Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 7A: Two- and Three-Dimensional Green’s Functions

7A.1Introduction

We discussed Green’s function briefly (Section 7.2). The one-dimensional Green’s function of the Laplace equation, with Dirichlet boundary conditions, is the solution of the differential equation

(7A.1a)$$\frac{{\text{d}}^{2}G}{\text{d}{x}^2}=-\mathrm{\delta }\left(x-x^{\prime }\right),0<x<L,$$

subject to the boundary conditions

(7A.1b)$$G=0,x=0,$$

(7A.1c)$$G=0,x=L,$$

and is shown to be

(7A.2a)$$G=G_1=x\frac{L-x^{\prime }}{L},0<x<x^{\prime },$$

(7A.2b)$$G=G_2=\frac{x^{\prime }\left(L-x\right)}{L},x^{\prime }<x<L.$$

For an arbitrary input f(x), the response y(x) can be obtained from a superposition integral

(7A.3)$$y\left(x\right)=\underset{0}{\overset{L}{{\displaystyle \int }}}f\left(x^{\prime }\right)G\left(x\text{,}{x}^{\prime }\right)\text{d}{x}^{\prime }.$$

The differential equation for y is

(7A.4a)$$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=-f\left(x\right),0<x<L,$$

and the boundary conditions are

(7A.4b)$$y=0,x=0,$$

(7A.4c)$$y=0,x=L.$$

Equation 7A.3 for y is the solution to Equation 7A.4, where ...