 Solution

Substitute the traveling normal mode $u_j=u\exp [i(kja-\omega t)]$. Combining the opposite-neighbour phases givesso the positive-frequency branch isFor positive spring constants this is a stable acoustic branch. At long wavelengths,The next-nearest-neighbour bond spans twice the distance, hence contributes the factor four. Relative to $G=0$, positive $G$ increases the sound speed by $\sqrt{1+4G/K}$. The chain still has one acoustic branch and no optical branch.

The group speed near $k=0$ and the maximum frequency need not change in the same way. The zone-edge value is always ${\omega }_0(\pi /a)=2\sqrt{K/m}$, because the next-nearest-neighbour sites then move together. For $G>K/4$ the frequency maximum moves inside the zone, with $\cos (ka)=-K/(4G)$; the small-$k$ sound speed nevertheless keeps increasing.