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.)
- Chia-Shun Yih, Nonlinear gravity-wave groups, Quart. Appl. Math. 47 (1989), no. 1, 167–184. MR 987905, DOI https://doi.org/10.1090/S0033-569X-1989-0987905-2
- Mark J. Ablowitz and Harvey Segur, On the evolution of packets of water waves, J. Fluid Mech. 92 (1979), no. 4, 691–715. MR 544892, DOI https://doi.org/10.1017/S0022112079000835
- V. D. Djordjević and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech. 79 (1977), no. 4, 703–714. MR 443555, DOI https://doi.org/10.1017/S0022112077000408
- Chia-Shun Yih, A solitary group of two-dimensional deep-water waves, Quart. Appl. Math. 45 (1987), no. 1, 177–183. MR 885180, DOI https://doi.org/10.1090/S0033-569X-1987-0885180-0
Chia-Shun Yih, Nonlinear gravity-wave groups, Quart. Appl. Math. 47, 167–184 (1989)
Mark J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech. 92, 691–715 (1979)
V. D. Djordjević and Larry G. Redekopp, On two dimensional packets of capillary-gravity waves, J. Fluid Mech. 79, 703–714 (1977)
Chia-Shun Yih, A solitary group of two-dimensional deep-water waves, Quart. Appl. Math. 45, 177–183 (1987)
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Article copyright:
© Copyright 1990
American Mathematical Society