= Solution
For an element $dx$, the vertical components of the approximately constant <tension> are $Ty_x$ at its two ends. Their difference is $Ty_{xx}dx$ to first order in the small slope. Its inertial response is $\rho\,dx\,y_{tt}$. Thus <Newton's second law> gives
$$
\boxed{y_{tt}-c^2y_{xx}=0,\qquad c=\sqrt{T/\rho}.}
$$
Here $\rho$ is mass per unit length. Small slopes, negligible gravity, uniform tension and negligible bending stiffness or dissipation are the assumptions in this <wave equation on a string>.
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