  We consider the propagation of flexural waves across a nearly flat, thin
membrane, whose stress-free state is curved. The stress-free configuration is
specified by a quenched height field, whose Fourier components are drawn from a
Gaussian distribution with power law variance. Gaussian curvature couples the
in-plane stretching to out-of-plane bending. Integrating out the faster
stretching modes yields a wave equation for undulations in the presence of an
effective random potential, determined purely by geometry. We show that at long
times/lengths, the undulation intensity obeys a diffusion equation. The
diffusion coefficient is found to be frequency dependent and sensitive to the
quenched height field distribution. Finally, we consider the effect of coherent
backscattering corrections, yielding a weak localization correction that
decreases the diffusion coefficient proportional to the logarithm of the system
size, and induces a localization transition at large amplitude of the quenched
height field. The localization transition is confirmed via a self-consistent
extension to the strong disorder regime.
