= Solution
Consider a string element of equilibrium length $dx$. Its <mass> is $\rho\,dx$, where $\rho$ is mass per unit length. Under the small-slope approximation, the <tension> is approximately uniform and its transverse component at an endpoint is $T\,\partial y/\partial x$. The two endpoint <forces> have net transverse component
$$
T\left[\frac{\partial y}{\partial x}(x+dx,t)-\frac{\partial y}{\partial x}(x,t)\right]\simeq T\frac{\partial^2y}{\partial x^2}\,dx.
$$
<Newton's second law> gives $\rho\,dx\,y_{tt}=T\,dx\,y_{xx}$. Cancelling the element length yields
$$
\boxed{\frac{\partial^2y}{\partial t^2}-c^2\frac{\partial^2y}{\partial x^2}=0,\qquad c=\sqrt{\frac{T}{\rho}}}.
$$
This is the one-dimensional <wave equation> for transverse displacement. Its derivation neglects gravity, bending stiffness and nonlinear changes in <tension> and segment length; the given small slope makes those omissions consistent to first order in the displacement.
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