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

Static Potential of Point Sources

Representation of Point Sources

Superposition of Potentials

A Green’s Function

Green’s Function Conditions V

Green’s Function for a General Linear Operator

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

3-D Point Source Representation in Various Coordinate Systems

2-D Line Source Representation in Various Coordinate Systems

Cylindrical Coordinate Example x y z dz z x y

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

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

Example: Scalar Point Source in a Rectangular Waveguide

Point Source in a Waveguide, cont’d

 Key result!  Key observation!

Point Source in a Waveguide, cont’d

2D Sources

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

“Proof” of Claim

“Proof” of Claim (cont.)

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

Example: Green’s Function for 2D Wave Equation

“Proof” of Claim

“Proof” of Claim (cont.)

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

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

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

The Addition Theorem x y

Solution of the Line Source Scattering Problem

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