Introduction to linear partial differential equations : A hyperbolic equation: the wave equation

A hyperbolic equation: the wave equation
Introduction to linear partial differential equations

Author : Gérard DEBEAUMARCHÉ

Publication date: October 10, 1999 | Lire en français

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2. A hyperbolic equation: the wave equation

2.1 Vibrating string equation

This was one of the first partial differential equations to be demonstrated.

It was studied in the first half of the 18th century by d'Alembert : $\frac{\partial^{2} u}{\partial t^{2}} = v^{2} \frac{\partial^{2} u}{\partial x^{2}}$

where v denotes the velocity of wave propagation in the string and u (x, t ) the ordinate of the point of abscissa x on the string at time t (this ordinate being...

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A hyperbolic equation: the wave equation