Substitute the traveling normal mode

u_{j} = u exp [i (kja - ω t)]

. Combining the opposite-neighbour phases gives

m ω^{2} = 2 K (1 - cos ka) + 2 G (1 - cos 2 ka),

so the positive-frequency branch is

ω_{0} (k) = 2 \frac{K sin ^{2} ( ka /2 ) + G sin ^{2} ( ka )}{m}, ∣ k ∣ \leq π / a .

For positive spring constants this is a stable acoustic branch. At long wavelengths,

ω_{0}^{2} ≃ \frac{a ^{2} ( K + 4 G )}{m} k^{2}, c_{s} = a \frac{K + 4 G}{m} .

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

1 + 4 G / 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

ω_{0} (π / a) = 2 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 / (4 G)

; the small-

k

sound speed nevertheless keeps increasing.

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