If we fourier transform the wave equation, or alternatively attempt to
find solutions with a specified harmonic behavior in time
, we convert it into the following *spatial* form:
(+ k^2) &phis#phi;() = -&rho#rho;_&omega#omega;&epsi#epsilon;_0
(for example, from the wave equation above, where
,
, and
by
assumption). This is called the *inhomogeneous Helmholtz equation*
(IHE).

The Green's function therefore has to solve the PDE:
(+ k^2) G(,_0) = &delta#delta;(- _0)
Once again, the Green's function satisfies the *homogeneous*
Helmholtz equation (HHE). Furthermore, clearly the Poisson equation is the
limit of the Helmholtz equation. It is straightforward
to show that there are *several* functions that are good candidates
for
. They are:
G_0(,_0) & = &-(k|- _0|)4&pi#pi;|- _0|

G_+(,_0) & = &-e^+ik|- _0|4&pi#pi;|- _0|

G_-(,_0) & = &-e^-ik|- _0|4&pi#pi;|- _0|

As before, one can add arbitrary bilinear solutions to the HHE,
to any of
these and the result is still a Green's function. In fact, these forms
are related by this sort of transformation and superposition:
G_0(,_0) = 12(G_+(,_0) + G_-(,_0))
or
G_+(,_0) & = & F(,_0) + G_0(,_0)

& = & -i(k|- _0|)4&pi#pi;|-
_0| + G_0(,_0)
etc.

In terms of any of these:
&phis#phi;() & = & &chi#chi;_0() - 1&epsi#epsilon;_0&int#int;_V &rho#rho;(_0)
G(,_0)d^3x_0

& = & &chi#chi;_0() + 14 &pi#pi;&epsi#epsilon;_0&int#int;_V
&rho#rho;(_0)e^ik|- _0||- _0|d^3x_0
where
as usual.

We name these three basic Green's functions according to their *asymptotic time dependence* far away from the volume
. In this
region we expect to see a time dependence emerge from the integral of
e.g.
&phis#phi;(,t) &sim#sim;e^ik r - i&omega#omega;t
where
. This is an *outgoing spherical wave*.
Consequently the Green's functions above are usually called the
*stationary wave*, *outgoing wave* and *incoming wave*
Green's functions.

It is essential to note, however, that *any* solution to the IHE can
be constructed from *any* of these Green's functions! This is
because the *form* of the solutions always differ by a homogeneous
solution (as do the Green's functions) themselves. The main reason to
use one or the other is to keep the form of the solution *simple and
intuitive*! For example, if we are looking for a
that is
supposed to describe the *radiation* of an electromagnetic field
from a source, we are likely to use an outgoing wave Green's function
where if we are trying to describe the *absorption* of an
electromagnetic field by a source, we are likely to use the incoming
wave Green's function, while if we are looking for stationary (standing)
waves in some sort of large spherical cavity coupled to a source near
the middle then (you guessed it) the stationary wave Green's function is
just perfect.

[As a parenthetical aside, you will often see people get carried away in
the literature and connect the outgoing wave Green's function for the
IHE to the retarded Green's function for the Wave Equation (fairly done
- they are related by a contour integral as we shall see momentarily)
and argue for a causal interpretation of the related integral equation
solutions. However, as you can *clearly* see above, not only is
there no breaking of time symmetry, the resulting descriptions are all
just different ways of viewing the same solution! This isn't completely
a surprise - the process of taking the Fourier transform symmetrically
samples all of the past and all of the future when doing the time
integral.

As we will see when discussing radiation reaction and causality at the
very end of the semester, if anything one gets into trouble when one
assumes that it is *always* correct to use an outgoing wave or
retarded Green's function, as the actual field at any point in space at
any point in time is time reversal invariant in classical
electrodynamics - absorption and emission are mirror processes and *both* are simultaneously occurring when a charged particle is being
accelerated by an electromagnetic field.]