Scattering of a spherical wave by a thin hard strip

This paper is concerned with a theoretical solution to the problem of scattering of a spherical wave by a strip. The strip is infinitely thin, infinite in length and of width 2a. The problem is first brought into the wave space through a spatial Fourier transform of the wave equation and of the boundary conditions on the strip. The Fourier transform is taken with respect to the co-ordinate axis parallel to the edges of the strip. Using the boundary conditions on the strip leads to an integral equation of the first kind, the unknown of which is the discontinuous potential jump across the strip. This latter is expanded into some suitable functions and the coefficients of the series expansion are thereafter determined from an infinite system of equations. The system's matrix is found to be mainly diagonal and tests on the stability of the numerical calculations suggest the significant number of equations in the system be limited to approximately ka + 5, with k being the wavenumber. Finally, after solving the system of equations and going back to the scattered field, the expression of this latter is made from an infinite series over some infinite double integrals whose approximate evaluation is made with the help of the two-dimensional stationary phase method. This treatment corresponds to the far field case. A further consideration of the right side of the system of equations leads to an improved value of the scattered field. Comparisons are made to an approximated prediction of the scattered field by using the Biot and Tolstoy exact theory of diffraction of a spherical wave by a hard wedge. The implementation of this approach to the strip requires the further consideration of the multiple diffraction between its edges for improving the calculated value of the scattered field. Some numerical examples are treated with discussions on their results.