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

For one transverse displacement $\psi$ the density isIts variation is $\delta S=\int dtdx[\rho {\psi }_t\delta {\psi }_t-F{\psi }_x\delta {\psi }_x-B{\psi }_{xx}\delta {\psi }_{xx}]$. Integrating once in time and twice in space gives the bulk coefficientThe spatial boundary contribution is $[(-F{\psi }_x+B{\psi }_{xxx})\delta \psi -B{\psi }_{xx}\delta {\psi }_x]$ integrated over time. ThusInsert $\psi \propto e^{i(kx-\omega t)}$ to obtainFor compression $F<0$, sufficiently long waves have ${\omega }^2<0$ and grow exponentially. Positive bending rigidity stabilizes short wavelengths; without it the compressed small-slope model would have no short-wavelength regularization.