Prof. David R. Jackson ECE Dept. Spring 2014 Notes 8 ECE 6341 1

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Prof. David R. JacksonECE Dept.

Spring 2014

Notes 8

ECE 6341

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Cylindrical Wave Functions

Helmholtz equation:2 2 0k

, , orz zz A F

2 2 22

2 2 2 2

1 10k

z

Separation of variables:

let , , ( ) ( )z R Z z

Substitute into previous equation and divide by .

3

Cylindrical Wave Functions (cont.)2 2 2

22 2 2 2

1 10k

z

let

, , ( ) ( )z R Z z

2 2 22

2 2 2 2

1 10

R R ZZ Z RZ R k R Z

z

22

1 10

R R Zk

R R Z

Divide by

4

Hence, f (z) = constant = - kz2

22

1 1Z R Rk

Z R R

( )f z ( , )g

(1)

or

22

1 10

R R Zk

R R Z

Cylindrical Wave Functions (cont.)

5

Hence 2z

Zk

Z

( ) ( ) ,sin( ),cos( )zjk zz z zZ z h k z e k z k z

Next, to isolate the -dependent term, multiply Eq. (1) by 2 :

2 2 2 22

1 10z

R Rk k

R R


6

2 2 2 1z

R Rk k

R R

( )f ( )g

(2)

Hence

Hence, 2constant

( ) ,sin( ),cos( )jh e so


7

From Eq. (2) we now have

2 2 2 2 1z

R Rk k

R R

First, multiply by R and collect terms:

2 2 2 2 2 0zR R k k R R

The next goal is to solve this equation for R().


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Define

Then,

Next, define

2 2 2zk k k

22 2 0R R k R

( ) ( )

x k

R y x

( ) ( )dR dy dx

R y x kd dx d

2( ) ( )R p y x k

Note that

and


9

Then we have

Bessel equation of order

Two independent solutions:

2 2 2 0x y xy x y

( ) ( ) ( )y x AJ x BY x

( ), ( )J x Y x

Hence

( ) ( ), ( )R J k Y k Therefore


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( ) ( ), ( )R J k Y k

Summary


,sin( ),cos( )je

( ) ,sin( ),cos( )zjk zz zZ z e k z k z

, , ( ) ( )z R Z z

2 2 2zk k k

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References for Bessel Functions

M. R. Spiegel, Schaum’s Outline Mathematical Handbook, McGraw-Hill, 1968.

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Government Printing Office, Tenth Printing, 1972.

N. N. Lebedev, Special Functions & Their Applications, Dover Publications, New York, 1972.

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Properties of Bessel Functions

0 1 2 3 4 5 6 7 8 9 100.6

0.4

0.2

0

0.2

0.4

0.6

0.8

11

0.403

J0 x( )

J1 x( )

Jn 2 x( )

100 x

x

Jn (x)

n = 0

n = 1

n = 2

(0)J is finite

0

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Bessel Functions (cont.)

0 1 2 3 4 5 6 7 8 9 107

6

5

4

3

2

1

0

10.521

6.206

Y0 x( )

Y1 x( )

Yn 2 x( )

100 x

x

Yn (x)

n = 0

n = 1

n = 2

(0)Y is infinite

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Small-Argument Properties (x 0):

( )J x Ax

( ) , 0Y x Bx

0 ( ) ln , 0Y x C x


For order zero, the Bessel function of the second kind behaves as ln rather than algebraically.

The order is arbitrary here, as long as it is not a negative integer.

1, 2, 3,...

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Note:Bessel equation is unchanged by

is a always a valid solution

n

( ) ( ), ( )y x J x J x

( )J x

Two linearly independent solutions


These are linearly independent when is not an integer.

1 2( ) , ( ) 0J x A x J x A x x

as

Non-Integer Order:

16

(This definition gives a “nice” asymptotic behavior as x .)

( ) cos( ) ( )( )

sin( )

J x J xY x


17

( ) ( 1) ( )nn nJ x J x

( ) lim ( )n nY x Y x

= n

(They are no longer linearly independent.)

In this case,


Symmetry property

( ) ( 1) ( )nn nY x Y x

Integer Order:

18


2

0

1( )

! ! 2

k k

k

xJ x

k k

! 1z z

Frobenius solution†:

( ) cos( ) ( )( )

sin( )

J x J xY x

- 1, -2, …

†Ferdinand Georg Frobenius (October 26, 1849 – August 3, 1917) was a German mathematician, best known for his contributions to the theory of differential equations and to group theory (Wikipedia).

…- 2, -1, 0, 1, 2 …

19

Bessel Functions (cont.)From the limiting definition, we have, as n:

21

0

2

0

1 !2 1( ) ln

2 ! 2

1 11

! ! 2

k nn

n nk

k nk

k

n kx xY x J x

k

xk n k

k n k

1 1 11 ( 0)

2 3p p

p 0 0 where

(Schaum’s Outline Mathematical Handbook, Eq. (24.9))

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0 :x 1

( ) ~ 1, 2, 3,...2 !

J x x


From the Frobenius solution and the symmetry property, we have that

1( ) ~ ( 1) 0,1,2,....

2 !n n

n nJ x x n

n

1( ) ~ 0,1,2,....

2 !n

n nJ x x n

n

( ) ( 1) ( )nn nJ x J x

21

0

2( ) ~ ln , 0.5772156

2

xY x

1 2( ) ~ ( 1)! , 0Y x

x

1 2( ) ~ ( 1)!

1,2,3,.....

n

nY x nx

n

1 2( ) ~ 1 ( 1)!

1,2,3,.....

nn

nY x nx

n

cos 1( ) ~ , 0

sin ! 2

(2 1) / 2

xY x

n

n


0 :x

( ) ( 1) ( )nn nY x Y x

(To derive this, see the eqs. on slides 18 and 20.)

22

x

2( ) ~ cos

2 4

2( ) ~ sin

2 4

J x xx

Y x xx


Asymptotic Formulas

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Hankel Functions

1

2

( ) ( ) ( )

( ) ( ) ( )

H x J x jY x

H x J x jY x

x As

( )1 2 4

( )2 2 4

2( ) ~

2( ) ~

j x

j x

H x ex

H x ex

Incoming wave

Outgoing wave

These are valid for arbitrary order .

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Fields In Cylindrical Coordinates

1 1

1 1

E A Fj

H A Fj

ˆ ˆz zA z A F z F or

We expand the curls in cylindrical coordinates to get the following results.

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zA

21E

j z

1H

21E

j z

1H

22

2

1zE k

j z

0zH

TMz:

TMz Fields

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zF

1E

21H

j z

1E

21H

j z

0zE 2

22

1zH k

j z

TEz:

TEz Fields

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Prof. David R. Jackson ECE Dept. Spring 2014 Notes 8 ECE 6341 1