Use Of Fourier Series In The Analysis Of Discontinuous Periodic Structures | Original |

Don’t fear the jump. Embrace the Fourier series—just remember to keep enough harmonics to capture the edge.

Even with jumps, the Fourier coefficients (\varepsilon_m) decay as (1/m) (for a step change). Meanwhile, the electric field or pressure wave is assumed to follow Bloch’s theorem: Don’t fear the jump

The surprising answer is that when analyzing physical structures with abrupt changes—think square waves, step-index optical fibers, digital signals, or phononic crystals. Meanwhile, the electric field or pressure wave is

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? So why would anyone want to use Fourier

[ E(x) = e^{i k x} \sum_{n=-\infty}^{\infty} E_n , e^{i n K x} ]