Existence and Stability of Patterns Arising from Square Wave Forcing of the Damped Mathieu Equation

In this work the nature and stability of patterns arising from parametric squarewave forcing of an inviscid fluid layer of infinite depth are investigated. Specifically the case of vertically shaken fluids is considered. Beginning with the non-linear PDE’s of the Zhang-Viñals Model of a fluid surface under small perturbations, it is shown how a linear, second order ODE damped Mathieu equation arises from a linear stability analysis. This analysis is performed for several different square wave forcing functions. It is shown both analytically and in the neutral stability curves that square wave forcing can be reduced to delta function forcing in the appropriate limit. In addition to this, the effects of larger forcing times on the neutral stability curves are examined. The effect of fluid parameters on the neutral stability curves is also explored. Lastly, a numerical solver is developed to observe the different patterns that are generated by several configurations of square wave forcing.