The performance of the second order local approximate DtN boundary condition, suggested by the authors in a previous work, is investigated analytically when employed for solving high-frequency exterior Helmholtz problems. This study proves that, in the high frequency regime, the reflected waves at the artificial boundary decay faster than $1/{(ka)^{15/8}}$ where $k$ is the wavenumber and $a$ is the semi-major axis of this boundary. Numerical results illustrate the accuracy and the efficiency of the proposed absorbing boundary condition when used for solving acoustic scattering problems in a domain-based formulation.