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An integral equation for solitary surface gravity waves of finite amplitude

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Abstract

The problem of a solitary surface gravity wave in a flow of an inviscid incompressible fluid in a channel of constant depth is considered. The problem is solved in two-dimensional formulation. The wave moves at a constant speed. In a coordinate system moving along with the wave, the flow is stationary. Its mathematical model is reduced to a boundary value problem for a strip in the complex potential plane. This is converted to a boundary value problem for a half-plane by conformal mapping. The solution is obtained using a Cauchy-type integral for the density of which a nonlinear integral equation is derived. Its solution is found with the Galerkin method and the Newton–Raphson technique. The calculated results are compared with the experimental data and the calculations by other researchers. The lower limit of the speed of a solitary wave is found. The advantage of the proposed method is the simplicity of the resulting integral equation, which makes it possible to effectively apply numerical methods of solution.

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  1. Tula State University, Tula, Russia

    Igor M. Lavit

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  1. Igor M. Lavit
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Correspondence to Igor M. Lavit.

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Software appendices to the paper with instructions on how to use them are available at: https://drive.google.com/drive/folders/1gNnRHoqMZ1xEeiZltODz7CBr4cCAHDHV?usp=sharing. These software applications allow you to get all the numerical results given in the paper.

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Communicated by Ivan Egorov.

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Lavit, I.M. An integral equation for solitary surface gravity waves of finite amplitude. Theor. Comput. Fluid Dyn. 36, 821–844 (2022). https://doi.org/10.1007/s00162-022-00620-3

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