Spectral properties of radiation for the Helmholtz equation with a random coefficient

We consider the Helmholtz equation [Δ + k₀²(1 + q(x, w)) ]u = − f in R² and R³, where the coefficient q(x, w) is a Gaussian random field. For any open ball B that includes the support of x → q(x, w), we approximate and characterize spectrally the source-to-measurement map f → u|_δB. To this end we first analyze the case with a deterministic coefficient q(x), and here discover and quantify a ’spectral leakage’ effect caused by the presence of the medium. We compare the predicted forward operator spectrum with the spectrum achieved in a Finite Element Method implementation of an example radiation problem. Our results are applicable in the analysis of the robustness of solution of inverse source problems in the presence of deterministic and random media.