1. ECE 6341
Spring 2016
Prof. David R. Jackson
ECE Dept.
Notes 8

2. Cylindrical Wave Functions
Helmholtz equation:
Separation of variables:
let
Substitute into previous equation and divide by .

3. Cylindrical Wave Functions (cont.)
Divide by 
let

4. Cylindrical Wave Functions (cont.)
(1) or
Hence, f (z) = constant = - kz2

5. Cylindrical Wave Functions (cont.)
Hence
Next, to isolate the  -dependent term, multiply Eq. (1) by  2 :

6. Cylindrical Wave Functions (cont.)
Hence
(2)
Hence, so

7. Cylindrical Wave Functions (cont.)
From Eq. (2) we now have
The next goal is to solve this equation for R().
First, multiply by R and collect terms:

8. Cylindrical Wave Functions (cont.)
Define
Then,
Next, define
Note that
and

9. Cylindrical Wave Functions (cont.)
Then we have
Bessel equation of order 
Two independent solutions:
Hence
Therefore

10. Cylindrical Wave Functions (cont.)
Summary

11. 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.

12. Properties of Bessel Functions
Jn (x) x

13. Bessel Functions (cont.)
Yn (x) x

14. Bessel Functions (cont.)
Small-Argument Properties (x  0):
For order zero, the Bessel function of the second kind Y0 behaves as ln(x) rather than algebraically.

15. Bessel Functions (cont.)
Non-Integer Order:
Two linearly independent solutions
Bessel equation is unchanged by
Note:
is a always a valid solution
These are linearly independent when  is not an integer.

16. Bessel Functions (cont.)
Symmetry property
The functions Jn and J-n are no longer linearly independent.

17. Bessel Functions (cont.)
Frobenius solution†:
This is valid for any  (including  = n).
†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).

18. Bessel Functions (cont.)
Definition of Y
  …- 2, -1, 0, 1, 2 …
(This definition gives a “nice” asymptotic behavior as x  .)
For integer order:

19. Bessel Functions (cont.)
From the limiting definition, we have, as   n:
(Schaum’s Outline Mathematical Handbook, Eq. (24.9))
where

20. Bessel Functions (cont.)
Example
Prove:
Denote:

21. Bessel Functions (cont.)
Example (cont.)
Plot of  function
(from Wikipedia)
Note that

22. Bessel Functions (cont.)
Example (cont.)
Hence

23. Bessel Functions (cont.)
Example (cont.)
Hence, we have so

24. Bessel Functions (cont.)
From the Frobenius solution and the symmetry property, we have that

25. Bessel Functions (cont.)

26. Bessel Functions (cont.)
Asymptotic Formulas

27. Hankel Functions Incoming wave Outgoing wave
These are valid for arbitrary order .

28. Fields In Cylindrical Coordinates
We expand the curls in cylindrical coordinates to get the following results.

29. TMz Fields
TMz:

30. TEz Fields
TEz: