  This paper deals with bathymetry-oriented optimization in the case of long
waves with small amplitude. Under these two assumptions, the free-surface
incompressible Navier-Stokes system can be written as a wave equation where the
bathymetry appears as a parameter in the spatial operator. Looking then for
time-harmonic fields and writing the bottom topography as a perturbation of a
flat bottom, we end up with a heterogeneous Helmholtz equation with impedance
boundary condition. In this way, we study some PDE-constrained optimization
problem for a Helmholtz equation in heterogeneous media whose coefficients are
only bounded with bounded variation. We provide necessary condition for a
general cost function to have at least one optimal solution. We also prove the
convergence of a finite element approximation of the solution to the considered
Helmholtz equation as well as the convergence of discrete optimum toward the
continuous ones. We end this paper with some numerical experiments to
illustrate the theoretical results and show that some of their assumptions
could actually be removed.
