Several years ago Dean (1948) showed how to solve the linearized potential problem of water waves passing over a submerged circular cylinder. He discovered the remarkable fact that there is no reflexion from the cylinder; the transmitted waves have the same amplitude as the incident waves, but they suffer a phase shift in passing the cylinder. Soon after the publication of Dean's paper, Ursell (1950) investigated the problem anew. He placed the solution on a rigorous basis, supplied a uniqueness proof, and developed a form of the solution with which it was reasonable to perform calculations.

A theory is given for the non-linear transfer of energy from gravity waves on water to capillary waves. When a progressive gravity wave approaches its maximum steepness it develops a sharp crest, at which the surface tension must be locally important. This gives rise to a travelling disturbance which produces a train of capillary waves ahead of the crest, i.e. on the foward face of the gravity wave. The capillary waves, once formed, then take further energy from the gravity wave through the radiation stresses, at the same time losing energy by viscosity.

The steepness of the capillary waves is calculated and found to be in substantial agreement with some observations by C. S. Cox. An approximate expression for the ripple steepness near the crest of the gravity wave is $(2 \pi|3)e^{-g|6T^\prime k^2},$ where T’ is the surface tension constant and K Is the curvature at the wave crest. The ripple steepness also varies with distance from the wave crest.

Under favourable circumstances the dissipation of energy by the capillary waves can be many times the dissipation in the gravity waves. The capillary waves may therefore play a significant role in the generation of waves by wind, in that they tend to delay the onset of breaking.