Optimal shape of an underwater moving bottom generating surface waves ruled by a forced Korteweg-de Vries equation
journal contributionposted on 02.10.2018, 14:48 by Jeremy Dalphin, Ricardo Lopes-BarrosRicardo Lopes-Barros
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.
R. B. acknowledges the support of Science Foundation Ireland under grant 12/IA/1683.
- Mathematical Sciences