= Long-range harmonic-chain dispersion
{title2=$\omega^2(k)=4M^{-1}\sum_{n>0}C_n\sin^2(nka/2)$}
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 $\sum n^2C_n$ converges and the chain is stable, its long-wavelength speed is $a\sqrt{\sum n^2C_n/M}$. A truly long-range interaction need not have a finite second moment or a finite linear acoustic slope.
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