  An alternative way of visualizing electromagnetic waves in matter and of
deriving the Finite Difference Time Domain method (FDTD) for simulating
Maxwell's equations for one dimensional systems is presented. The method uses
d'Alembert's splitting of waves into forward and backward pulses of arbitrary
shape and allows for grid spacing and material properties that vary with
position. Constant velocity of waves in dispersionless dielectric materials,
partial reflection and transmission at boundaries between materials with
different indices of refraction, and partial reflection, transmission, and
attenuation through conducting materials are derived without recourse to
exponential functions, trigonometric functions, or complex numbers. Placing
d'Alembert's method on a grid is shown to be equivalent to FDTD and allows for
a simple and visual proof that FDTD is exact for dielectrics when the ratio of
the spatial and temporal grid spacing is the wave speed, a straightforward way
to incorporate reflectionless boundary conditions, and a derivation that FDTD
retains second order accuracy when the grid spacing varies with position and
the material parameters make sudden jumps across layer boundaries.
