Boundary controllability of the one-dimensional wave equation with rapidly oscillating density

We consider the one-dimensional wave equation with periodic density rho of period epsilon --> 0 in a bounded interval. By a counterexample due to Avellaneda, Bardos and Rauch we know that the exact controllability property does not hold uniformly as epsilon --> 0 when the control acts on one of the extremes of the interval. The reason is that the eigenfunctions with wavelength of the order of epsilon may have a singular behavior so that their total energy cannot be uniformly estimated by the energy observed on one of the extremes of the interval. We give partial controllability results for the projection of the solutions over the subspaces generated by the eigenfunctions with wavelength larger and shorter than epsilon. Both results are sharp. We use recent results on the asymptotic behavior of the spectrum with respect to the oscillation parameter epsilon, the theory of nonharmonic Fourier series and the Hilbert uniqueness method (HUM).