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.
 A: 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.
