AbstractThat X-rays focused by reflection from concave surfaces exhibit anomalous image broadening was shown by Ehrenberg using a line source. Even flat nonfocusing surfaces of many different materials produced an anomalous striated reflection. According to Ehrenbevg, the common cause for the observed broadening in the focused image and the striated pattern in the nonfocused reflection is a periodic surface structure. In later years, Eliot, using a point source of X-rays reflected from a concave surface, observed a striated pattern at a position well beyond the focal plane. It is to be noted that Eliot's experiment combined elements of Ehrenberg's two experiments—a focusing surface and observations made far from any focal plane, real or virtual. Eliot observed his striations with fused silica surfaces but not with obsidian. Recently, Yoneda, using X-rays collimated by a Soller slit and incident on a plane surface, observed an anomalous line reflection not previously reported.Various experiments performed by the present authors have attempted to duplicate some of the above situations as closely as possible, either directly with X-rays or in an analogous manner with visibic light. It will be shown that satisfactory explanations can be formulated from previously unsuspected diffraction phenomena, recently confirmed by experiment as well as by a simple experimental oversight. In spite of the dilemma presented, either horn eliminates all previous limitations due to surface conditions, so that the tolerable surface roughness is once again determined by the well-known Rayleigh criterion rather than by any periodic surface irregularities introduced through the polishing process or by nature.