Axisymmetric capillary waves

Capillary waves propagating at a constant velocity along a cylindrical jet are calculated numerically. Fully nonlinear solutions are presented. It is shown that there is a two-parameter family of solutions. This family includes the two-dimensional waves of Crapper [J. Fluid. Mech. 2 (1957) 532-540] as a particular case. As the amplitude increases, the waves ultimately reach a limiting configuration with a trapped bubble at their troughs. As the amplitude decreases, some of the solutions approach a uniform stream whereas others approach a static configuration. A linear theory and a collocation scheme are presented to describe these two limiting behaviors.