Research in nonlinear water waves

Satisfactory progress has been made with the continuing studies of the effect of a thin drift layer on the stability and shape of capillary gravity waves. The problem about the counting for the integro-differential equation formulation of the shape of finite amplitude waves of permanent form has been resolved. The equations now seem to describe a well-posed problem, and the task now is to calculate solutions for problems of physical relevance. The calculation of stability for smooth profiles has been carried out. This proved to be more difficult than anticipated but rather interesting. It uncovered new features of the way in which the Rayleigh equation (inviscid Orr-Sommerfeld equation) describes the onset of instability. The work was extended to three-dimensional disturbances, and it was shown that the Squire theorem still applies, so that the most unstable disturbances, are two dimensional. Further studies of the results indicate that the new instabilities may not be closely relevant to the generation of waves by wind, as the surface drift velocities are not likely to be large enough. However, there is a possibility that there may be some relevance to situations where the shear layers are generated mechanically.