  The Faraday wave experiment is a classic example of a system driven by
parametric forcing, and it produces a wide range of complex patterns, including
superlattice patterns and quasipatterns. Nonlinear three-wave interactions
between driven and weakly damped modes play a key role in determining which
patterns are favoured. We use this idea to design single and multi-frequency
forcing functions that produce examples of superlattice patterns and
quasipatterns in a new model PDE with parametric forcing. We make quantitative
comparisons between the predicted patterns and the solutions of the PDE.
Unexpectedly, the agreement is good only for parameter values very close to
onset. The reason that the range of validity is limited is that the theory
requires strong damping of all modes apart from the driven pattern-forming
modes. This is in conflict with the requirement for weak damping if three-wave
coupling is to influence pattern selection effectively. We distinguish the two
different ways that three-wave interactions can be used to stabilise
quasipatterns, and present examples of 12-, 14- and 20-fold approximate
quasipatterns. We identify which computational domains provide the most
accurate approximations to 12-fold quasipatterns, and systematically
investigate the Fourier spectra of the most accurate approximations.
