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The Faraday wave experiment is a parametric system in which there is the generation of standing waves on the surface of a fluid when its container is oscillated vertically. It is also a convenient and flexible system to study pattern-forming instabilities and can form attractive patterns such as stripes, squares, hexagons, quasipatterns and superlattice patterns. Previous studies on selection of superiattice patterns have focused on three-wave interactions between excited modes and weakly damped modes. A problem with the previous approach is that the weakly damped modes only have an influence strong enough to explain stability of the superlattice patterns when the modes are very weakly damped. This severely restricts the parameter range over which this theory might be expected to be correct. We treat this as a codimension-two problem, where the primary and weakly damped modes are both considered to be close to marginally stable. At the codimension-two point, the system is governed by an 18 dimensional set of amplitude equations, 12 modes on a circle with a radius 1 and 6 modes on a circle with a radius 1/√7 We compute the main equilibrium points (stripes, rhombs: hexagons and superlattice patterns) and analyse their stability. We have computed coefficients from a model PDE that allows two critical wavenumbers, and demonstrated excellent agreement between the amplitude equation prediction and which patterns are stable in the PDE. Finally, we have compared our new approach to superlattice stability to the earlier work, and demonstrated that, for the model PDE, the predictions for superlattice stability are greatly improved by the new approach.

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