Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 4 / a

An element with unperturbed coordinate length $dx$ has mass $\rho dx$ and transverse speed squared ${\dot{\psi }}_y^2+{\dot{\psi }}_z^2$. Its actual length isAt fixed tension the extra elastic energy to this order is $F(ds-dx)$. Subtracting it from the kinetic energy givesHere prime means ${\partial }_x$ and dot means ${\partial }_t$. The approximation retains quadratic transverse energy and neglects higher-order slope effects and longitudinal motion. The resulting field equations are $\rho {\ddot{\psi }}_i=F{\psi }_i^{\prime \prime }$ for $i=y,z$, with wave speed $\sqrt{F/\rho }$ when $F>0$.