Next-nearest-neighbour monatomic-chain frequency maximum (${\omega }_{\max }/[2\sqrt{C_1/M}]=(1+4\alpha )/(4\sqrt{\alpha })$)

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 $\alpha \le 1/4$ its maximum on $0\le x\le 1$ 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.