Plane Waves

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)

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)

for arbitrary smooth   and   is the most general solution of the 1-dimensional wave

equation. Any waveform that preserves its shape and travels along the  -axis at speed   is a

solution to the one dimensional wave equation (as can be verified directly, of course). How

boring! These particular harmonic solutions have this form (verify this).]

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)

as the condition for a solution. Then a real   leads to the plane wave solution indicated

above, with  , which is the most familiar form of the solution (but not the only one)!

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)

Which basically means for a real unit vector   that   and   are perpendicular to  , the

direction of propagation! A plane electromagnetic wave is therefore a transverse wave. This

seems like it is an important thing to know, and is not at all a mathematical conclusion of the

wave equation per se.

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

(9.23)

(the   cancels,  ). This means that   and   have the same phase if   

is real9.4

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)

and 

(9.25)

where   and   are constants that may be complex. It is worth noting that 

(9.26)

have the same dimensions and that the magnitude of the electric field is greater than that of the

magnetic field to which it is coupled via Maxwell's Equations by a factor of the speed of light in

the medium, as this will be used a lot in electrodynamics.

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 real9.5 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)

So,   must be orthogonal to   and the difference of their squares must be one. For

example: 

(9.35)

works, as do infinitely more More generally (recalling the properties of hyberbolics functions): 

(9.36)

where the unit vectors are orthogonal should work for any  .

Thus the most general   such that   is 

(9.37)

where (sigh)   and   are again, arbitrary complex constants. Note that if   is complex, the

exponential part of the fields becomes: 

(9.38)

This inhomogeneous plave wave exponentially grows or decays in some direction while

remaining a ``plane wave'' in the other (perpendicular) direction.

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.