Where is the "wave information" hidden in this ODE

In the $i$th row of the differential system, the term $(x_{i+1} - 2 x_{i} + x_{i-1})/\Delta\xi^2$ is an order-2 central finite-difference approximation of the space derivative $\partial^2 x/\partial \xi^2$, where $\xi$ is a space coordinate such that $x_i(t) \simeq x(i\, \Delta \xi ,t)$. Thus, the differential system $\ddot{x_i} + K x_i = 0$ may be viewed as a finite-difference spatial discretization of the wave equation $$ \frac{\partial^2 x}{\partial t^2} - c^2 \frac{\partial^2 x}{\partial \xi^2} = 0 \, , $$ which speed of sound in $\xi$-$t$ coordinates is $c = \Delta \xi$ /s.

Now, let us assume that $x = \exp\left({\text{i}(\omega t - k \xi)}\right)$ is a monochromatic wave. Injecting this Ansatz in the wave equation, we obtain the expression of the physical wave number $k = \omega/c$, i.e. the dispersion relation. Injecting the Ansatz $x = \exp\left({\text{i}(\omega t - \kappa \xi)}\right)$ in the corresponding discrete equation $$ \ddot{x}_i - c^2 \frac{x_{i+1} - 2 x_{i} + x_{i-1}}{{\Delta\xi}^2} = 0 \, , $$ we obtain the relation $$ \kappa \Delta\xi = \arccos\left( 1 - \frac{(k\Delta\xi)^2}{2} \right) $$ satisfied by the numerical wave number $\kappa$. A series expansion as $k\Delta\xi \to 0$ gives the numerical dispersion relation $$ \kappa \simeq k + \frac{k^3 \Delta\xi^2}{24} + O\left(k^4 \Delta\xi^3\right) . $$

We may decompose $x(t)$ into a linear combination of eigenvectors, $x(t)=\sum_{k=1}^n\alpha_k(t)\phi_k$.

Each coefficient satisfies $\ddot\alpha_k(t)+\omega_k\alpha_k(t)=0$, so $\alpha_k(t)=a_k\exp(j\sqrt{\omega_k}t)+b_k\exp(-j\sqrt{\omega_k}t)$ for some complex $a_k,b_k$. I'm using $j=\sqrt{-1}$ because we are already using $i$ for the spatial index.

Similarly, each eigenvector can be written as $\phi_k^{(i)}=\frac1j\bigl(\exp(j\kappa_ki)-j\exp(-j\kappa_ki)\bigr)$, where $\kappa_k=\pi(2k-1)/(2n+1)$.

Multiplying them together, each eigenmode $\alpha_k(t)\phi_k^{(i)}$ can be expressed as a sum of terms of the form $c\exp\bigl(j(\pm\sqrt{\omega_k}t\pm\kappa_ki)\bigr)$, or equivalently, $c\exp\bigl(j\kappa_k\bigl(\pm i\pm(\sqrt{\omega_k}/\kappa_k)t\bigr)\bigr)$. Thus each eigenmode is a superposition of travelling waves of wavenumber $\kappa_k$ and velocity $\pm\sqrt{\omega_k}/\kappa_k$.

In particular, for $k\ll n$, we have $\sqrt{\omega_k}/\kappa_k\approx1$, consistent with the analysis from the wave equation. However, there is some dispersion at higher wavenumbers, which explains the trailing high-frequency oscillations.