= Solution
For the wave scheme, use $v_m^n=G^n e^{im\theta}$. After multiplying by $G$, the recurrence becomes
$$
G^2+(4\mu s-2\rho)G+1=0,
\qquad
s=\sin^2\left(\frac\theta2\right).
$$
Equivalently,
$$
G+G^{-1}=2(\rho-2\mu s).
$$
Because the two roots have product one, both lie on the unit circle precisely when
$$
|\rho-2\mu s|\leq1.
$$
This must hold for every $s\in[0,1]$. At $s=0$ it requires $\rho\leq1$. Since the question assumes $1\leq\rho\leq2$, stability is possible only for $\rho=1$. The condition at $s=1$ then becomes
$$
|1-2\mu|\leq1,
$$
or $0\leq\mu\leq1$. With the stipulated $\mu>0$, the <centered three-level wave scheme> is stable exactly when
$$
\boxed{\rho=1,\qquad 0<\mu\leq1}.
$$
For every $\rho>1$, sufficiently long-wavelength modes have a real root larger than one, whatever the value of $\mu$.
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