When a periodic structure has a discontinuity—like a sudden change in material properties or a sharp pulse in time—a standard differential equation approach often fails. At the point of the jump, the derivative doesn't exist in the classical sense.
The analysis of discontinuous periodic structures involves solving Maxwell's equations or other governing equations with periodic boundary conditions. However, the presence of discontinuities makes it challenging to find an analytical solution. This is where Fourier series come into play. When a periodic structure has a discontinuity—like a