Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 2 / c / Solution

With the specified ratios, the two factors in the modal amplitude become $\sin (n\pi /5)$ and $\sin (n\pi /14)$. Hence a normal mode is unexcited precisely when one of these factors is zero:That includes all multiples of five or fourteen, with multiples of seventy counted once. The mode $n=0$ is not a fixed-end string mode.