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

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

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

Hence, f (z) = constant = - k z 2 (1) or 4 Cylindrical Wave Functions (cont.)

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

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

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

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

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

10 Summary Cylindrical Wave Functions (cont.)

References for Bessel Functions 11  M. R. Spiegel, Schaum’s Outline Mathematical Handbook, McGraw-Hill,  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,  N. N. Lebedev, Special Functions & Their Applications, Dover Publications, New York, 1972.

Properties of Bessel Functions x J n ( x ) n = 0 n = 1 n = 2 12

Bessel Functions (cont.) x Y n ( x ) n = 0 n = 1 n = 2 13

Small-Argument Properties ( x  0 ): Bessel Functions (cont.) For order zero, the Bessel function of the second kind behaves as ln rather than algebraically. 14 The order is arbitrary here, as long as it is not a negative integer.

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

(This definition gives a “nice” asymptotic behavior as x  .) Bessel Functions (cont.) 16

 = n (They are no longer linearly independent.) In this case, Bessel Functions (cont.) Symmetry property 17 Integer Order:

Bessel Functions (cont.) Frobenius solution † :   - 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). 18   …- 2, -1, 0, 1, 2 …

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

Bessel Functions (cont.) From the Frobenius solution and the symmetry property, we have that 20

Bessel Functions (cont.) 21 (To derive this, see the eqs. on slides 18 and 20.)

Bessel Functions (cont.) Asymptotic Formulas 22

Hankel Functions Incoming wave Outgoing wave 23 These are valid for arbitrary order .

Fields In Cylindrical Coordinates 24 We expand the curls in cylindrical coordinates to get the following results.

TM z : TM z Fields 25

TE z : TE z Fields 26