Plane waves can propagate in any direction. Any superposition of these waves, for all possible , is also a solution to the wave equation. However, recall that and are not independent, which restricts the solution in electrodynamics somewhat.

To get a feel for the interdependence of
and
, let's
pick
so that e.g.:

(9.17) | |||

(9.18) |

which are plane waves travelling to the right or left along the -axis for any complex , . In one dimension, at least, if there is no dispersion we can construct a fourier series of these solutions for various that converges to any well-behaved function of a single variable.

[Note in passing that:

(9.19) |

If there is dispersion (velocity a function of frequency) then the
fourier superposition is no longer stable and the last equation no
longer holds. *Each* fourier component is still an exponential, but
their velocity is different, and a wave packet spreads out it
propagates. We'll look at this shortly to see how it works for a very
simple (gaussian) wave packet but for now we'll move on.

Note that
and
are connected by having to satisfy
Maxwell's equations even if the wave is travelling in just one direction
(say, in the direction of a unit vector
); we cannot choose the
wave amplitudes separately. Suppose

where , , and are constant vectors (which may be complex, at least for the moment).

Note that applying
to these solutions in the HHE
leads us to:

(9.20) |

This has mostly been ``mathematics'', following more or less directly
from the wave equation. The same reasoning might have been applied to
sound waves, water waves, waves on a string, or ``waves'' of
nothing in particular. Now let's use some *physics* in the spirit
suggested in the last section of the Syllabus and see what it tells us
about the *particular* electromagnetic waves that follow from
Maxwell's equations turned into the wave equation. These waves all
satisfy *each* of Maxwell's equations separately.

For example, from Gauss' Laws we see e.g. that:

(9.21) |

or (dividing out nonzero terms and then repeating the reasoning for ):

(9.22) |

Repeating this sort of thing using one of the the curl eqns (say,
Faraday's law) one gets:

(9.23) |

*If*
is real (and hence a unit vector), then we can
introduce three real, mutually orthogonal unit vectors
and use them to express
the field strengths:

(9.24) |

(9.25) |

(9.26) |

These relations describe a wave propagating in the direction
. This
follows from the (time-averaged) Poynting vector (for any particular
component pair):

(9.27) | |||

(9.28) | |||

(9.29) | |||

(9.30) |

Now, kinky as it may seem, there is no real^{9.5} reason
that
cannot be complex (while remains real!)
As an exercise, figure out the complex vector of your choice such that

(9.31) |

Since I don't really expect you to do that (gotta save your strength
for the real problems later) I'll work it out for you. Note that this
is:

(9.32) |

(9.33) |

(9.34) |

(9.35) |

(9.36) |

Thus the **most general**
such that
is

(9.37) |

(9.38) |

Fortunately, nature provides us with few sources that produce this kind of behavior (Imaginary ? Just imagine!) in electrodynamics. So let's forget it for the moment, but remember that it is there for when you run into it in field theory, or mathematics, or catastrophe theory.

Instead we'll concentrate on kiddy physics descriptions of polarization when is a real unit vector, continuing the reasoning above.