Derivative Tracing: An Educational Numerical and Graphical Method for Visualizing Electromagnetic Wave Propagation

Abstract

In this study, we present a Derivative Tracing method to strengthen students’ conceptual understanding of electromagnetic (EM) wave propagation directly from Maxwell’s equations. We aim to develop a graphical and numerical approach that lets students explore EM wave behavior on graph paper or with basic programming tools. We translate the abstract mathematical structure of Maxwell’s equations into visual and computational representations, bridging theoretical electromagnetics and classroom understanding. The method first rewrites Maxwell’s equations in a simplified spatial-derivative form. We then specify boundary conditions for the electric and magnetic fields. Next, a simple numerical scheme updates these fields over time to illustrate the mechanisms that drive EM wave propagation. Derivative Tracing matches the standard FDTD results within 5% while remaining much simpler and more classroom-friendly. Relative to conventional FDTD and FEM, it emphasizes conceptual clarity and educational accessibility, making it suitable for teaching electromagnetics in undergraduate laboratories. We further apply the approach to several scenarios, including wave propagation in transmission lines, lossy media, and vacuum. Overall, the method offers an accessible framework that supports deeper understanding of EM wave dynamics without advanced simulation software or high computational resources.