  We study weakly stable hyperbolic boundary problems with highly oscillatory
coefficients that are large, $O(1)$, compared to the small wavelength $\eps$ of
oscillations. Such problems arise, for example, in the study of classical
questions concerning the stability of Mach stems and compressible vortex
sheets. For such applications one seeks to prove energy estimates that are in
an appropriate sense "uniform" with respect to the small wavelength $\eps$, but
the large oscillatory coefficients are a formidable obstacle to obtaining such
estimates. In this paper we analyze a simplified form of the linearized
problems that are relevant to the above stability questions, and obtain results
that are both positive and negative. On the one hand we identify favorable
structural conditions under which it is possible to prove uniform estimates,
and then do so by a new approach. We also construct examples showing that large
oscillatory coefficients can give rise to an instantaneous \emph{multiple}
amplification of the amplitude of solutions relative to data; for example,
boundary data of a given amplitude $O(1)$ can \emph{immediately} give rise to a
solution of amplitude $O(\frac{1}{\eps^K})$, where $K>1$.\footnote{Examples of
first-order amplification, where $K=1$, are well-known \cite{MA,CG}.} We use
the examples of multiple amplification to confirm the optimality of our uniform
estimates when the favorable structural conditions hold. When those conditions
do not hold, we explain how multiple amplification of infinite order may rule
out useful estimates.
