TABLE OF CONTENTS The cross wave instability shown in Fig. 6.15 is parametrically excited by the progressive waves generated by a planar wavemaker at a subharmonic frequency of the wavemaker frequency (Bowline et al., 1999 and Hudspeth et al., 2005). Parametrically excited standing cross waves that oscillate in a direction transverse to the wavemaker forcing with crests perpendicular to the wavemaker may be analyzed by the generalized Melnikov method (GMM) and by the Liapunov characteristic exponents. The GMM is a global perturbation analysis about a separatrix and about a manifold of fixed points that are connected by separatrices for higher dimensional nonlinear dynamical systems (Wiggins, 1988, Sec. 4), The Wiggins-Holmes (1987) generalization of the Melnikov method (Melnikov, 1963 and Arnold, 1978) to higher dimensions may be applied to parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Hamiltonian for these cross waves in an inviscid fluid is homomorphic to the Hamiltonian for a parametrically excited pendulum that is an example of a Floquet oscillator that may be approximated by the Mathieu equation (Berge et al., 1984). The Luke Lagrangian density function (Luke, 1967) for surface gravity waves with surface tension contains three generalized coordinates (or, equivalently three-degrees-of-freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The neutral Floquet stability diagram (Jordan and Smith, 1987, Fig. 9.2,...
