 Long-range harmonic-chain dispersion

For a translation-invariant monatomic chain with symmetric pair force constants, $M{\ddot{u}}_p={\sum }_{n>0}{C}_n(u_{p+n}+u_{p-n}-2u_p)$. Substituting a traveling normal mode gives the displayed even, reciprocal-periodic phonon dispersion relation. If $\sum n^2{C}_n$ converges and the chain is stable, its long-wavelength speed is $a\sqrt{\sum n^2{C}_n/M}$. A truly long-range interaction need not have a finite second moment or a finite linear acoustic slope.