For an harmonic plane wave impinging on a perfectly reflecting smooth plane the total field, incident and reflected, satisfying on this plane a Dirichlet or Neumann boundaray condition, has an integral representation that we extend to the specular reflection from a perfectly reflecting rough plane. To make this generalization possible, some constraints must be imposed on the wavelength of the incident field and on the rough amplitude to make the diffuse field negligible so that only the coherent field is important and we may use the fact that the coherent power is identical to that of a smooth surface. This generalized integral representation supplies an approximation of the coherent field valid far from the rough plane. We limit the discussion to acoustic, TE, TM electromagnetic wave incident on 1D-perfectly reflecting rough planes with roughness described by zig-zag functions piecewise linear with opposite slop on adjacent intervalls.
2. DeSanto, J. A. and P. A. Martin, "On the derivation of boundary integral equation for scattering by an infinite one-dimensional rough surface," Acous. Soc. Am., Vol. 102, 67-77, 1997.
3. DeSanto, J. A. and P. A. Martin, "On the derivattion of boundary integral equation for scattering by an infinite two-dimensional rough surface," J. Math. Phys., Vol. 39, 894-912, 1998.
4. Saillard, M. and A. Sentenac, "Rigorous solution for electromagnetic scattering from rough surfaces," Waves Random Media, Vol. 11, R103-R137, 2001.
5. Soriano, G. and M. Saillard, "Scattering of electromagnetic waves from two-dimensional rough surfaces with an impedance pproximation," J. Opt. Soc. Am. A, Vol. 18, 124-133, 2001.
6. Chandler-Wilde, S. N. and B. A. Zhang, "Uniqueness result for scattering by infinite rough surfaces," SIAM J. Appl. Math., Vol. 58, 1774-1790, 1998.
7. Tsang, L., A. K. Jin, and K. H. Ding, Scattering of Electromagnetic Waves: Theory and Application, Wiley, New York, 2000.
8. Tsang, L. and A. K. Jin, Scattering of Electromagnetic Waves: Advanced Topics, Wiley, New York, 2001.
9. Murakami, H., "Laminated composite plate theory with improved in-plane response," J. Appl. Mech., Vol. 53, 661-666, 1986.
10. Hillion, P., "Diffraction of scalar waves at plane apertures: A different approach," Journal of Electromagnetic Waves and Applications, Vol. 14, 1677-1687, 2000.
11. Ishimaru, A., Wave Propagation and Scattering in Random Media, Vol. 2, Academic Press, New York, 1978.
12. DeBoor, C., A Practical Guide to Splines, Springer, New York, 1978.