Consider a one-dimensional chain of $2 N \gg 1$ atoms, each of mass $m$. Impose periodic boundary conditions. The forces between neighbouring atoms are modelled as springs, with alternating spring constants $\lambda$ and $\alpha \lambda$. In equilibrium, the separation between the atoms is $a$.

Denote the position of the $n^{\text {th }}$atom as $x_{n}(t)$. Let $u_{n}(t)=x_{n}(t)-n a$ be the displacement from equilibrium. Write down the equations of motion of the system.

Show that the longitudinal modes of vibration are labelled by a wavenumber $k$ that is restricted to lie in a Brillouin zone. Find the frequency spectrum. What is the frequency gap at the edge of the Brillouin zone? Show that the gap vanishes when $\alpha=1$. Determine approximations for the frequencies near the centre of the Brillouin zone. Plot the frequency spectrum. What is the speed of sound in this system?