Two-mode rhomboidal states in driven surface waves

Two-mode rhomboid patterns are generated experimentally via two-frequency parametric forcing of surface waves. These patterns are formed by the simple nonlinear resonance: k^→'₁ - k^→₂ = k^→₁ where k₁ and k₂(=k'₂) are concurrently excited eigenmodes. The state possesses a direction-dependent time dependence described by a superposition of the two modes, and a dimensionless scaling delineating the state’s region of existence is presented. We also show that 2n-fold quasipatterns naturally arise from these states when the coupling angle between k^→₂ and k^→'₁ is 2π/n.