  We formulate a new family of high order on-surface radiation conditions to
approximate the outgoing solution to the Helmholtz equation in exterior
domains. Motivated by the pseudo-differential expansion of the
Dirichlet-to-Neumann operator developed by Antoine et al. (J. Math. Anal. Appl.
229:184-211, 1999), we design a systematic procedure to apply
pseudo-differential symbols of arbitrarily high order. Numerical results are
presented to illustrate the performance of the proposed method for solving both
the Dirichlet and the Neumann boundary value problems. Possible improvements
and extensions are also discussed.
