1. D. R. Wilton ECE Dept. ECE 6382 Green’s Functions in Two and Three Dimensions

2. Static Potential of Point Sources

3. Representation of Point Sources

4. Superposition of Potentials

5. A Green’s Function

6. Green’s Function Conditions V

7. Green’s Function for a General Linear Operator

8. A Source-Weighted Superposition over the Unit Source Response Provides a General Solution

9. 3-D Point Source Representation in Various Coordinate Systems

10. 2-D Line Source Representation in Various Coordinate Systems

11. Cylindrical Coordinate Example x y z dz z x y

12. Example: A Simple Static Green’s Function with Boundary Conditions --- Charge over a Ground Plane 1 [C] z z -1 [C]

13. Static Green’s Function with Boundary Conditions (cont.) z

14. Example: Scalar Point Source in a Rectangular Waveguide

15. Point Source in a Waveguide, cont’d

16.  Key result!  Key observation!

17. Point Source in a Waveguide, cont’d

18. 2D Sources

19. Example: Green’s Function for 2D Poisson’s Equation

20. “Proof” of Claim

21. “Proof” of Claim (cont.)

22. Solution Is Easily Extended to 2D Sources Off the z-Axis x y z Line source

23. Example: Green’s Function for 2D Wave Equation

24. “Proof” of Claim

25. “Proof” of Claim (cont.)

26. Extension to 2D Sources Off the z-Axis x y z Line source

27. Summary of Common 2D, 3D Greens Functions Line source x y z Point source These Green’s functions are actually fundamental solutions since there are no imposed boundary conditions

28. Line Source Illumination of a Circular Cylinder x y a Line source

29. The Addition Theorem x y

30. Solution of the Line Source Scattering Problem

31. Interpretation as a Green’s Function Line source x y a