Solution
= Solution
For the growing <normal mode> in two deep layers,
$$
\sigma^2(k)=\frac{Ak-\gamma k^3}{\rho_1+\rho_2},\qquad A=g(\rho_1-\rho_2).
$$
It vanishes at the endpoints of the unstable band and has its unique interior maximum where $A-3\gamma k^2=0$. Using $k_c=\sqrt{A/\gamma}$ gives
$$
\boxed{\sigma_m=\left[\frac{2A k_c}{3\sqrt3(\rho_1+\rho_2)}\right]^{1/2}.}
$$
This is the maximum exponential <growth rate>, not the real oscillation frequency.