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Model Equations for Gravity‐Capillary Waves in Deep Water
Author(s) -
Akers Benjamin,
Milewski Paul A.
Publication year - 2008
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2008.00409.x
Subject(s) - euler equations , dispersion relation , physics , quadratic equation , euler's formula , mechanics , gravity wave , dispersion (optics) , capillary wave , wave packet , surface tension , classical mechanics , wave propagation , mathematics , mathematical analysis , geometry , thermodynamics , optics , quantum mechanics
The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two‐dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gravity‐capillary dispersion relation, but preserving exactly either a quadratic energy or a momentum. Solitary wavepacket waves are calculated for each model. Each model supports the elevation and depression waves known to exist in the Euler equations. The stability of these waves is discussed, as is the dynamics resulting from instabilities and solitary wave collisions.