# Importance of Sine Waves And Maxwell's Equations

Sine waves, by a huge margin, are the most important waveform in electronics - we measure a circuit's frequency response with sine waves and represent all other signals through sine waves, with the help of the Fourier transform. However, Fourier transform on itself does not make sine waves special - after all, there are other ways to decompose a signal into a bunch of orthogonal functions (wavelet transform, for instance).

So there must be a fundamental physical reason for sine waves being so important. I would imagine that this reason is the fact that the electromagnetic wave equation (which can be readily derived from Maxwell's equations) is a second-order diff eq, so a sinusoid is its solution - that's why sine waves do not disperse in transmission lines and that's why a "frequency component" that has a particular propagation velocity in a medium, is a sinusoid.

Is the reasoning above correct? In a fantastical world where the equation of electric signal propagation was, for example, a third-order diff eq, would sine waves be as important as in our reality (I know it's a bit ridiculous to ask what would happen if the fundamental physical laws were different, but still)?

• I speculate that complex exponentials and sinusoids (i.e. exponentials, sinusoids, and products thereof) will still be important, as they are eigenfunctions of the differential operator no matter how many times it is applied. I do not, however, have the appropriate mathematical background to make a formal claim or answer out of it. Jul 21 '20 at 22:05
• Related on Physics.SE: Why use Fourier series instead of Taylor? Jul 21 '20 at 22:17
• The fact that a sinusoid is a single "pure" frequency is probably the reason it's so heavily used. Imagine the problems if our power was 240v/120vAC 60Hz squarewave... Jul 21 '20 at 22:35
• Jul 21 '20 at 23:39
• Does this answer your question? Why is sine wave preferred over other waveforms? Jul 22 '20 at 9:06

If you take the Maxwell equations, they can be combined to derive the wave equation for light. Since the orthogonal basis for the solution space of the wave equation is the complex exponentials, we can see that sine waves are intimately related to the Maxwell equations here. When wave propagation is defined by the Maxwell equations / the wave equation, they always must be able to expressed as linear combinations of complex exponential. This is really the reason why sine waves don't disperse. Any other wave forms contain superpositions of the sine waves that travel at different speeds, causing dispersion.

What's said here isn't much different from what you've already put together, but it is definitely a fundamental physical reason for sine wave popularity.

Regarding the fantastical third world: I think that sine waves would still be important because the solutions for linear third order differential equations are still exponentials. That said, it would also be important to consider (real) exponentials, which are sort of the other extremely fundamental function that is seen everywhere. Depending on the third order differential equation, there could also be exponential solutions, which could be a very important part of the fantastical world.

Extending the thoughts on Maxwell, when such waves attempt to propagate INTO a sea of electroncs, the differential_equation solution is not a sin, but the erfc(T,X).

And a distributed_element model is appropriate.

Some years back, I actually measured the speed_of_propagation of a fast edge into and thru standard thickness (35 microns, 1.4 mils) PCB foil.

The speed was about 150 nanoSeconds to penetrate and exit the other side.

That is about 1,000,000 X slower than the speed of light.

And to my delight, the famous E&M by Jackson did the math, predicting the same value.

Again, for such transient and real_world time results, the "erf" and "erfc" functions are needed.