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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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 \[ 0 \le \beta < 0.1547\] or \[ \beta > \frac {1}{2},\] 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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Article copyright: © Copyright 1990 American Mathematical Society