Nonlinear Surface Waves of a Fluid in Rectangular Containers Subjected to Vertical Periodic Excitations

Nonlinear surface waves of an inviscid incompressible fluid in a rigid rectangular container subjected to vertical periodic excitations are investigated. A method of analysis utilizing projections transforms the nonlinear boundary value problem into sets of nonlinear ordinary equations. These equations are then studied with the method of averaging. For the case of external resonance without internal resonance, a second order theory has been developed, which gives results for three-dimensional waves. The depth of the fluid is shown to have a critical role in the qualitative nature of the motion. For the cases of external resonance in the presence of internal resonances of superharmonic and subharmonic types, the problem is reduced to the study of a fourth order system of nonlinear differential equations that are similar to the Lorenz equations when transformed to Cartesian coordinates. A detailed study of bifurcation phenomena of the superharmonic case is presented. Numerical simulations show that there are at least three different bifurcation sequences leading to chaos, including two period doubling sequences and one bifurcation sequence starting from a homoclinic orbit.