinates
The Helmholtz equation in cylindrical coordinates is
By separation of variables, assume
. We have
.
The only possible solution of the above is
where
,
and
are constants of ,
and .
and
satisfy
.
The final solution for a give set of
,
and
can be expressed as
,
where
is the Bessel function of the form
.
The exact values of
, ,
and the forms of the harmonic
functions and the Bessel function are determined by the boundary
conditions.
In general,
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Note:
1. Choose
, if
included.
2.
Choose
, if
included.
3.
Choose integer
.
, if the space contain all range of
Likewise, the corresponding solutions for
follow.
and
, that is,
are as
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The Circular Waveguide
1.
a.
B. C.
where
is the roots of
b.
Cutoff frequency:
c.
Wave impedance:
a.
B. C.
.
2.
where
.
b.
Cutoff frequency:
c.
Wave impedance:
is the roots of
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3.
4.
Always degenerate(
First mode:
).
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Higher Order Modes of Coaxial Lines
mode B. C.:
mode B. C.:
Dominant:
.
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Homework #7 Problem 5.7
Circular wave:
Radial Waveguide
Radial wave:
Parallel-plate radial waveguide
B. C.:
at
and
TM to z mode:
TE to z mode:
Phase constant:
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Note:
Wave impedance:
Note:
5.
6.
For real
7.
For
8.
propagation, evanescent.
First modes:
.
a.
b.
c.
,
are complex function of .
,
is imaginary,
is also imaginary, not
: predominantly resistive
: predominantly reactive
Dominate mode:
. Only
transmission-line mode.
Inward wave:
and
exist. TEM,
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Outward wave:
Wedge Radial Waveguides
Assume no variation in z.
B. C.:
at
and
TM to z mode:
TE to z mode:
Dominant mode:
, only
and
mode.
Inward:
, Outward:
, TEM, transmission-line
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The Circular Cavity
TM to z mode (
):
TE to z mode (
):
Dominant mode:
1.
:
. Shorted radial waveguide mode
.
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2.
3.
:
If
. Shorted circular waveguide mode
.
, the second resonance is 1.59 times the first
resonant frequency. For rectangular cavity of small height,
1.58.
4.
Q of
mode:
For the same height-to-diameter ratio, the circular cavity has an 8.3%
higher Q than the rectangular cavity. This is to be expected, since the
volume-to-area ratio is higher for a circular cylinder than for a square
cylinder.
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Other Guided Waves
Two dielectric Circular Waveguides
Assume z-directed propagation waves. Hybrid modes exist.
In dielectric 1:
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In dielectric 2:
B. Cs.:
continues at
.
A linear equation of unknowns A, B, C, and D. For not trivial
solution, the determinant must be zero, thus solving
and
Partially filled circular waveguides
Must satisfy: finite at
,
at
,
Dielectric-rod waveguides (optical fibers)
Must satisfy: finite at
, decay for
.
at
.
.
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where
.
Dominant mode: the lowest
mode. Zero cut-off frequency.
Coated conductor waveguides
Must satisfy:
at
,
at
where
, decay for
.
.
Dominant mode: the lowest
TM mode. Zero cut-off frequency.
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Chap 6. Spherical Wave Functions
Spherical Wave Functions satisfying Helmholtz equation (
).
: spherical Bessel function.
1.
Zero-th order:
2.
Higher order: polynomials of
3.
Only
4.
For out-going waves and
is finite at
Alternatively, for
,
times
or
.
.
, use
.
and
can be chosen as
two independently solutions. All solutions have singularity at
except
with
integer. Also,
for
.
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For TM or TE to z analysis, we have
Alternatively, consider TM or TE to
Note:
analysis.
does not satisfy Helmholtz equation.
Then,
where
The electric and magnetic fields can be computed by
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The Spherical Cavity
For TE to r, choose
B.C.:
at
. We have
where
are the p-th zero of
.
Similarly, for TM to r, choose
.
Then,
where
are the p-th zero of
Resonant frequencies:
Note:
.
.
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The first mode:
Degeneracies:
example:
Orthogonality Relationships
From Green’s Identity
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Let
,
then,
For
Legendre Polynomial Expansion(Fourier-Legendre Series)
Let (assume
)
then,
Define tesseral harmonics as
,
then the spherical wave function
and
can be written as
.
We have
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Since
When
,
A two-dimensional Fourier-Legendre series can be obtained for a
function
on a spherical surface as
Then,
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Space as a Waveguide
TM to r:
Then,
TE to r:
Then,
Note:
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Other Radial Waveguides
Conical Waveguide
B. C.:
Solution space:
TM to r:
To satisfy the B. C.,
TE to r:
To satisfy the B. C.,
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Biconical Waveguide
B. C.:
Solution space:
Since
is not included, use
TM to r:
To satisfy the B. C.,
TE to r:
To satisfy the B. C.,
for
.
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Dominant (transmission line) Mode: TEM (
Note:
)
gives zero fields and is not chosen.
Then,
Wedge Waveguides
B. C.:
Solution space:
TM ro r:
TE ro r:
No spherical TEM mode, but has cylindrical TEM mode.
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Horn Waveguide
TM ro r:
TE ro r:
Biconical Cavity
(Shorted transmission line)
Resonant frequencies:
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Source of Spherical Waves
For a z-directed current source
and
Wave Transformation
Consider a plane wave propagating in z-direction:
Solution space:
1. Independent of
2.
included:
3.
: m=0.
.
included:
, n integer.
Then,
Differentiate both sides n times at
Also we establish the identity:
Scattering by Spheres
, we have
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Assume an x-polarized ztraveling plane wave incident
on a PEC sphere with radius a.
Then,
and
.
Using wave transformation, we have
From
,
Derive
be expressed as
from
and use the following identity,
.
We have
.
Similarly,
In order to match the boundary condition at
, that is,
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The form of the scattered field must be the same as the incident field
except the Bessel functions must represent out-going waves.
Therefore,
By applying the B. C., we have
At far field,
and only retain
Back-scattered field
Consider
Calculate effective area by
1.
Small
:
term dominant and
, we have
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(Good
approximation for
)
This is Rayleigh scattering law.
2. Large :
3.
Physical optics approximation.
Others: resonance region.
Consider the fields scattered by small sphere.
Use small argument approximation of Bessel
functions, we have
.
Therefore,
dominates. At far field from small sphere,
Comparing above to the field radiated by electric and magnetic
dipoles, the scattered field is the field of an x-directed electric dipole
and a y-directed magnetic dipole as formulated below:
In general, the scattered field of any small body can be expressed in
terms of an electric dipole and a magnetic dipole. For a conducting
body, the magnetic dipole may vanish, but the electric dipole always
exist.
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Dielectric Sphere
For small dielectric sphere, at far field, the equivalent x-directed
electric dipole and y-directed magnetic dipole are
Also, the field inside the sphere is uniform. This results are the same
as D. C. case. This is called quasi-static approximation.
EMT 134
Appendix: Legendre Functions
Legendre equation:
where
is the Legendre function of order n.
Associated Legendre equation:
where
is the Legendre function of order n and m and