A numerical study of scattering from an impenetrable sinusoidal boundary is carried out by means of the Rayleigh theory, implemented by a collocation technique. This leads to highly ill-conditioned linear systems of equations the solutions of which are of dubious significance. It is shown, by employing either an interval or stochastic arithmetic method, that: 1) when the maximal surface slope $\sigma$ is small (large) the boundary condition error $\varepsilon$ decreases (increases) and the number of significant digits of $\varepsilon$ decreases as the order of the linear system increases and/or $\sigma$ increases so that a point is reached beyond which the computations have no meaning, this being at the outset for large $\sigma$; 2) the values of $\varepsilon$ computed by the two methods are identical over the significant digits shared by them for all orders of the linear system; 3) the interval method is more pessimistic than the stochastic arithmetic method as concerns the accuracy of $\varepsilon$.