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New Analytical Solution for Nonlinear Shallow Water-Wave Equations

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Abstract

We solve the nonlinear shallow water-wave equations over a linearly sloping beach as an initial-boundary value problem under general initial conditions, i.e., an initial wave profile with and without initial velocity. The methodology presented here is extremely simple and allows a solution in terms of eigenfunction expansion, avoiding integral transform techniques, which sometimes result in singular integrals. We estimate parameters, such as the temporal variations of the shoreline position and the depth-averaged velocity, compare with existing solutions, and observe perfect agreement with substantially less computational effort.

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Notes

  1. The first few zeros of the function \(J_{0}(z)\) are: \(z_1=2.405\), \(z_2=5.520\), \(z_3=8.654\), \(z_4=11.792\), ....

  2. The first few zeros of the function \(J_1(w)\) are: \(w_1=0\), \(w_2=3.832\), \(w_3=7.016\), \(w_4=10.174\), ....

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Acknowledgements

This work is partially supported through the grant provided by Middle East Technical University with the project no: BAP-08-11-DPT2002K120510. We acknowledge the partial support from the Scientific and Technological Research Council of Turkey project no. 109Y387 of the joint research program between Turkey and Greece. This contribution was also partially supported by the project ASTARTE (Assessment, STrategy And Risk Reduction for Tsunamis in Europe), Grant 603839, 7th FP (ENV.2013.6.4-3).

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Authors and Affiliations

  1. Department of Civil Engineering, Adana Science and Technology University, 01180, Adana, Turkey

    Baran Aydin

  2. Department of Engineering Sciences, Middle East Technical University, 06800, Ankara, Turkey

    Utku Kânoğlu

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  1. Baran Aydin
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  2. Utku Kânoğlu
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Correspondence to Baran Aydin.

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Aydin, B., Kânoğlu, U. New Analytical Solution for Nonlinear Shallow Water-Wave Equations. Pure Appl. Geophys. 174, 3209–3218 (2017). https://doi.org/10.1007/s00024-017-1508-z

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