- Optical branch: $\omega^2\approx
- 2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\
- Acoustic branch: $\omega^2\approx
- \frac{C/2}{M_1+M_2}k^2a^2$\\
+ $\Rightarrow$\\
+ $\sqrt{(\frac{1}{M_1}+\frac{1}{M_2})^2-
+ \frac{k^2a^2}{M_1M_2}}=$
+ $(\frac{1}{M_1}+\frac{1}{M_2})
+ \sqrt{1-\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}
+ \stackrel{Taylor}{\approx}
+ (\frac{1}{M_1}+\frac{1}{M_2})
+ (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\
+ $\omega \approx \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+ \sqrt{1\pm (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})}$\\
+ $\stackrel{{\color{red}+}}{\rightarrow}
+ \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+ \sqrt{2-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}
+ \stackrel{Taylor}{\approx}
+ \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}\sqrt{2}
+ (1-\frac{1}{2}\frac{1}{4}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\
+ $\stackrel{{\color{blue}-}}{\rightarrow}
+ \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+ \sqrt{\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}=
+ \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+ \sqrt{\frac{1}{2}\frac{1}{M_1M_2(1/M_1+1/M_2)^2}}ka$\\
+ {\color{red}Optical branch}: $\omega\stackrel{ka\ll 1}{\approx}
+ \sqrt{2C\left(\frac{1}{M_1}+
+ \frac{1}{M_2}\right)}$\\
+ {\color{blue}Acoustic branch}: $\omega\stackrel{ka\ll 1}{\approx}
+ \sqrt{\frac{C/2}{M_1+M_2}}ka$\\