The method of small perturbations is applied to the problem of plane-wave scattering from a soft circular surface with sinusoidal roughness. Rayleigh-theory and extinction-theorem perturbative solutions of arbitrary order are developed, and they are compared to each other as well as to an exact solution. A numerical study yields quantitative information about effects associated with the use of the Rayleigh hypothesis, about the merit of higher-order solutions, and about the reliability of the error criteria that measure a posteriori the conformity of perturbative solutions to the boundary condition. It is shown that the accuracy of perturbative solutions depends not only on the height, but also on the slope, of surface corrugations. Moreover, it is shown that the Rayleigh hypothesis does not affect the accuracy of far-field perturbative calculations. The mean extinction error is proposed as a measure for the error of the extinction-theorem perturbative solution.
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