An integral equation for solitary surface gravity waves of finite amplitude

• Published in: Theoretical and computational fluid dynamics Vol. 36; no. 5; pp. 821 - 844
• Main Author: Lavit, Igor M.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2022; Springer; Springer Nature B.V
• Subjects: Aquatic reptiles; Boundary value problems; Classical and Continuum Physics; Computational Science and Engineering; Conformal mapping; Coordinate systems; Coordinates; Engineering; Engineering Fluid Dynamics; Fluid flow; Galerkin method; Gravity waves; Incompressible flow; Incompressible fluids; Integral equations; Mathematical models; Methods; Numerical methods; Original Article; Solitary waves; Surface gravity waves; Wave propagation; Russia; Solitary wave; Nonlinear integral equation; Incompressible inviscid fluid; The Galerkin method
• ISSN: 0935-4964; 1432-2250
• DOI: 10.1007/s00162-022-00620-3

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. Graphic abstract