The Faraday instability (1831) occurs when a fluid layer is vertically oscillated at frequency ω. Each surface mode with wavenumber k obeys a Mathieu equation: ẍ + 2γẋ + ω₀²(1 − A cos ωt)x = 0, where ω₀²=gk+σk³/ρ from gravity and surface tension. When ω ≈ 2ω₀/n (parametric resonance tongues), modes grow exponentially. The standing waves appear at half the drive frequency — a subharmonic response.