On the accelerations in the free surface of nonlinear gravity-capillary waves

The real and apparent accelerations in the free surface of nonlinear gravity-capillary waves are calculated over a wide range of wavelengths. It is shown that surface tension has hardly any effect on the results previously obtained for pure gravity waves, for wavelengths down to 20 cm in length, over the range of steepness considered. At the other extreme, we present for pure capillary waves, exact analytic results valid for all steepnesses up to and including the highest wave. This yields some intriguing conclusions including the fact that the horizontal component of the real acceleration has the opposite sign over most of the surface of the highest wave to that which it does over the surface of linear waves. We also consider the effect of gravity on this solution as well as the case when both restoring forces are of equal importance.

For short waves considered here, the behavior of all the accelerations in the nonlinear wave is nonsinusoidal with maximum values considerably in excess of those predicted by linear theory.