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How the linear wave equation is a direct consequence of Newton's second law applied to any element of a string carrying a travelling wave?

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    $\begingroup$ Try to analyse a small infinitesimal length of the string. Draw its free body diagram and use Newton's second law. You will surely get something out of that. $\endgroup$
    – user243267
    Dec 1 '19 at 16:04
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The wave equation is not a consequence of analyzing a vibrating string. But rather analyzing a string leads to a result that fits the wave equation. Analysis of many other situations lead to results that fit the wave equation

See

https://physics.stackexchange.com/a/110842/45664

Derivation of 1D wave equation

In the second reference is a way to derive the wave equation using only $f(x-ct)$ and $f(x+ct)$, not requiring any physical situation at all.

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