Value Proposition of Fourier Analysis?

Question

I am a software engineer trying to wrap his head around Fast Fourier Transform (FFT). Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of the algorithm/operations, and in fact will likely just use an open source math library to do most of the heavy lifting for me. But there's something fundamental here that I just want to understand.

According to Wikipedia:

Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and 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:

For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.

So here we have two examples:

1. Decomposing a general function into component trig functions to study heat transfer; and
2. Decomposing a sampling of a musical note into component frequencies

So I get the what here; that is, what FFT/Fourier Analysis is attempting to do (decomposing a function into component functions), and like I said, I can handle the how later (either coding up FFT myself or using a library), but what I'm stuck on is the why.

Why do this? Using one of the examples above, why decompose a sound sample (musical note) into component frequencies? What is there to be gained/learned from doing this? What extra information does this expose for analytical purposes? Why can we better understand heat transfer by using FFT to decompose some "heat function" into smaller/component trig functions?

Answer 1

Differential equations such as the one that describes heat flow are often easily solvable when they have a sinusoidal (or complex exponential) source term. So, physicists solve it in the easy case and then make use of the Fourier transform to turn the general problem (with a source that is not sinusoidal) into a sum of easy problems.

In other words, thanks to the Fourier Transform, the solution of a linear differential equation with a non-sinusoidal source term (hard to solve) can be written as a sum of solutions of linear differential equation with sinusoidal source terms (easy to solve).

Answer 2

Sound is moving waves in matter. Sin and Cos are waves too.

Heat transfer is often modelled with differential equations. Complex exponentials (which are the basis functions in the Fourier Transform) are eigenfunctions to the operation of differentiation.