  We investigate the relationship between the linear surface wave instabilities
of a shallow viscous fluid layer and the shape of the periodic,
parametric-forcing function (describing the vertical acceleration of the fluid
container) that excites them. We find numerically that the envelope of the
resonance tongues can only develop multiple minima when the forcing function
has more than two local extrema per cycle. With this insight, we construct a
multi-frequency forcing function that generates at onset a non-trivial harmonic
instability which is distinct from a subharmonic response to any of its
frequency components. We measure the corresponding surface patterns
experimentally and verify that small changes in the forcing waveform cause a
transition, through a bicritical point, from the predicted harmonic
short-wavelength pattern to a much larger standard subharmonic pattern. Using a
formulation valid in the lubrication regime (thin viscous fluid layer) and a
WKB method to find its analytic solutions, we explore the origin of the
observed relation between the forcing function shape and the resonance tongue
structure. In particular, we show that for square and triangular forcing
functions the envelope of these tongues has only one minimum, as in the usual
sinusoidal case.
