Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains

Potential non-uniqueness of boundary representations of the Helmholtz equation underscores the importance of investigating continuous boundary-based and domain-based formulations, which is the main purpose of this work. Uniqueness properties of the solutions of boundary integral equations are reviewed. We analyze formulations for domain-based computation that are derived by the DtN method, which imposes a relation between the function and its normal derivative on an artificial boundary. The DtN formulation is shown to possess non-reflective boundary conditions and to give rise to exact (and thereby unique) solutions. In practical implementation the DtN map is often truncated. The truncated DtN operator fails to completely inhibit reflection of higher modes, resulting in loss of uniqueness at characteristic wave numbers of higher harmonics. However, simple expressions that determine a sufficient number of terms in the operator for unique solutions at any given wave number are derived. We prove that a local approximation of the boundary conditions restores uniqueness for all wave numbers, and derive its three-dimensional version.