Numerical investigation on special modes with narrow amplification diagram in harbor oscillations

  • Xiouzhou MaEmail author
  • Zhenjun Zheng
  • Xiang Zhang
  • Guohai Dong


In the studies of harbor oscillations, some modes with extremely narrow amplification diagram are significantly common. An extended mild-slope equation and a fully nonlinear Boussinesq equation are used to study those modes (termed as extreme modes in this paper). The difference between the extreme modes for various harbor shapes, the influence of nonlinearity, and the formation mechanism is comprehensively investigated. The obtained results show that the entrance width has a significant effect on extreme modes, which causes a variation in the extreme modes for different harbor shapes. The smaller a harbor entrance is, the more severe the response of the extreme modes will be. Wave nonlinearity can mitigate the response of extreme modes significantly by transporting the energy to superharmonic components, thereby reducing the time required to reach a steady state. The wave energy inside the harbor increases very slowly because of the unusually weak flow field through the entrance excited by the extreme modes, thus resulting in an exceedingly long time required for the development of the extreme modes. Only a few special wave frequencies can result in such a flow field. Therefore, for extreme modes, the widths of their resonant peaks in the amplification diagram are narrow. Hence, extreme modes can have a significant effect on the harbor only under remarkably ideal conditions. During the plan-shape design of a harbor based on numerical simulation, attention can be focused on ordinary modes, and extreme modes can be ignored to a certain extent.


Boussinesq model Extended mild-slope equation Extreme modes Harbor oscillations Hybrid finite element method Wave energy flux 


Funding information

This work was financially supported by the National Key R&D Program of China (2017YFC1404200) and the Fundamental Research Funds for the Central Universities (Grant No. DUT18ZD214).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Xiouzhou Ma
    • 1
    Email author
  • Zhenjun Zheng
    • 1
  • Xiang Zhang
    • 1
  • Guohai Dong
    • 1
  1. 1.The State Key Laboratory of Coastal and Offshore EngineeringDalian University of TechnologyDalianChina

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