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We investigate the relationship between the linear surface wave instabilitiesof a shallow viscous fluid layer and the shape of the periodic,parametric-forcing function (describing the vertical acceleration of the fluidcontainer) that excites them. We find numerically that the envelope of theresonance tongues can only develop multiple minima when the forcing functionhas more than two local extrema per cycle. With this insight, we construct amulti-frequency forcing function that generates at onset a non-trivial harmonicinstability which is distinct from a subharmonic response to any of itsfrequency components. We measure the corresponding surface patternsexperimentally and verify that small changes in the forcing waveform cause atransition, through a bicritical point, from the predicted harmonicshort-wavelength pattern to a much larger standard subharmonic pattern. Using aformulation valid in the lubrication regime (thin viscous fluid layer) and aWKB method to find its analytic solutions, we explore the origin of theobserved relation between the forcing function shape and the resonance tonguestructure. In particular, we show that for square and triangular forcingfunctions the envelope of these tongues has only one minimum, as in the usualsinusoidal case.Comment: 12 pages, 10 figure

Forcing function control of Faraday wave instabilities in viscous shallow fluids