Fourier Series Real life application

Question

Is there any real life example of Fourier Series(NOT TRANSFORM) from either Physics or Engineering? It can be in trigonometric form or exponential form, cause I'm going to show a conversion from one form to another.

Answer 1

The original motivation for Fourier series was to approximate any periodic function, such as square waves and triangle waves, by sines and cosines. So they naturally come up in acoustics, where the multitude of sounds you hear through your headphones are built up from these sinusoidal functions, which are themselves absurdly simple to generate.

Yet Fourier series are not perfect and their approximation of a square wave is an example. Even after including a lot of sines and cosines, there still remains an overshoot/undershoot at the transition between high and low amplitudes:

This is the Gibbs phenomenon. The study of these overshoots, and how well a Fourier series can approximate any function in general, leads to the most important relation in digital signal processing: the Nyquist–Shannon sampling theorem. This has implications in, for example, capturing audio from a microphone to store in a digital format and then playing it to an audience: how well is that sound?

Fourier series underpin the answers to these questions. While the Fourier transform represents a procedure to convert between the continuous-wave and discrete samples, the series itself remains relevant to the analysis of how well this transformation goes both ways.

Other applications of the actual Fourier series in physics and engineering include the general heat equation (which was the problem Fourier himself solved with the series that got named after him), vibrational modes of structural elements in buildings, quantum harmonic oscillators and generally any place where some function repeats itself over and over.