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The wave equation

The wave equation describes disturbances that propagate at a fixed speed $c$ , as on a string. d'Alembert's solution shows that every solution is a sum of two waves travelling in opposite directions without changing shape.

u_{tt} = c^2u_{xx}

the wave equation

u = F(x-ct) + G(x+ct)

d'Alembert's solution: two waves travelling in opposite directions

Symbols

c

wave speed

m/s

Example

$u_{tt}=25u_{xx}$ : $c=\sqrt{25}=5$ m/s. A string with $L=2$ m and $c=2$ m/s has mode 2 with angular frequency $\omega_2=\dfrac{c\cdot2\pi}{L}=2\pi\approx6.28$ rad/s.

Unlike the heat equation, wave solutions are not damped - they oscillate without dying out.

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Part of Laplace, Fourier and PDEs: Partial differential equations.