Lectures on Electromagnetism, Charged Particles, and Quantum Wells. Number one.

I remember going into my Electromagnetism class at Texas A&M with a lot of justified apprehension. I had been exposed to Maxwell's Equations in both their derivative and intregral forms, see https://www.allaboutcircuits.com/technical-articles/maxwells-equations-in-present-form/ for a well formed expression of both forms of these equations ALONG WITH the constituative relationships that are necessary to derive the amplitudes of the resultant solutions. Thanks to TI for a bit of clean erudition at the expense of some good PR. Now, I got this bit of curvy hieroglyph tossed at me RIGHT AFTER I had been exposed to surface integrals, which in theory should have been enough to define the boundary conditions of the surface and contour integrals used here to describe the general formulation of these governing equations in light of the Taylor series expansion AND a bit of inference about surfaces ALWAYS being defineable as a contiguous and collapsing series of TRIANGLES whose limit would equal that of ANY smooth surface. For all the good THAT would have done me.

Inevitably ANY lecturer resorts to promulgating FIRST an infinite perfectly conducting plane of some sort whose surface electric field strength can be assumed to be zero because the conductor can support INFINITE current but that is unnecessary because the strength of the IMPOSSIBLE TO IMPLEMENT forcing current is finite (please don't comment on the field strengths of such a current across an infinite perfectly conducting plane) and the resulting field propagates in a straight plane PERFECTLY off the conductor with derivable wavelength. He or she goes from there to rectangular cavity conductors, cylindrical cavity (or dialectric filled cavity) waveguides, then to half and quarter wavelength antennas and stops about there. Radars are treated in another course where their energy is modeled along more optical lines and finally come actual light (LED, Laser, and fiber) treatments.

Look, these boundary condition first driven solutions to Maxwell's equations leave me stretched and puffy eyed. Let's look at this again. Even a child is exposed at some point in their early adolescence (and probably before) to the idea that light has a wave nature. Photons and stuff that propagate out from their source in an expanding front of dissipating energy that loses strength about linearly as it propagates outward. We all pretty much get that. Lets start with waves (I know, there's the equally important magnetic component as well, but that propagates along in lockstep with the time varying electric field, so let's just note that it's there and is ALSO perfectly linear in it's interactions with other magnetic fields.) and formalize them into a cosine as a first step. So here we are, a wave can be mathematically expressed as Acos(Bx), where A is an amplitude and B is an angular frequency.

The next step is to look at sums of waves. With a small loss of generality lets look first at a sum of equal amplitude waves, Acos(Bx) and Acos(Cx). there is a funny geometric property that is EASY to prove that demonstrates that this expression is equal to cos([(B+C)/2]x)(cos((B-C)x) + cos(C-B)x). Modulation in signal processing terms. The difficulty of unequal amplitude waves at the same frequency can be addressed by expressing the greater amplitude wave as a sum of two positive waves at the same frequency and phase and the amplitude of the lesser wave as a sum a positive and negative amplitude wave at the same phase and deriving the coupling appropriatley. Here we have a clue how to conduct our analysis, but we need the aid of a function that will take us out of the discrete arena and into the continuous one, and that function is the gaussian function and the tool that will work the transform is the convolution. The convolution tool is SO useful to the signals processor that it aught to be called the GREAT ENABLER. https://www.britannica.com/science/convolution-mathematics

Now we have to introduce a bit of non-ideality to our analysis. A cosine function CANNOT exist in reality, because it would extend infinitely backwards and infinitely forwards in time and, in the case of the case of electromagnetic waveforms, infinitely backwards and forwards in space as well. Any electromagnetic phenomenea that did that would have infinite energy. This can be demonstrated by the introduction of the cosine transform, https://en.wikipedia.org/wiki/Sine_and_cosine_transforms See how a perfect cosine of ANY finite amplitude will explode in energy content, energy defined as the ability to do work. Note that a cosine transform is just a continuous mapping of the convolution of a function with cosines of continuos frequency. Remember that the formula used here represents electric field strength of a propagating wave. The Link from electric field to the ability to do work is made via the Lorentz field law https://en.wikipedia.org/wiki/Lorentz_force. To finitely bound the waveforms under consideration here we will consider functions made of sums, sums occasionally infinite BUT STILL DISCRETE, of cosine functions multiplied by a gaussian function. Spectrally this will look like a cosine spike convolved with a gaussian function. A cosine function up for a long time will look specterlly like a tight band of cosine energy concentrated about a cosine midpoint.

That's the end of this lecture.