Hydrodynamics of a membrane bound to a substrate

We briefly review recent theoretical work on the dynamical equilibrium shape fluctuations of a membrane interacting with a substrate. Solving the hydrodynamical equations we determine the damping rate γ(q) as a function of the wave‐vector q. For small q , we find universally γ∼q 2 . In an intermediate q ‐range, the damping rate behaves like either γ∼q 6 (monotonic damping) or γ∼q 2 (non‐monotonic damping) depending on the relative sizes of the mean separation and the parallel correlation length, which are both determined by the adhesion potential. For large q , one recovers the behavior γ∼q 3 of a free membrane. A numerical solution of the dispersion relation is presented for a membrane subject to electrostatic repulsion in a linear attractive potential.