Geometrically-induced localization of flexural waves on thin warped physical membranes

TL;DR

This paper investigates how geometric curvature in thin membranes influences flexural wave localization, revealing a transition from diffusion to localization driven by disorder strength and system size.

Contribution

It introduces a geometric wave equation with a random potential from membrane curvature, demonstrating a localization transition and weak localization effects in flexural waves.

Findings

Diffusion of undulation intensity at long times/lengths.

Frequency-dependent diffusion coefficient affected by membrane geometry.

Localization transition induced by large amplitude quenched height fields.

Abstract

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.