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

Author

Listed:
  • Jeremy Dalphin

    (UMR 2071 CNRS-Universidad de Chile)

  • Ricardo Barros

    (Loughborough University)

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.

Suggested Citation

  • Jeremy Dalphin & Ricardo Barros, 2019. "Optimal Shape of an Underwater Moving Bottom Generating Surface Waves Ruled by a Forced Korteweg-de Vries Equation," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 574-607, February.
    • Handle: RePEc:spr:joptap:v:180:y:2019:i:2:d:10.1007_s10957-018-1400-8
    DOI: 10.1007/s10957-018-1400-8
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