🔎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, $c=1 / \sqrt{\varepsilon_0 \mu_0}$.

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