A time-domain approach to band formation in periodic media

Band formation in periodic media is a central topic in solid-state physics and is typically introduced using Bloch's theorem as an eigenvalue problem for infinite periodic systems. Although elegant, this approach assumes an idealized infinite lattice, shifts attention away from real-space wave dynamics, and presents band structures as static outcomes rather than emergent features of wave propagation. As a result, students often do not connect band gaps with familiar phenomena such as reflection, transmission, and interference, creating a disconnect between formal band theory and observable wave behavior. We address this problem by discussing how band gaps occur from wave propagation in finite periodic systems. A broadband wavepacket is launched into a layered medium using a staggered-grid finite-difference space-time algorithm, and the transmission spectra are used to identify band gaps. The Bloch dispersion relation and the exponential decay of the wave field within band gap regions are extracted, thus establishing a connection between real-space wave propagation and the Bloch description of periodic media.