Where $h$ is the out-of-plane displacement, $\rho$ the mass density and $\kappa$ the bending rigidity.
I tried to proceed by using the Euler-Lagrange equations with the substitution:
$$h=h_o \exp\left[-i(\omega t + qa)\right]$$
as per standard-solid-state phonon treatments, but I don't know how to deal with the $(\nabla^2 h)$ term following the above substitution (perhaps that is the incorrect part).
I know the answer should turn out to be:
Note: Many of the papers I look at point to the following paper for the answer - but Google gives me nothing for it. But I feel the answer is simple and I am not doing the math correctly. There are various more complicated treatments in the literature, including QM formulations, none of which I can follow very well.
I. M. Lifshitz, Zh. Eksp. Teor. Fiz, vol. 22, p. 475, 1952.