Summary of the analysis

We can draw three important conclusions:

1. The key parameter in the formulas is $p=\omega\Delta t$ (dimensionless)

1. Period of oscillations: $P=2\pi/\omega$
2. Number of time steps per period: $N_P=P/\Delta t$
3. $\Rightarrow\ p=\omega\Delta t = 2\pi/ N_P \sim 1/N_P$
4. The smallest possible $N_P$ is 2 $\Rightarrow$ $p\in (0,\pi]$
2. For $p\leq 2$ the amplitude of $u^n$ is constant (stable solution)
3. $u^n$ has a relative phase error $\tilde\omega/\omega \approx 1 + \frac{1}{24}p^2$, making numerical peaks occur too early