Dalphin, Jeremy
Lopes-Barros, Ricardo
Optimal shape of an underwater moving bottom generating surface waves ruled by a forced Korteweg-de Vries equation
It is well known since Wu & Wu (1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves propagating ahead of the disturbance in procession. One possible new application of this phenomenon
could very well be surfing competitions, where in a controlled environment, such as a pool, waves can be generated with the use of a translating bottom. In this paper, we use the forced Korteweg-de Vries equation to investigate the shape of the moving body capable of generating the highest first
upstream-progressing solitary wave. To do so, we study the following optimization
problem: maximizing the total energy of the system over the set of non-negative square-integrable bottoms, with uniformly bounded norms and compact supports. We establish analytically the existence of a maximizer saturating the norm constraint, derive the gradient of the functional, and then
implement numerically an optimization algorithm yielding the desired optimal
shape.
Shape optimization;Existence theory;Optimal control;Surface wave generation;Numerical simulation;Forced Korteweg-de Vries equation;Finite-difference methods
2018-10-02
https://repository.lboro.ac.uk/articles/Optimal_shape_of_an_underwater_moving_bottom_generating_surface_waves_ruled_by_a_forced_Korteweg-de_Vries_equation/9376664