Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 1 / b / iv / Solution

Now $A_0=0$, so every mass is again at rest after the pulse and $z=0+O({ϵ}^2)$. Put $k=ϵA_1>0$. The final transverse coordinates areThe unit vectors along $x=y$ and $x=-y$ are eigenvectors with eigenvalues $1-k$ and $1+k$, respectively. Consequently an initial circle becomes, to first order, an ellipse compressed along the line $x=y$ and stretched along $x=-y$. This persistent deformation is displacement memory.