According to wikipedia and other sources, there are no longitudinal electromagnetic waves in free space. I'm wondering why not.
Consider an oscillating charged particle as a source of EM waves. Say its position is given by $x(t) = sin(t)$. It is clear that at any point on the $x$ axis, the magnetic field is zero. But there is still a time-varing electric field (more or less sinusoidal in intensity, with a "DC offset" from zero), whose variations propagate at the speed of light. This sounds pretty wave-like to me. Why isn't it? Is there perhaps a reason that it can't transmit energy?
A very similar question has already been asked, but it used a "rope" analogy, and I feel that the answers overlooked the point that I'm making.