Past exam of the mathematical methods II course of the University of Oxford / 2014 / 11 / a / Solution

For an element $dx$, the vertical components of the approximately constant tension are $T{y}_x$ at its two ends. Their difference is $T{y}_{xx}dx$ to first order in the small slope. Its inertial response is $\rho dx{y}_{tt}$. Thus Newton's second law givesHere $\rho$ 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.