# The Wave Equation-Derivation and Physical Applications and Wave Speed Determination

The wave equation, given by $$\partial\_t^2 u-c^2 \nabla^2 u=f$$, is a crucial partial differential equation in physics defined by its second-order time derivative, which enables the description of propagating waves unlike the diffusion equation. Its application depends on the physical context: for transversal waves on a string or a membrane, the equation arises from applying Newton's second law under a small deviation approximation ( $$|\nabla u| \ll 1$$ ), with the wave speed squared determined by the ratio of tension to density ( $$c^2=S / \rho\_{\ell}$$ or $$c^2=\sigma / \rho\_A$$ ). In contrast, for electromagnetic fields, the wave equation for the magnetic field $B$ is a direct and exact consequence of Maxwell's equations, where the wave speed is precisely the speed of light, $$c=1 / \sqrt{\varepsilon\_0 \mu\_0}$$.

### :clapper:Demos

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### :bubbles:Cue Column

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