  We consider a linear system that consists of a linear wave equation on a
horizontal hypersurface and a parabolic equation in the half space below. The
model describes longitudinal elastic waves in organic monolayers at the
water-air interface, which is an experimental setup that is relevant for
understanding wave propagation in biological membranes. We study the scaling
regime where the relevant horizontal length scale is much larger than the
vertical length scale and provide a rigorous limit leading to a
fractionally-damped wave equation for the membrane. We provide the associated
existence results via linear semigroup theory and show convergence of the
solutions in the scaling limit. Moreover, based on the energy-dissipation
structure for the full model, we derive a natural energy and a natural
dissipation function for the fractionally-damped wave equation with a time
derivative of order 3/2
