We consider the numerical solution of linear systems arising in acoustic control. More specifically, we consider preconditioned iterative methods for solving KKT systems arising in PDE-constrained optimization where the underlying PDE is the Helmholtz equation. An approach based on an approximation of the Schur complement is considered, with attention given to existing results for the case of a positive definite PDE operator and how such results extend to the indefinite case. As this approach requires repeated solves of the forward problem, we discuss the effect of a choosing a preconditioner that instead solves a damped problem, known as a shifted Laplacian preconditioner. We discuss how this impacts the choice of solver for the forward problem and the overall consequences on solving the inverse problem.