Optimal Shape of an Underwater Moving Bottom Generating Surface Waves Ruled by a Forced Korteweg-de Vries Equation

Jeremy Dalphin () and Ricardo Barros ()
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Jeremy Dalphin: UMR 2071 CNRS-Universidad de Chile
Ricardo Barros: Loughborough University

Journal of Optimization Theory and Applications, 2019, vol. 180, issue 2, No 12, 574-607

Abstract: Abstract It is well known since Wu and Wu (in: Proceedings of the 14th symposium on naval hydrodynamics, National Academy Press, Washington, pp 103–125, 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.

Keywords: Shape optimization; Existence theory; Optimal control; Surface wave generation; Numerical simulation; Forced Korteweg–de Vries equation; Finite-difference methods; Primary 49K20; Secondary 35Q53; 49M29; 65M06; 76B15; 49J45; 49J50 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10957-018-1400-8

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