The frequency shifts of third sound resonances during free decay are observed to have a quadratic dependence on the amplitude of the wave motion. An unambiguous measurement of amplitude allows for quantitative comparisons to predictions for the frequency shifts based on a variety of nonlinear influences acting within the superfluid⁴He film and resonator. Dispersive terms play a minor role in this analysis. We conclude that nonlinear terms dictated by a straight forward application of classical hydrodynamics are applicable to superfluid helium films.