The incommensurate modulation evident in the diffraction pattern of the superconductor Bi₂Sr_3-xCa_xCu₂O_8+y consists of almost sinusoidally varying displacements of up to 0.4 Å of the Bi and Sr atoms in the a- and c-directions of the unit cell, and of up to 0.3 Å of the Cu atoms in the c direction only. Thus, a newly discovered feature of the Bi₂Sr_3-xCa_xCu₂O_8+y structure is sizable Cu displacement, which is related to static wave formation in the Cu-O sheets. Reported thermal parameters give evidence that similar distortions occur on cooling of the thallium-containing superconductors.