= Next-nearest-neighbour monatomic-chain frequency maximum
{title2=$\omega_{\max}/[2\sqrt{C_1/M}]=(1+4\alpha)/(4\sqrt\alpha)$}
Let $\alpha=C_2/C_1>0$ and $x=\sin^2(ka/2)$. The squared frequency in units $4C_1/M$ is $(1+4\alpha)x-4\alpha x^2$. For $\alpha\le1/4$ its maximum on $0\le x\le1$ is at the zone edge and equals one. For $\alpha>1/4$ it moves inside the zone to $x=(1+4\alpha)/(8\alpha)$, with the displayed frequency ratio. The small-wavevector speed instead increases by $\sqrt{1+4\alpha}$; maximizing the frequency and finding the sound speed are different operations.
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