4 The Wave Equation in One Dimension

The vibrating string was the first partial differential equation to be solved, by d’Alembert in 1747. Its solution is completely explicit, and each of its features is a property of hyperbolic equations in general: disturbances move at a finite speed, energy is conserved, singularities are carried along characteristics without being smoothed, and boundaries reflect waves. This section works them out for \begin {equation} \label {eq:wave1d} u_{tt} = c^2u_{xx} , \end {equation} which describes small transverse vibrations of a string of tension \(T\) and linear density \(\rho \) with \(c = \sqrt {T/\rho }\), as well as sound in a pipe and signals on a transmission line.

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