Effects of wavelength ratio on wave modelling

Abstract

The efficacy of perturbation approaches for short–long wave interactions is examined by considering a simple case of two interacting wave trains with different wavelengths. Frequency-domain solutions are derived up to third order in wave steepness using two different formulations: one employing conventional wave-mode functions only, and the other introducing a modulated wave-mode representation for the short-wavelength wave. For long-wavelength wave steepness and short-to-long wavelength ratio ε1 and ε3 respectively, the two results are shown to be identical for ε1 [Lt ] ε3 < 0.5. As ε1 approaches ε3, the conventional wave-mode approach converges slowly and eventually diverges for ε1 [Gt ] ε3. The loss of convergence is because the linear phase of conventional wave-mode functions is ineffective for modelling the modulated phase of the short wave. As expected, this difficulty can be removed by using a modulated wave-mode function for the short wave. On the other hand, for relatively large ε3 ∼O(1), the conventional wave-mode approach converges rapidly while the slowly varying interaction between the two waves cannot be accurately predicted by the present modulated wave-mode approach. These findings have important implications to (time-domain) numerical simulations of the nonlinear evolution of ocean wave fields, and suggest that a hybrid wave model employing both conventional (for large-ε3 interactions) and modulated (for small-ε3 interactions) wave-mode functions should be particularly effective.

Modeling extreme waves based on equations of potential flow with a free surface

“…The reason on this restriction is: the small waves overrun the surface of large waves. The amplitudes of wave disturbances with large wave numbers attenuate with depth so quickly, that they become insignificant at the depth of the order of dominant wave height [64]. In this case, restoring the high-order modes on a free surface becomes inaccurate.…”

Weak or strong nonlinearity: the vital issue

“…But the changes in wavelength of the short wave, as it responds to the changes in "effective g" on the surface of the long wave, will produce a cumulative phase change of the short wave (over a quarter-wavelength of the long wave, say), which can be made as large as we please, by making the wavelength ratio k 2 /k 1 sufficiently large, and also keeping sufficient steepness k 1 a 1 in the long wave. We can conclude [51] that Stokes' expansion must diverge, and thus the weakly nonlinear theory must break down. It is not clear, however, whether the waves will break.…”

“…This is clearly incorrect (the exact computations in Figs. 11 and 12 show that two waves of length ratio 2:1 will, when combined, break well before either would individually), and will lead the methods of [51] to produce a non-breaking model of a breaking wave.…”

“…The phenomenon is also recognised in the mathematical literature, where at high ratios there are "short-wave riding on long-wave" models [48][49][50] in which the short-wave is transported onto the surface of the long one, and responds to the "effective g" on its surface. Formally [51], if we write the 2nd order potential (4.2) in the form (4.5) we see that the first term in the brackets {} produces no phase change in the 1st order short-wave potential, and the second term produces a phase change which can never exceed 90 degrees. But the changes in wavelength of the short wave, as it responds to the changes in "effective g" on the surface of the long wave, will produce a cumulative phase change of the short wave (over a quarter-wavelength of the long wave, say), which can be made as large as we please, by making the wavelength ratio k 2 /k 1 sufficiently large, and also keeping sufficient steepness k 1 a 1 in the long wave.…”

Statistical properties of nonlinear one-dimensional wave fields

“…The reason for this limitation restriction: the small waves overrun the surfaces of large waves. The amplitudes of wave disturbances with large wave numbers attenuate with depth quickly, becoming insignificant for the depth of dominant wave height (Zhang et al, 1993). In this case, restoring the high-order modes on a free surface becomes inaccurate.…”