(6) Similarly, using the fifth-order expansion, the asymptotes to Stokes third-order theory are *H/L*0 <

theory to be expanded by adding successively smaller areas to the domain of linear theory in Figure II-1-20

until the breaking limit is reached. The fifth-order Stokes theory gets close enough to the breaking limit, and

higher order solutions may not be warranted. Laitone (1962) suggests a shallow-water limit on Stokes' theory

by setting the Ursell number *U*R equal to 20. For an Ursell number of approximately 20, Stokes' theory

approaches the cnoidal theory.

Figure II-1-20 may be used to establish the boundaries of regions where a particular wave theory should be

used. Stokes (1847) noted that this parameter should be small for long waves. An alternative, named the

(8) Limits of validity of the nonlinear (higher-order) wave theories established by Cokelet (1977) and

Williams (1981), are shown in Figure II-1-21. Regions where Stokes waves (short waves) and cnoidal and

II-1-58

Water Wave Mechanics

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