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

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

Cylindrical Wave Functions (cont.) Divide by  let

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

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

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

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:

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

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

Cylindrical Wave Functions (cont.) Summary

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.

Properties of Bessel Functions Jn (x) x

Bessel Functions (cont.) Yn (x) x

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.

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.

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

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).

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

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

Bessel Functions (cont.) Example Prove: Denote:

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

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

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

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

Bessel Functions (cont.)

Bessel Functions (cont.) Asymptotic Formulas

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

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

TMz Fields TMz:

TEz Fields TEz: