The propagation and decay of acoustic waves in exterior domains in an essential ingredient in the study of fluid–structure interaction. A strategy must be devised to compute solutions over domains which are unbounded. Exact impedance conditions at an artificial external boundary are specified by the DtN method, yielding an equivalent problem that is suitable for domain-based computation. The DtN boundary condition is non-reflective, giving rise to exact (and thereby unique) solutions. The truncated DtN operator, which is employed in practice, fails to inhibit the reflection of higher modes, so that non-unique solutions may occur at their harmonics. Simple expressions determine a sufficient number of terms in the truncated operator for unique solutions at any given wave number. There are three characteristic length scales in the computational problem: the radius of the artificial boundary, the geometry of the body (represented by the internal boundary) and the mesh size. Numerical studies examine the dependence of the conditioning of finite element coefficient matrices on the number of terms in the truncated DtN operator vs. the wave number non-dimensionalized by each of the length scales. Analytic results regarding the number of terms sufficient for unique solutions are confirmed. As long as this criterion is respected, no upper limit on the allowable wave number is detected. A local approximation of the boundary conditions restores uniqueness for all wave numbers.