If you’ve ever studied Fourier series, you likely remember the core idea: any periodic function can be broken down into a sum of simple sine and cosine waves. But then came the catch—the series often struggles with discontinuities , producing that infamous 9% overshoot known as the Gibbs phenomenon. So why would anyone want to use Fourier series on discontinuous problems?
[ E(x) = e^{i k x} \sum_{n=-\infty}^{\infty} E_n , e^{i n K x} ] If you’ve ever studied Fourier series, you likely
The surprising answer is that when analyzing physical structures with abrupt changes—think square waves, step-index optical fibers, digital signals, or phononic crystals. If you’ve ever studied Fourier series