Kelvin-Helmholtz instabilities across periodic plates

We consider the linear stability of two inviscid fluids, in the presence of gravity, sheared past each other and separated by an flexible plate. Conditions for exponential growth of velocity perturbations are found as functions of the flexural rigidity of the plate and the shear rate. This Kelvin-Helmholtz instability is then analysed in the presence of plates with spatially periodic (with period $a$) flexural rigidity arising from, for example, a periodic material variation. The eigenvalues of this periodic system are computed using Bloch's Theorem (Floquet Theory) that imposes specific Fourier decompositions of the velocity potential and plate deformations. We derive the nonhermitian matrix whose eigenvalues determine the dispersion relation. Our dispersion relation shows that plate periodicity generally destabilises the flow, compared to a uniform plate with the same mean flexural rigidity. However, enhanced destabilisation and stabilization can occur for disturbances with wavelengths near an even multiple of the plate periodicity. The sensitivity of flows with such wavelengths arises from the nonpropagating, ``Bragg reflected'' modes coupled to the plate periodicity through the boundary condition at the plate.