Abstract : In this thesis we compute the Green's function of the Laplace and Helmholtz equations in a two- and three-dimensional half-space with an impedance boundary condition. For the computations we use a partial Fourier transform, the limiting absorption principle, and some special functions that appear in mathematical physics. The Green's function is then used to solve a compactly perturbed impedance half-space wave propagation problem numerically by using integral equation techniques and the boundary element method. The knowledge of its far field allows stating appropriately the required radiation condition. Expressions for the near and far field of the solution are given, whose existence and uniqueness are briefly discussed. For each case a benchmark problem is solved numerically. The physical and mathematical background is extensively exposed, and the theory of compactly perturbed impedance full-space wave propagation problems is also included. The herein developed mathematical techniques are then applied to the computation of harbor resonances in coastal engineering. Likewise, they are applied to the computation of the Green's function for the Laplace equation in a two-dimensional half-plane with an oblique-derivative boundary condition.