1. Prof. David R. Jackson ECE Dept. Spring 2014 Notes 13 ECE 6341 1

2. Plane Wave Expansion The goal is to represent a plane wave in cylindrical coordinates as a series of cylindrical waves (to help us do scattering problems). Generating function: (Schaum’s Outline Eq. (24.16)) x y 2

3. Plane Wave Expansion (cont.) Let Hence 3

4. Plane Wave Expansion (cont.) Generalization: where or “Jacobi-Anger Expansion” 4 Note: the plane wave and the scattered field must have the same k z.

5. Alternative Derivation Multiple by e -jm   and integrate over  Note that Let Hence 5 Note: The plane-wave field on the LHS is finite on the z axis.

6. Alternative Derivation Hence or 6 Note: It is not obvious, but a m should be a constant (not a function of  ).

7. Alternative Derivation (cont.) Hence or Identity (adapted from Schaum’s Mathematical Handbook Eq. (24.99)): 7 We then have

8. Scattering by Cylinder A TM z plane wave is incident on a PEC cylinder. ii TM z 8 x z a

9. Scattering by Cylinder (cont.) Let To find A 1 : 9

10. Scattering by Cylinder (cont.) Hence or For denote 10

11. Scattering by Cylinder (cont.) where To solve for, first put into cylindrical form (Jacobi-Anger identity): Assume the following form for the scattered field: 11

12. Scattering by Cylinder (cont.) At Both will be satisfied if 12

13. Scattering by Cylinder (cont.) Hence This yields or Wethen have 13

14. Scattering by Cylinder (cont.) Note: We were successful in solving the scattering problem using only a TM z scattered field. This is because the cylinder was perfectly conducting. For a dielectric cylinder, the scattered field must have BOTH A z and F z. 14