  The boundary element method is an efficient algorithm for simulating acoustic
propagation through homogeneous objects embedded in free space. The
conditioning of the system matrix strongly depends on physical parameters such
as density, wavespeed and frequency. In particular, high contrast in density
and wavespeed across a material interface leads to an ill-conditioned
discretisation matrix. Therefore, the convergence of Krylov methods to solve
the linear system is slow. Here, specialised boundary integral formulations are
designed for the case of acoustic scattering at high-contrast media. The
eigenvalues of the resulting system matrix accumulate at two points in the
complex plane that depend on the density ratio and stay away from zero. The
spectral analysis of the Calder\'on preconditioned PMCHWT formulation yields a
single accumulation point. Benchmark simulations demonstrate the computational
efficiency of the high-contrast Neumann formulation for scattering at
high-contrast media.
