  In this paper we present and analyse a high accuracy method for computing
wave directions defined in the geometrical optics ansatz of Helmholtz equation
with variable wave number. Then we define an "adaptive" plane wave space with
small dimensions, in which each plane wave basis function is determined by such
an approximate wave direction. We establish a best $L^2$ approximation of the
plane wave space for the analytic solutions of homogeneous Helmholtz equations
with large wave numbers and report some numerical results to illustrate the
efficiency of the proposed method.
