Diffraction by a strip at almost grazing angle

The problem of high-frequency diffraction by a soft strip at almost grazing incidence is considered. By using the parabolic equation method, and variable separation in elliptical coordinates, we derive the two terms asymptotic approximation of the solution. First we consider the boundary layer near the surface of the strip and derive an asymptotic representation for the velocities on the surface. Then we apply Green׳s formula to derive the asymptotic representation for the far field. Both asymptotic representations in the boundary layer and for the far field are expressed in the form of rapidly converging integrals containing Whittaker or Coulomb wave functions. The approximation for the total scattering cross-section is checked to match to known asymptotic results: the physical optics approximation for the not too small angles of incidence on one side and the asymptotic expression for the limiting case of grazing incidence on the other side. Simple approximations for the total scattering cross-section in powers of the scaled angle are derived.