Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 7 / iii / Solution

For the proposed vector, the first horizontal component of $Kq$ is $-k(\alpha +\beta )/2$. Since the corresponding component of $q$ is zero, a normal mode requires $\beta =-\alpha$. The vertical components then give $Kq=2kq$. With an arbitrary nonzero overall amplitude, choose $\alpha =1$ to obtainThe analogous pattern invariant under reflection across the vertical axis isDirect multiplication confirms $K{q}_x=2k{q}_x$. The two patterns give independent eigenvectors with the same eigenvalue, as expected from the square's rotation symmetry.