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The Solutions of Wave Equation in Cylindrical CoordinatesThe 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 harmonicfunctions and the Bessel function are determined by the boundaryconditions.

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 , that is,

.

Likewise, the corresponding solutions for and are asfollow.

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The Circular Waveguide

1.

a. B. C. where

is the roots of .

b. Cutoff frequency:

c. Wave impedance:

2.

a. B. C. where is the roots of

.

b. Cutoff frequency:

c. Wave impedance:

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3. Always degenerate( ).4. 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 waveguideB. 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 , are complex function of .7. For , is imaginary, is also imaginary, not

propagation, evanescent.8. First modes: .

a. : predominantly resistiveb. : predominantly reactivec. Dominate mode: . Only and exist. TEM,

transmission-line mode.

Inward wave:

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Outward wave:

Wedge Radial WaveguidesAssume no variation in z.B. C.: at and

TM to z mode:

TE to z mode:

Dominant mode: , only and , TEM, transmission-linemode.

Inward: , Outward:

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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. : . Shorted circular waveguide mode .

3. If , 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 thevolume-to-area ratio is higher for a circular cylinder than for a squarecylinder.

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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 trivialsolution, the determinant must be zero, thus solving and .

Partially filled circular waveguidesMust satisfy: finite at , at , at .

Dielectric-rod waveguides (optical fibers)Must satisfy: finite at , decay for .

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where .Dominant mode: the lowest mode. Zero cut-off frequency.

Coated conductor waveguidesMust satisfy: at , at , decay for .

where .Dominant mode: the lowest TM mode. Zero cut-off frequency.

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Chap 6. Spherical Wave FunctionsSpherical Wave Functions satisfying Helmholtz equation ( ).

: spherical Bessel function.

1. Zero-th order:

2. Higher order: polynomials of times or .

3. Only is finite at .

4. For out-going waves and , use .

Alternatively, for , and can be chosen astwo 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 analysis.

Note: 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 afunction on a spherical surface as

Then,

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Space as a WaveguideTM to r:

Then,

TE to r:

Then,

Note:

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Other Radial WaveguidesConical WaveguideB. C.: Solution space: TM to r:

To satisfy the B. C.,

TE to r:

To satisfy the B. C.,

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Biconical WaveguideB. C.: Solution space:

Since is not included, use for .TM to r:

To satisfy the B. C.,

TE to r:

To satisfy the B. C.,

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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 WaveguideTM 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 : m=0.2. included: .3. included: , n integer.Then,

Differentiate both sides n times at , we have

Also we establish the identity:

Scattering by Spheres

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Assume an x-polarized z-traveling plane wave incidenton a PEC sphere with radius a.Then,

and

.

Using wave transformation, we have

From , be expressed as

Derive 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 fieldexcept the Bessel functions must represent out-going waves.Therefore,

By applying the B. C., we have

At far field, and only retain , we have

Back-scattered fieldConsider

Calculate effective area by

1. Small : term dominant and

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(Good

approximation for )This is Rayleigh scattering law.

2. Large :

Physical optics approximation.3. Others: resonance region.

Consider the fields scattered by small sphere.Use small argument approximation of Besselfunctions, we have

.

Therefore, dominates. At far field from small sphere,

Comparing above to the field radiated by electric and magneticdipoles, the scattered field is the field of an x-directed electric dipoleand a y-directed magnetic dipole as formulated below:

In general, the scattered field of any small body can be expressed interms of an electric dipole and a magnetic dipole. For a conductingbody, the magnetic dipole may vanish, but the electric dipole alwaysexist.

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Dielectric Sphere

For small dielectric sphere, at far field, the equivalent x-directedelectric dipole and y-directed magnetic dipole are

Also, the field inside the sphere is uniform. This results are the sameas D. C. case. This is called quasi-static approximation.

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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