The wave equation is a partial differential equation that characterizes how waves propagate. The analysis can be quite complex, but we will examine a few simple examples of solutions to the wave equation for special cases. These solutions can describe a large number of phenomena in musical acoustics from vibrating strings to columns of air.
First the basic physics of traveling waves and the traveling-wave decomposition are introduced. Then the types of reflections found in rigidly-terminated vibrating strings are considered and a number of musically-relevant initial conditions are investigated. Students learn how to predict the behavior of a a rigidly-terminated vibrating string given any initial conditions. Finally, students sketch the traveling-wave decomposition of a vibrating string for three given initial conditions.