  Thin elastic sheets supported on compliant media form wrinkles under lateral
compression. Since the lateral pressure is coupled to the sheet's deformation,
varying it periodically in time creates a parametric excitation. We study the
resulting parametric resonance of wrinkling modes in sheets supported on
semi-infinite elastic or viscoelastic media, at pressures smaller than the
critical pressure of static wrinkling. We find distinctive behaviors as a
function of excitation amplitude and frequency, including (a) a different
dependence of the dynamic wrinkle wavelength on sheet thickness compared to the
static wavelength; and (b) a discontinuous decrease of the wrinkle wavelength
upon increasing excitation frequency at sufficiently large pressures. In the
case of a viscoelastic substrate, resonant wrinkling requires crossing a
threshold of excitation amplitude. The frequencies for observing these
phenomena in relevant experimental systems are of the order of a kilohertz and
above. We discuss various experimental implications of the results.
