  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.
