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Green's Functions for the Wave Equation

As by now you should fully understand from working with the Poisson equation, one very general way to solve inhomogeneous partial differential equations (PDEs) is to build a Green's function11.1 and write the solution as an integral equation.

Let's very quickly review the general concept (for a further discussion don't forget WIYF ,MWIYF). Suppose ${\cal D}$ is a general (second order) linear partial differential operator on e.g. $\rm I \hspace{-.180em} R^3$ and one wishes to solve the inhomogeneous equation:

\begin{displaymath}
{\cal D} f(\mbox{\boldmath$x$}) = \rho(\mbox{\boldmath$x$})
\end{displaymath} (11.26)

for $f$.

If one can find a solution $G(\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0)$ to the associated differential equation for a point source function11.2:

\begin{displaymath}
{\cal D} G(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0) = \delta(\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0)
\end{displaymath} (11.27)

then (subject to various conditions, such as the ability to interchange the differential operator and the integration) to solution to this problem is a Fredholm Integral Equation (a convolution of the Green's function with the source terms):
\begin{displaymath}
f(\mbox{\boldmath$x$}) = \chi(\mbox{\boldmath$x$}) + \int_{...
...x$},\mbox{\boldmath$x$}_0)
\rho(\mbox{\boldmath$x$}_0) d^3x_0
\end{displaymath} (11.28)

where $\chi(\mbox{\boldmath$x$})$ is an arbitrary solution to the associated homogeneous PDE:
\begin{displaymath}
{\cal D} \left[ \chi(\mbox{\boldmath$x$}) \right]= 0
\end{displaymath} (11.29)

This solution can easily be verified:

$\displaystyle f(\mbox{\boldmath$x$})$ $\textstyle =$ $\displaystyle \chi(\mbox{\boldmath$x$}) + \int_{\rm I \hspace{-.180em} R^3} G(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)
\rho(\mbox{\boldmath$x$}_0) d^3x_0$ (11.30)
$\displaystyle {\cal D} f(\mbox{\boldmath$x$})$ $\textstyle =$ $\displaystyle {\cal D}\left[ \chi(\mbox{\boldmath$x$})\right] +
{\cal D} \int...
...G(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0) \rho(\mbox{\boldmath$x$}_0) d^3x_0$ (11.31)
$\displaystyle \rho(\mbox{\boldmath$x$}_0) d^3x_0$     (11.32)
$\displaystyle {\cal D} f(\mbox{\boldmath$x$})$ $\textstyle =$ $\displaystyle 0 +
\int_{\rm I \hspace{-.180em} R^3} {\cal D} G(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0) \rho(\mbox{\boldmath$x$}_0) d^3x_0$ (11.33)
$\displaystyle {\cal D} f(\mbox{\boldmath$x$})$ $\textstyle =$ $\displaystyle 0 +
\int_{\rm I \hspace{-.180em} R^3} \delta(\mbox{\boldmath$x$}-\mbox{\boldmath$x$}_0) \rho(\mbox{\boldmath$x$}_0) d^3x_0$ (11.34)
$\displaystyle {\cal D} f(\mbox{\boldmath$x$})$ $\textstyle =$ $\displaystyle \rho(\mbox{\boldmath$x$})$ (11.35)

It seems, therefore, that we should thoroughly understand the ways of building Green's functions in general for various important PDEs. I'm uncertain of how much of this to do within these notes, however. This isn't really ``Electrodynamics'', it is mathematical physics, one of the fundamental toolsets you need to do Electrodynamics, quantum mechanics, classical mechanics, and more. So check out Arfken, Wyld, WIYF , MWIYFand we'll content ourselves with a very quick review of the principle ones we need:



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Next: Poisson Equation Up: Radiation Previous: Quickie Review of Chapter   Contents
Robert G. Brown 2014-08-19