(139) |

(140) |

(141) |

(142) |

(143) |

(144) |

The above equation, which reduces to Poisson's equation in the limit
, and is called *Helmholtz's equation*, is
linear, so we may attempt a Green's function method of solution. Let us
try to find a function
such that

(145) |

(146) |

(147) |

(148) |

(149) |

Reconstructing
from Eqs. (2.106), (2.110), and
(2.112), we obtain

(150) |

(151) |

Now, the real space Green's function for the
inhomogeneous wave equation (2.103) satisfies

(152) |

(153) |

(154) |

(155) |

The real space Green's function specifies the response of the system
to a point source at position which
appears momentarily at time . According to the *retarded Green's
function* the response consists of a spherical wave, centred
on , which propagates forward in time.
In order for the wave to reach position at time it must
have been emitted from the source at at the *retarded
time*
. According to the
*advanced Green's function* the response consists of a
spherical wave, centred
on , which propagates backward in time. Clearly, the advanced
potential is not consistent with our ideas about causality, which
demand that an effect can never precede its cause in time. Thus, the
Green's function which is consistent with our experience is

(156) |

In conclusion, the most general solution of the inhomogeneous wave equation
(2.103) which satisfies sensible boundary conditions at infinity and
is consistent with causality is

(157) |

(158) |

With the benefit of hindsight, we can see that the Green's function

(159) |

(160) |

(161) |

(162) |

Let us suppose that there are two solutions of Eq. (2.126) which satisfy
the boundary condition (2.125) and revert to the unique
Green's function
for Poisson's equation (2.113) in the limit
. Let us
call these solutions and , and let us form the difference
. Consider a surface which is a sphere of arbitrarily
small radius centred on the origin. Consider a second surface
which is a sphere of arbitrarily large radius centred on the
origin. Let denote the volume enclosed by these surfaces.
The difference function satisfies the homogeneous Helmholtz equation,

(163) |

(164) |

(165) |

Equation (2.127) can be written

(166) |

(167) |

(168) |

(169) |

(170) | |||

(171) |

where

(172) |

The large behaviour of the is clearly inconsistent
with the Sommerfeld radiation condition (2.125). It follows that
all of the in Eq. (2.132) are zero. The most general solution can
now be expressed in the form

(173) |

(174) |

(175) |

(176) |

Let us now consider the surface integral (2.129). Since we are interested in
the limit
we can replace by the first term of
its expansion in (2.136), so

(177) |