Royal Society Publishing

Limiting Gravity Waves in Water of Finite Depth

J. M. Williams


Progressive, irrotational gravity waves of constant form exist as a two-parameter family. The first parameter, the ratio of mean depth to wavelength, varies from zero (the solitary wave) to infinity (the deep-water wave). The second parameter, the wave height or amplitude, varies from zero (the infinitesimal wave) to a limiting value dependent on the first parameter. For limiting waves the wave crest ceases to be rounded and becomes angled, with an included angle of 120 degrees. Most methods of calculating finite-amplitude waves use either a form of series expansion or the solution of an integral equation. For waves nearing the limiting amplitude many terms (or nodal points) are needed to describe the wave form accurately. Consequently the accuracy even of recent solutions on modern computers can be improved upon, except at the deep-water end of the range. The present work extends an integral equation technique used previously in which the angled crest of the limiting wave is included as a specific term, derived from the well known Stokes corner flow. This term is now supplemented by a second term, proposed by Grant in a study of the flow near the crest. Solutions comprising 80 terms at the shallow-water end of the range, reducing to 20 at the deep-water end, have defined many field and integral properties of the flow to within 1 to 2 parts in 10$^{6}$. It is shown that without the new crest term this level of accuracy would have demanded some hundreds of terms while without either crest term many thousands of terms would have been needed. The practical limits of the computing range are shown to correspond, to working accuracy, with the theoretical extremes of the solitary wave and the deep-water wave. In each case the results agree well with several previous accurate solutions and it is considered that the accuracy has been improved. For example, the height:depth ratio of the solitary wave is now estimated to be 0.833 197 and the height:wavelength ratio of the deep-water wave to be 0.141 063. The results are presented in detail to facilitate further theoretical study and early practical application. The coefficients defining the wave motion are given for 22 cases, five of which, including the two extremes, are fully documented with tables of displacement, velocity, acceleration, pressure and time. Examples of particle orbits and drift profiles are presented graphically and are shown for the extreme waves to agree very closely with simplified calculations by Longuet-Higgins. Finally, the opportunity has been taken to calculate to greater accuracy the long-term Lagrangian-mean angular momentum of the maximum deep-water wave, according to the recent method proposed by Longuet-Higgins, with the conclusion that the level of action is slightly above the crest.