> Fourier argued that the distribution of heat through the rod could be written as a sum of simple waves.

How do you even begin to think of such things? Some people are wired differently.

I don't know/remember the historical derivation, but the story you might get in a class goes something like:

Energy can't be created or destroyed, so it follows a continuity equation: du/dt + dq/dx = 0. Roughly, the only way for energy to change in time is by coming from somewhere in space. There are no magic sources/sinks (a source or sink would be a nonzero term on the right).

Then you have Fourier's law/Newton's law of cooling: heat flows proportional to temperature difference, from high to low: q = -du/dx.

Combining these, you get the heat equation: du/dt = d^2 u/dx^2.

Now if you're very fancy, you can find deeper reasons for this, but otherwise if you're in engineering analysis class, just guess that u(t,x)=T(t)X(x). i.e. it cleanly factors along time/space.

But then T'(t)X(x)=X''(x)T(t), so T'(t)/T(t) = X''(x)/X(x). But the left and right are functions of different independent variables, so they must be constant. So you get X''= λX for some lambda. But then from calc1, X is sin/cos.

Likewise T' = λ T so T is e^-λt from calc 1.

Then since it's a linear differential equation, the most general solution (assuming it splits the way we guessed) is a weighted sum of any allowable T(t)X(x), so you get a sum of exponentially decaying (in time) waves (in space).

It didn't come completely out of nowhere, Euler and Bernoulli had looked at trigonometric series for studying the elastic motion of a deformed beam or rod. In that case, physical intuition about adding together sine waves is much more obvious. https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_t...

Other mathematicians before Fourier had used trigonometric series to study waves, and physicists already understood harmonic superposition on eg a vibrating string. I don't have the source but I believe Gauss even noted that trigonometric series were a solution to the heat equation. Fourier's contribution was discovering that almost any function, including the general solution to the heat equation, could be modelled this way, and he provided machinery that let mathematicians apply the idea to an enormous range of problems.

I think he was very familiar with differential equations and series expansions and the "wild west" stage of calculus in general. The frontier of cool and interesting mathematics has moved a lot in 200 years.