I assume we're talking about electromagnetic waves in the classical limit here, so that gives me an excuse to avoid thinking about the quantum effects. So all we have for rules is the set of Maxwell equations. We can mathematically write any EM wave pattern as a combination of plane waves, each with a particular frequency and wavelength and direction. We aren't interested in the transverse part, so let's ask what would happen if you had a purely longitudinal wave of the electric (E)
and magnetic (B
) fields. That would look like, for example, a uniform E
field pointing in the z direction in the xy plane, with the strength of E
varying sinusoidally along z. There are two big problems here. First, that E
has a divergence, which requires some charge to be present, not just propagating fields. So this is not a propagating EM wave. Second, to find out how much B
is changing, Maxwell tells us to just take the curl of E
, times some constant. However, here curl(E
)=0. Therefore B
isn't changing in time. That's another reminder that this is not a propagating wave.
So Maxwell's equations tell us that only transverse EM waves can exist.
(published on 07/07/11)