Out-of-Plane Phonons I am trying to derive the out-of-plane phonon dispersion relation for a membrane. As far as I can tell, one of the simplest ways to do so is with a Lagrangian of the form:
$$L_{bending}=\frac{1}{2}\rho \dot{h}^2-\frac{1}{2}\kappa(\nabla^2h)^2$$
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:
$$\omega^2=\frac{\kappa q^4}{\rho}$$
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.
 A: Here we go :
Lets considere the lagrangian density of our free membrane. For convenience, we will use a cartesian set of coordinates, so that :
$$\mathcal{L}=\frac{\rho}{2}\dot{h}-\frac{\kappa}{2}\left[{\partial_x}^2 h+{\partial_y}^2 h\right]^2$$
Apparently, this lagrangian depends on a quite large number of variables :
$$\mathcal{L}(x,y,t,h,\dot{h},\partial^2_x h,\partial^2_y h)$$
But still nothing to be afraid of! :)
One can show (see this) that for this type of lagrangian density, Euler-Lagrange equation is :
$$\sum_{\alpha\beta}\frac{\partial}{\partial\alpha}\left(\frac{\partial\mathcal{L}}{\partial(\partial_\alpha h)}\right)-\frac{\partial^2}{\partial\alpha\partial\beta}\left(\frac{\partial\mathcal{L}}{\partial(\partial^2_{\alpha\beta} h)}\right)=\frac{\partial\mathcal{L}}{\partial h}$$
where $(\alpha,\beta)\in\{x,y,t\}$
Computing this equation for our membrane lagrangian, it follows the differential equation :
$$\rho\ddot{h}+\kappa\left(\partial^4_x h + \partial^4_y h\right)=0$$
Then, simply by taking the (space-time) Fourier transform $\tilde{h}$ of $h$ so that :
$$h(\vec{r},t)=\sum_\vec{q}\int\frac{d\omega}{2\pi} e^{i(\vec{q}\cdot\vec{r}-\omega t)}\tilde{h}(\vec{q},\omega)$$
where $\vec{r}=(x,y)$, and replacing it in the previous equation, you obtain the dispersion relation :
$$\omega^2=\frac{\kappa q^4}{\rho}$$
For a quantum treatment, it feels like you will have to compute the associated hamiltonian $\mathcal{H}$ of $\mathcal{L}$ and then quantized it, using the correspondence $\{\cdot\}\rightarrow\frac{1}{i\hbar}\left[\cdot\right]$ between Poisson brackets and commutators.
