Nonlinear gravity-wave groups

Groups of gravity waves of permanent form in deep water are investigated. The analysis provides a systematic procedure for determining the form of the group to any order of approximation, and a calculation is carried to the third order of the amplitude at least and, where it matters, to the fourth order. Closed formulas for the phase velocity $c$ of the basic waves and the group velocity ${c_g}$ are obtained. Inspection of the analytic procedure reveals that these formulas remain intact for all subsequent calculations to any order of approximation. These formulas are in terms of the group wavenumber $\varepsilon$ which, to the attained order of approximation, is found to be proportional to the amplitude $a$ and the square of the basic wavenumber $k$, but is, for any assigned $k$, a power series in $a$. It is found that $c$ increases and ${c_g}$ decreases with $\varepsilon$, in such a way that $2c{c_g} = g/k$, where $g$ is the gravitational acceleration. The results are compared with the corresponding ones obtained by the cubic-Schrödinger-equation (CBE) approach, and wherever comparison is possible there is agreement. The CBE approach, however, does not give the variation of ${c_g}$ with the amplitude.