Horizontal particle velocities in long waves

Abstract

A general expression is derived for the horizontal particle velocities in long waves. The derivation is based on a polynomial expansion in the vertical coordinate z which results in an expression for u in terms of the surface elevation η and its derivatives. The result is valid for high waves but is in the preliminary form not directly applicable. It is shown, however, that for waves of relatively small amplitude to depth ratio the result corresponds to the well‐known cnoidal wave theory. The horizontal velocities predicted by that theory are then critically examined, and it is shown that there are conflicting properties, which imply that no cnoidal theory can be satisfactory for waves higher than 30–40% of the depth of water. Instead, a simple formula based on a parabolic truncation of the above‐mentioned polynomial in z is derived for the velocity under the crest of arbitrary high waves. The results are compared both with experiments and with stream function theory, and, particularly, the latter gives remarkably good agreement even for the highest waves.