Department of Physics, Emory University, Atlanta, Georgia 30322~Received 14 November 1996!Surface waves produced by parametric excitation experience a transition between simply connected wavesand droplet-ejecting waves when the applied forcing exceeds a threshold level. Our measurements on a numberof different fluids indicate that low-viscosity fluids have threshold accelerations which depend on only surfacetension and forcing frequency while high viscosity fluids have thresholds which depend on only viscosity andforcing frequency. Models for the transition based on the condition that the wavelength, waveheight ratioapproaches a fixed value at the transition are consistent with our observations.@S1063-651X~97!05207-0#PACS number~s!: 47.20.Ma, 47.55.Dz, 68.10.2m, 92.10.Hm