Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Nonlinear groups of gravity-capillary waves


Author: Chia-Shun Yih
Journal: Quart. Appl. Math. 48 (1990), 581-599
MSC: Primary 76B15
DOI: https://doi.org/10.1090/qam/1074974
MathSciNet review: MR1074974
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Abstract: Nonlinear groups of gravity-capillary waves in deep water are investigated by a systematic direct approach that can be applied to nonlinear groups of other dispersive waves. Two formulas in closed form expressing the variations of the phase velocity $ c$ of the basic waves and of their group velocity $ {c_g}$ with the amplitude of the waves are obtained. These are in terms of the wavenumber $ \varepsilon $ of the envelope and $ {\varepsilon ^2}$ can be determined by the present approach as a power series in $ {a^2}$, if $ 2a$ represents the amplitude of the waves. To the order of approximation achieved here, $ {\varepsilon ^2}$ is determined as a multiple of $ {a^2}$. If $ k$ is the wavenumber of the basic waves, $ g$ is the gravitational acceleration, $ \rho $ is the density of the fluid, $ \hat T$ is surface tension, and $ \beta = \hat T{k^2}/\rho g$, then wave groups are possible for

$\displaystyle 0 \le \beta < 0.1547$

or

$\displaystyle \beta > \frac{1}{2},$

although the analysis is valid only when $ \beta $ is not near $ \frac{1}{2}$. The phase velocity increases with the amplitude in the former interval for $ \beta $ and decreases with the amplitude in the latter interval. The group velocity $ {c_g}$ decreases with the amplitude in the former interval for $ \beta $, or for $ \frac{1}{2} < \beta < 1$, but increases with the amplitude if $ \beta > 1$. When the results of this paper are compared with the results of previous authors, wherever comparison is possible, complete agreement is found. (Previous authors did not give the variation of $ {c_g}$ with the amplitude.)

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DOI: https://doi.org/10.1090/qam/1074974
Article copyright: © Copyright 1990 American Mathematical Society


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