## Harmonics of Vibrating Strings

In 1753, Daniel Bernoulli viewed the solutions as a superposition of simple vibrations, or harmonics. Such ideas stretch back to the Pythagoreans study of the vibrations of strings, which led to their program of a world of harmony. This chapter introduces several generic partial differential equations and sees how such equations lead naturally to the study of boundary value problems for ordinary differential equations. It determines the form of the series expansion and the Fourier coefficients in these cases. The chapter presents the derivation of the Fourier series representation for a general interval. It describes the solutions of the heat equation and wave equation on a finite interval to obtain an initial value Green’s function. The main difference from the solution of the heat equation is the form of the time function. The chapter seeks to solutions of initial-boundary value problems involving the heat equation and the wave equation.