This paper considers the scattering of an electromagnetic wave from a simplistic model of a random rough surface. The model surface is an ensemble of systems each of which consists of two conducting, parallel, Rayleigh hemicylinders of infinite length and equal radius on an infinite conducting plane. This is the simplest multiple-scattering case of the N-hemicylinder problem, and is examined to gain insights into the approximation methods used to study the more general case. The separation distance between the protrusions varies randomly from one member of the ensemble to another except that the hemicylinders are not permitted to overlap. This surface is illuminated by a plane wave whose electric field vector is parallel to the hemicylinder axes, i.e., having TM polarization. The exact solution for the ensemble average of the square of the scattering amplitude shows that multiple scattering makes a significant contribution to the correction term introduced by the no-overlap condition. The correction term is about a factor of 2 greater than that calculated by first-order perturbation theory, which neglects multiple scattering. This contradicts the tenet that perturbation theory is correct in the limit of small (i.e., Rayleigh) scatterers. We also use a recently developed variational principle to study scattering from the random rough-surface model and find a correction term in essential agreement with the exact result.
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