The 1D Wave Equation

doi:10.1117/3.2250614.ch4

Book: Computing the Flow of Light: Nonstandard FDTD Methodologies for Photonics Design

Author(s): James B. Cole, Saswatee Banerjee

Published: 2017

https://doi.org/10.1117/3.2250614.ch4

Author Affiliations +

Abstract

In a uniform medium, Maxwell’s equations reduce to the wave equation, and analytic solutions are derived by solving the wave equation subject to the appropriate boundary conditions. Although the 1D wave equation may seem simple, when boundary and initial conditions are specified and when sources are included, its analytical solutions are far from trivial and sometimes do not even exist. Before moving on to FDTD for the wave equation, we develop some analytical solutions to which numerical solutions can be compared. Topics: • Derivation and various forms of wave equation • Analytical solutions • Extra topics: relativity from the wave equation • Green’s function and associated mathematics