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For example the current version of the Fourier analysis article on Wikipedia says the study is:

[…] named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

And as mentioned in e.g. Origin of the Fourier transform (1878), the book often cited re. Fourier series has a title that translates to Analytical Theory of Heat.

But I don't see any sines or cosines in the (rather simple-looking) heat equation.

Nowadays it seems that techniques based on Fourier's work are particularly famous through their use in Digital Signal Processing, in which FFTs/DFTs are often applied to data series that are directly related to sine waves. When I think of Fourier transforms I think of analyzing data that's full of signals and noise and rapid changes in levels, not something like a hot spot slowly averaging itself out through a chunk of metal.

There's another related question here, What are some good references elucidating the discovery/creation of Fourier Series?, but as that asked for references the answers there don't focus on specifically answering this question per se, and in some ways could be considered link-only answers relative to this question.

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    $\begingroup$ It is the same relation as between a system of first order linear ODE and the eigenvectors of the system matrix: in the eigenbasis the ODE reduce to an algebraic system. From the modern perspective (i.e. in hindsight), in any problem involving a linear operator one should automatically "see", or look for, its eigenvectors. Sines and cosines are the eigenfunctions of the second derivative, so they reduce solving the 1D heat equation to solving a (trivial) ODE. One gets Fourier series for problems on bounded intervals and Fourier transform on the entire real line. $\endgroup$
    – Conifold
    Jul 15 at 6:36
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    $\begingroup$ ... and, adding to @Conifold's comment, "separation of variables" in PDEs is a wonderful gadget, when it is available. $\endgroup$ Jul 15 at 21:15
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    $\begingroup$ Be sure to note the distinction between Fourier series and Fourier transforms. Fourier series appear naturally when solving the heat equation by separation of variables, which, in a direct sense, is the answer to the question. If you solve the heat equation, the answer is a linear combination of sines and cosines. As for Fourier's path to sines and cosines, I'll have to leave that to someone else $\endgroup$ Jul 23 at 16:10
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Not to sway others from providing a better answer from more interesting sources, but I think one clue can be found in the current version of the Wikipedia article for Joseph Fourier himself:

There were three important contributions in this work [i.e. The Analytic Theory of Heat], one purely mathematical, two essentially physical. In mathematics, Fourier claimed that any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. Though this result is not correct without additional conditions, Fourier's observation that some discontinuous functions are the sum of infinite series was a breakthrough. […]

Further simplifying and adding emphasis: "any function can be expanded into a series of sines". So even though heat transfer might not seem very "wavelike", functions representing it could still be analyzed in terms of trigonometric series. And perhaps doing so was useful to find or analyze solutions of the underlying differential equation.

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  • $\begingroup$ It should be noted that Fourier was wrong in claiming that every function can be decomposed in sine and cosines. A small portion yes but not the whole one. A function connected with heat is a function that can be decomposed. It goes to zero fast on both sides. $\endgroup$ Jul 18 at 20:06

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