When calculating the acoustical branch I get
\omega =\sqrt{2k_2/m} |sin(ka)| instead of \omega =\sqrt{k_2/m} |sin(ka)|
After taylor expanding the square root term of the formula I end up with:
m\omega^2 =k_1 + k_2 -(k_1 +k_2 cos(2ka)) = k_2(1-cos(2ka))
Using the trigonometric identity of (1-cos(2ka) )= 2sin(ka)^2 I get
m\omega^2 = k_2 2sin(ka)^2 \rightarrow \omega = \pm \sqrt{(2k_2/m)} |sin(ka)|
This is half of the mass I would expect.
Can someone please explain where I went wrong here?