In this paper, we are concerned with the error analysis for the nite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish an explicit priori error estimates in H1 and L2- norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modifcation due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results.