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.

我们考虑一个线性系统，该系统由水平超曲面上的线性波动方程和下方半空间中的抛物线方程组成。该模型描述了水-空气界面有机单层中的纵向弹性波，这是一种实验装置，与理解生物膜中的波传播有关。我们研究了相关的水平长度尺度远大于垂直长度尺度的缩放机制，并提供了一个严格的极限，从而导致了膜的部分阻尼波动方程。我们通过线性半群理论提供了相关的存在结果，并显示了在定标极限下解的收敛性。此外，基于完整模型的能量耗散结构，