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 bathymetry, i.e., the bottom topography, as a perturbation of a flat bottom, we end up with a heterogeneous Helmholtz equation with an impedance boundary condition. In this way, we study a 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 the 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 are necessary.