Dynamics of cross-waves in finite rectangular tanks

The paper examines the cross-waves emerging on the surface of the tank in a direction perpendicular to the direction of the wavemaker's motion. The difficulties of mathematical analysis of this phenomenon are related to the fact that the linearized motion equations do not describe the generation mechanism for such waves. The experimental observations of cross-waves in the experimental tank of the Institute of Hydromechanics of NAS of Ukraine indicate that they emerge due to the realization of parametric resonance conditions in the nonlinear system. The frequency ranges of wavemaker oscillations for which the cross-waves occur are quantitatively estimated. It should be mentioned that all previous studies in this domain were based on the methodologies that used Havelock's solution for the reservoir of infinite length or depth. On the contrary, here the generation of cross-waves is considered in the long rectangular tank with finite dimensions. To study their steady regimes, the Lame superposition method developed for solving elasticity theory problems was used for the first time. As shown, the considered waves in the finite tanks may be generated by the immediate motion of the wavemaker. The corresponding analytical solutions for the velocity potential are obtained, and the orders of magnitude are assessed for all its components by introducing the small parameter. The mathematical model is proposed to describe the process of the wavemaker's energy transmission into the cross-waves. The obtained differential equation is of the Mathieu type. Studying its coefficients reveals the possibility of the existence of cross-waves with non-zero amplitudes in the system under consideration. It is worth noting that all the addends forming the parametric term are proportional to the function of the forcing oscillation. Therefore, the generation of the cross-waves is possible only due to cyclic fluid motions excited by the wavemaker.