# Are microwaves sinusoidal or is that just a model?

I know that projectiles are parabolas because I can derive that from constant acceleration. And the height of a Ferris wheel rider vs. time is demonstrably a sine wave. What is the underlying thing that tells us microwaves are sinusoidal vs. some other periodic shape? I'm teaching trig.

• Do you mean electromagnetic frequencies in the GHz range, or the machines that heat up our tacos? – psitae Apr 4 '19 at 21:35
• Who says that microwaves are sine waves? Any microwave signal that is modulated with information is, in fact, not a sine wave. But sine waves are interesting because any periodic function can be expressed as an infinite sum of sines and cosines, and that fact enables powerful mathematical techniques for analyzing signals and periodic motions. – Solomon Slow Apr 4 '19 at 21:50
• Beth, please elaborate. You're teaching trig at the high school level? How is the mathematical function of a microwave related to this? – David White Apr 4 '19 at 22:27
• Microwaves can be sinusoidal but don’t have to be. – G. Smith Apr 4 '19 at 23:03
• @SolomonSlow That's an answer. You should post it. – FGSUZ Apr 4 '19 at 23:07

## 1 Answer

The wave equation for light tells us that solutions are sinusoidal:

$$\frac{\partial^2 E}{\partial t^2} = c^2\frac{\partial^2 E}{\partial x^2}$$

(here, in one dimensions with the electric field $$E(x, t)$$ a function of position $$x$$ and time $$t$$).

If you guess a sinusoidal solution:

$$E(x, t) = e^{i(kx-\omega t+\phi)}$$

(note:

$$e^{i(kx-\omega t+\phi)} = \cos{(kx-\omega t+\phi)}+i\sin{(kx-\omega t+\phi)}$$

is the most general solution that is sinusoidal in space and time)

and plug it in, you get:

$$-\omega^2E(x, t) = -c^2k^2E(x, t)$$

which means:

$$\omega = ck =2\pi c/\lambda$$

is the relation between frequency and wavelength ($$\lambda = 2\pi/k$$).

Hence you can specify a solution by its frequency $$\omega$$.

This represents and infinitely long wave with infinite duration. In the real world, boundary conditions matter, so this is not the actual solution. Nevertheless, the machinery of Fourier analysis allows us to build any solution, periodic or not, as a superposition of different frequencies.

• "The wave equation for light tells us that solutions are sinusoidal." Absolutely not! For the pure wave equation without any added dispersive terms, disturbances of any shape are allowed. The general one-dimensional solution is $E(x,t)=f(x-ct)+g(x-ct)$, where $f$ and $g$ are any (suitably smooth) functions. – Buzz Apr 5 '19 at 1:02
• One of the arguments should be $(x+ct)$, otherwise I agree. – Pieter Apr 5 '19 at 2:12
• I am at 10,500' feet, straight from sea level. Sorry. – JEB Apr 5 '19 at 14:13
• @Buzz did you read the "allows us to build any solution, periodic or not, as a superposition of different frequencies." part? – JEB Apr 11 '19 at 23:36