Threshold accelerations necessary to excite surface waves parametrically in a vertically vibrated fluid layer are presented and compared with a linear stability analysis of the full hydrodynamic equations. The effects of viscosity and finite depth are treated rigorously. The case of two-frequency forcing is considered in detail because of its relevance to recently reported pattern forming phenomena. Absolute agreement between theory and experiment, generally to within 2%, is obtained for kinematic viscosities in the range 10–50 cS (1 S= 1 cm 2/s) and three frequency ratios. A slight discrepancy is noted for single-frequency forcing at high frequencies.© 1996 The American Physical Society.