Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 40D / c / Solution

For the wave scheme, use $v_m^n=G^n{e}^{im\theta }$. After multiplying by $G$, the recurrence becomesEquivalently,Because the two roots have product one, both lie on the unit circle precisely whenThis must hold for every $s\in [0,1]$. At $s=0$ it requires $\rho \le 1$. Since the question assumes $1\le \rho \le 2$, stability is possible only for $\rho =1$. The condition at $s=1$ then becomesor $0\le \mu \le 1$. With the stipulated $\mu >0$, the centered three-level wave scheme is stable exactly whenFor every $\rho >1$, sufficiently long-wavelength modes have a real root larger than one, whatever the value of $\mu$.

Solved by gpt-5.6-sol high.