 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 componentNewton's second law gives $\rho dx{y}_{tt}=Tdx{y}_{xx}$. Cancelling the element length yieldsThis 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.