= Solution
Use a <Fourier mode> $u_m^n=\zeta^ne^{im\theta}$. Substitution gives
$$
\zeta^2-2[1-2\mu\sin^2(\theta/2)]\zeta+1=0.
$$
Its roots have product one. They lie on the <unit circle> for all spatial frequencies precisely when $|1-2\mu\sin^2(\theta/2)|\leq1$, giving
$$
\boxed{0<\mu\leq1,\qquad\Delta t/\Delta x\leq1.}
$$
For $\mu>1$, the Nyquist frequency has a real reciprocal root pair with one root of modulus greater than one, giving exponential growth and instability.
The endpoint convention needs care in this two-step wave scheme: repeated unit roots occur at zero frequency, and also at the Nyquist frequency when $\mu=1$. They are not power-bounded for arbitrary two independent displacement sequences with a mesh-independent difference. With the usual initialization from displacement and velocity,
$$
u^1=u^0+k v^0+\frac\mu2\Delta_h^{\rm unscaled}u^0,
$$
each Fourier component satisfies $\widehat u^n=\cos(n\eta)\widehat u^0+k[\sin(n\eta)/\sin\eta]\widehat v^0$, where $\cos\eta=1-2\mu\sin^2(\theta/2)$. Taking the limiting values at repeated roots and using $|\sin(n\eta)|\leq n|\sin\eta|$ gives $|\widehat u^n|\leq|\widehat u^0|+nk|\widehat v^0|$. This is a uniform finite-time stability bound in the physical displacement/velocity data and includes $\mu=1$. A strict root-condition test for arbitrary two-level data must record the double-root qualification rather than calling that endpoint unconditionally power-bounded.
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