= Periodic displacements in a harmonic chain
For a chain with lattice spacing $b$ and $N$ atoms, write $x_s=sb+u_s$. The periodic normal-mode condition is $u_N=u_0$, equivalently $x_N=x_0+Nb$ for unwrapped positions. The literal position condition $x_N=x_0$ is incompatible with the equilibrium $x_s=sb$ in that coordinate system. Fourier modes $k_r=2\pi r/(Nb)$ give $\omega_r=2\lambda|\sin(k_rb/2)|$ for spring coefficient $m\lambda^2$. The zero mode describes free center-of-mass translation and must be fixed or handled separately when defining an oscillator vacuum.
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