Exact Two-Dimensional Multi-Hump Waves on Water of Finite Depth with Small Surface Tension

This paper concerns the existence of multi-hump traveling waves propagating on the free surface of a two-dimensional water channel under the influence of gravity and small surface tension force. The fluid of constant density is assumed to be inviscid and incompressible and the flow is irrotational....

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2020-04, Vol.236 (1), p.451-503
Main Authors: Deng, Shengfu, Sun, Shu-Ming
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Elevation
Euler-Lagrange equation
Fluid dynamics
Fluid flow
Fluid- and Aerodynamics
Free surfaces
Incompressible flow
Infinity
Mathematical analysis
Mathematical and Computational Physics
Physics
Physics and Astronomy
Surface tension
Theoretical
Traveling waves
Wave propagation
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Summary:This paper concerns the existence of multi-hump traveling waves propagating on the free surface of a two-dimensional water channel under the influence of gravity and small surface tension force. The fluid of constant density is assumed to be inviscid and incompressible and the flow is irrotational. It was known that the exact governing equations, called Euler equations, possess a generalized solitary-wave solution of elevation that consists of a single crest (or hump) at the center and a much smaller oscillation at infinity. This paper provides the first proof of the existence of multi-hump waves using the Euler equations. It is shown that when the wave speed is near its critical value and the surface tension is small, the Euler equations have a two-hump solution which consists of two crests, that are spaced far apart, and a smaller oscillation at infinity. Moreover, the ideas and methods may be used to study 2 m -hump solutions.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-019-01474-6