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A rigorous integral formalism for the problem of scattering of electromagnetic radiation from a cylindrical, perfectly conducting rough surface of arbitrary shape is introduced. The computer code obtained from this theory enables us to show that the range over which the incident field affects the surface current density is of the order of the radiation wavelength. This phenomenon is explained using a new approximate theory, able to express the scattered field in the form of an integral whose integrand is known in closed form. Using the rigorous computer code, we show that the new approximate theory is better than the Kirchhoff approximation in the resonance region. Finally, it is shown that the phenomenon of short interaction range of the incident field permits the rigorous computation of the field scattered from a rough surface of arbitrary width.