Asymptotic techniques in wave propagation problems exhibit attractive features in computational aspects compared to numerical methods. However, they experience problems in complicated environments due to the transition regions associated with them. The transition region involving a shadow boundary in the downward refractive index profile has been investigated. The ray paths were determined solving the ray equation analytically. The Green's function due to a line source is constructed and evaluated using numerical integration (reference solution) and asymptotic techniques. Then, the field produced by the source, is synthesized, using superposition of beams generated from complex source point representation. The floating parameters such as beam width, number of beams and the width of the cone were tuned to get "optimum" solution which represented the total field in a very broad region very accurately. However, difficulties were encountered as the observer moved into the shadow region, the search for complex saddle point was not successful. Another difficulty is the numerical integration of the generalized ray integral for the observer located close to the shadow boundary. It was difficult to determine the steepest descent path numerically, due to rapid growth of the integrand. Over all it was demonstrated that the complex source point generated beams could be used to construct the wave fields in the inhomogeneous media.