Computational aspects of shortwave scattering by a smooth waveshaped grating under grazing incidence

The authors consider the following time-harmonic problem: a smooth waveshaped periodic boundary is illuminated by a plane electromagnetic wave. The surface is assumed to be perfectly conductive and the problem is treated in two dimensional case, so it can be separated in two scalar problems for E and H polarization. Therefore the wave field as a scalar function satisfies the reduced wave equation and Dirichlet´s or Neumann´s boundary condition according to polarization. All the numerical results presented hold for the case of Dirichlet´s boundary condition, although the asymptotic formulae can be extended to other boundary conditions. The problem is investigated under the following assumptions: i) wavelength of the incident plane wave is assumed to be small compared with the period of the boundary, height of the humps, and radius of curvature of the boundary. This means the shortwave approximation applies and both ideas and methods of the GTD can be used. ii) The grazing angle is small. This implies that only the tops of the humps of the boundary are illuminated while all other parts of the boundary remain unlit, i.e. in a deep shadow. Both these assumptions seem to be natural for the problem of radiolocation near the ocean surface. The latter one has attracted the authors´ attention and remains the main aim of their investigations. In present formulation the authors model the ocean surface as a wavy periodic boundary and reduce the problem to the 2D case. Thus they arrive at the classical problem of the grating theory which nevertheless can be regarded as the model one for the problem mentioned above