  The goal of this paper is to investigate the stability of the Helmholtz
equation in the high- frequency regime with non-smooth and rapidly oscillating
coefficients on bounded domains. Existence and uniqueness of the problem can be
proved using the unique continuation principle in Fredholm's alternative.
However, this approach does not give directly a coefficient-explicit energy
estimate. We present a new theoretical approach for the one-dimensional problem
and find that for a new class of coefficients, including coefficients with an
arbitrary number of discontinuities, the stability constant (i.e., the norm of
the solution operator) is bounded by a term independent of the number of jumps.
We emphasize that no periodicity of the coefficients is required. By selecting
the wave speed function in a certain \resonant" way, we construct a class of
oscillatory configurations, such that the stability constant grows
exponentially in the frequency. This shows that our estimates are sharp.
