We present new results on the stability of the Helmholtz equation with non-smooth and rapidly oscillating coefficients on bounded domains for the heterogeneous Helmholtz equation. Injectivity of the problem is proved for a large class of coefficients by the unique continuation principle, however, this does not give directly a coefficient-explicit energy estimate. In this talk, we will present a new theoretical approach for the one-dimensional problem and find that for a class of oscillatory and discontinuous coefficients, the stability constant (i.e., the norm of the solution operator applied to a r.h.s. in L^2) is bounded by a term independently of the number of discontinuities. We also show that problems with periodic media behave much more stable than the general case. We present examples of coefficients so that the solution has exponentially increasing local energy with respect to the frequency at any predetermined location inside the domain, showing that our estimates are sharp.