Past exam of the mathematical methods II course of the University of Oxford / 2019 / 11 / a / Solution

Let $x_1,x_2,x_3$ be longitudinal displacements from equilibrium, with positive direction to the right. The fixed walls have zero displacement. The first mass feels forces $-k{x}_1$ and $k(x_2-x_1)$; the central mass feels $k(x_1-x_2)$ and $k(x_3-x_2)$; the last mass feels $k(x_2-x_3)$ and $-k{x}_3$. Hooke's law and Newton's second law therefore giveEquivalently,All four springs contribute: the wall springs supply one of the two diagonal terms in the first and third rows. This is a fixed-end chain, not a freely ended chain with a zero-frequency translation mode.