= Solution
Take an ideal flexible <mechanical string> with constant <tension> $T>0$ and uniform <linear mass density> $\mu>0$. Let $x$ measure distance along its straight equilibrium position and let $y(x,t)$ be its small transverse <displacement>. Neglect <damping>, bending stiffness and external distributed <forces>, and assume $|y_x|\ll1$.
A segment of reference length $\Delta x$ has mass $\mu\Delta x$. The endpoint <tension> forces have transverse resultant
$$
T\sin\theta(x+\Delta x)-T\sin\theta(x)
\simeq T\bigl[y_x(x+\Delta x,t)-y_x(x,t)\bigr],
$$
since $\sin\theta\simeq\tan\theta=y_x$ for small slopes. <Newton's second law>, divided by $\Delta x$ and taken to the limit, gives
$$
\mu y_{tt}=Ty_{xx},\qquad
\boxed{y_{xx}=\frac1{c^2}y_{tt},\quad c=\sqrt{T/\mu}.}
$$
Here $t$ is time, the subscripts denote <partial derivatives>, and $c$ is the <wave speed>. The transverse <acceleration> is $y_{tt}$; $y_{xx}$ describes the small-slope curvature responsible for the net transverse <force>.
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