Second-harmonic resonance with Faraday excitation

Capillary-gravity waves in an inviscid liquid exhibit second- or sub-harmonic resonance at precise frequencies. When the container performs small periodic vertical vibrations, either wave may also experience Faraday (‘parametric’) excitation. Equations describing this situation are derived, incorporating slight detuning from two-wave and Faraday resonances. Similar equations arise in other physical contexts.With Faraday forcing of the wave with lower frequency, the evolution equations (without detuning) are transformable to the corresponding unforced equations, the general solution of which is known. With Faraday forcing of the wave with higher frequency no such simplification is possible. Here, various transformed equations are considered and numerical results elucidate their solutions. For some initial data, solutions remain bounded; but other initial values give unbounded solutions. We establish the form of the boundaries that separate these two classes.