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Explicit modified mild-slope equation for wave scattering by piecewise monotonic and piecewise smooth bathymetries

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Abstract

The mild-slope equation and modified mild-slope equation (MMSE) have played an important role in modeling the interaction between waves and bathymetries or structures. However, they are implicit equations whose coefficients are defined by an implicit dispersion relation. In this paper, for both two-dimensional bathymetries and three-dimensional axisymmetric bathymetries with piecewise monotonicity and piecewise second-order smoothness, we are able to transform the implicit MMSE into an explicit equation by introducing a new independent variable. Besides, an alternative form of the implicit MMSE is also transformed into an explicit equation under the same assumptions. These explicit equations make it much easier to obtain analytic solutions to the MMSE. As practical examples, two analytic solutions to the present explicit MMSE for wave reflection by a single linear slope and for wave scattering by a submerged circular truncated shoal are constructed.

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Acknowledgments

The authors wish to gratefully acknowledge the financial support from Guangxi Natural Science Foundation (2011GXNSFD018006), Scientific Research Foundation of Guangxi Universities (201102ZD014), and Guangxi Experiment Centre of Science and Technology (LGZXKF201110). They are obliged to Miss Dan-Juan Fu for her kind help in drawing Fig. 2. Finally, they would like to thank Professor Ye-Lin Ou and one anonymous referee for their valuable suggestions.

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Authors and Affiliations

  1. School of Sciences, Guangxi University for Nationalities, Nanning,  530006, Guangxi, People’s Republic of China

    Huan-Wen Liu & Xiao-Mei Zhou

  2. College of Mathematics and Information Science, Guangxi University, Nanning,  530004, Guangxi, People’s Republic of China

    Huan-Wen Liu

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  1. Huan-Wen Liu
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  2. Xiao-Mei Zhou
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Correspondence to Huan-Wen Liu.

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Liu, HW., Zhou, XM. Explicit modified mild-slope equation for wave scattering by piecewise monotonic and piecewise smooth bathymetries. J Eng Math 87, 29–45 (2014). https://doi.org/10.1007/s10665-013-9661-6

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  • DOI: https://doi.org/10.1007/s10665-013-9661-6

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