Periodic Structures - Use Of Fourier Series In The Analysis Of Discontinuous

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:

Let’s explore how engineers and physicists use Fourier series to model and solve real-world discontinuous periodic systems. Consider a perfect square wave—a signal that jumps instantly between +1 and -1. This is the poster child for discontinuity. Its Fourier series is: This is the poster child for discontinuity

The surprising answer is that when analyzing physical structures with abrupt changes—think square waves, step-index optical fibers, digital signals, or phononic crystals. Embrace the Fourier series—just remember to keep enough

Don’t fear the jump. Embrace the Fourier series—just remember to keep enough harmonics to capture the edge. e^{i n K x} ]

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