Dispersion curves, surface displacements, and displacement profiles over the plate thickness are numerically calculated for acoustic plate modes, propagating in fused and Y-rotated, X-cut quartz samples with thickness h/lambda in the range 0-4.5 (h, thickness; lambda, wavelength). The Anisimkin Jr.' modes with velocity v(n) close to that of longitudinal bulk wave v(L) and the dominant longitudinal displacement u1 distributed uniformly through the plate thickness are found in quartz crystals with cut angles mu = 120 degrees-140 degrees and plate thickness h/lambda = 0-0.36, 0.94-1.78, 2.32-3.08, and 3.64-4.44. The same modes in fused quartz are not found, except in a narrow region near zero thickness.