 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 ${\omega }_r=2\lambda |\sin (k_{r}b/2)|$ 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.