Statistics of a narrow band wave train

In practical applications the frequency spectrum is widely used rather than the action density. Define the wave number spectrum

$$F(\ensuremath{\vec{k}}) = \frac{\omega}{g}N(\ensuremath{\vec{k}})$$ (19)

and the frequency spectrum

$$E(\omega,\theta)d\omega d\theta = F(\ensuremath{\vec{k}})d\ensuremath{\vec{k}}.$$ (20)

Then, from Eq.(), we obtain for large time the following relationship between $kappa_{40}$ and frequency spectrum:

$$\kappa_{40} = \frac{12g}{m_0^2} \mathcal{P} \int d\theta_{1,2,3} d\omega_{1,2... ...qrt{\frac{\omega_4}{\omega_1\omega_2\omega_3}} \frac{E_1 E_2 E_3}{\Delta\omega}$$ (21)

where $\omega_4=\sqrt{g\vert\ensuremath{\vec{k}}_1+\ensuremath{\vec{k}}_2-\ensuremath{\vec{k}}_3\vert}$. Eq.() is valid for arbitrary two-dimensional frequency spectra. However, for operational purposes evaluation of a six-dimensional integral is far too time-consuming and in order to make progress we will make the simplifying assumption of the so-called narrow band approximation. This means, we concentrate on almost unidirectional waves that have a sharply peaked frequency spectrum.

In the narrow-band approximation the spectrum is mainly concentrated at $omega=omega_0$ and $theta=theta_0$ and falls of rapidly, much faster than the other terms in the integrand of Eq.(). In that event, we can approximate the transfer coefficient $T_{1,2,3,4}$ by its narrow band value $k_0^3$. In addition, the angular frequency $omega_4$ becomes independent of $theta$,

$$\omega_4=\left \{\vert\omega_1^2+\omega_2^2-\omega_3^2\vert\right\}^{1/2}.$$ (22)

In fact, apart from the spectra there is no $theta$-dependence. This therefore gives a considerable simplification. Introduce the one-dimensional frequency spectrum

$$E(\omega) =\int{\rm d}\theta E(\omega,\theta),$$ (23)

then

$$\kappa_{40}=\frac{12gk_0^3}{m_0^2}{\cal P}\int{\rm d}\omega_1{\rm... ...\omega_2\omega_3}} \frac{ E(\omega_1) E(\omega_2) E(\omega_3)} {\Delta \omega},$$ (24)

so in the narrow-band approximation only the evaluation of a three-dimensional integral is required. This is operationally feasible, but in practice, the resolution of the frequency spectrum is too coarse to give an accurate evaluation of the singular integral.

A further simplification may be achieved as follows. Approximate the one-dimensional spectrum by a Gaussian function

$$E(\omega)=\frac{m_0}{\sigma_{\omega}\sqrt{2\pi}}\; e^{ -\frac{1}{2}\nu^2},$$ (25)

with

$$\nu=\frac{\omega-\omega_0}{\sigma_{\omega}},$$ (26)

and $m_0$ is the surface elevation variance. Clearly, for small band-width there is a small parameter, namely

$$\Delta = \frac{\sigma_{\omega}}{\omega_0}$$ (27)

and in Eq.() all relevant frequencies, etc., are expanded in terms of $Delta$. The resulting expression for the kurtosis becomes

$$\kappa_{40}=\frac{24\epsilon^2}{\Delta^2}{\cal P}\int\frac{ {\rm ... ...{2}[\nu_1^2+\nu_2^2+\nu_3^2]}} {(\nu_1+\nu_2-\nu_3)^2 -\nu_1^2-\nu_2^2+\nu_3^2}$$ (28)

where we have introduced the steepness parameter $epsilon=k_0sqrt{m_0}$. Eq.() shows the important result that the kurtosis depends on the ratio of two small parameters, namely the integral steepness of the waves and the relative width of the frequency spectrum. The wave steepness reflects, of course, the importance of nonlinearity while the relative width represents the importance of dispersion (but in a spectral sense). The work of () and () has shown that these parameters play a key role in the evolution of deep-water gravity waves. Nonlinearity counteracts dispersion in such a way that focusing of wave energy may occur resulting in extreme wave events and as a consequence in large deviations from the Normal distribution of the surface elevation. In order to measure the relative importance of nonlinearity and dispersion, () introduced the Benjamin Feir Index (BF Index/$BFI$), defined as

$$BFI= \frac{\epsilon}{\Delta}\sqrt{2}.$$ (29)

The $sqrt{2}$-factor is included for historical reasons as according to () a random wave train becomes unstable if $BFI>1$.

In the Appendix we have evaluated the three dimensional integral exactly and introducing the BF Index one finds that the kurtosis depends on the square of $BFI$:

$$\kappa_{40}=\frac{\pi}{\sqrt{3}} BFI^2$$ (30)

Note that for a narrow-band, weakly-nonlinear wave train the BF Index is formally of ${cal O}(1)$. Therefore, the resonant and nonresonant interactions give rise to a much larger contribution to the kurtosis than the bound waves, as the latter contribution is only of the order of the square of the steepness. This follows immediately from the well-known expression for the surface elevation of a steady, narrow band wave train, correct to third order in amplitude,

$$\eta = \alpha \left(1 + \frac{1}{8}k_0^2\alpha^2\right)\cos\theta + \frac{1}{2}k_0\alpha^2\cos2\theta + \frac{3}{8}k_0^2\alpha^3\cos3\theta,$$ (31)

where $alpha$ is connected to the free wave normal variable $b$ through $alpha=bsqrt{omega/2g}$. Since in lowest order $b$, and hence $alpha$, obeys a linear equation, it is justified to assume that the first-order wave train $eta=alphacostheta$ obeys Gaussian statistics. Hence, $theta$ is uniformly distributed and $alpha$ obeys a Rayleigh distribution with width $m_0^{1/2}$ (, ,), and the statistical properties of a narrow band wave train may now readily be obtained. The Stokes wave model (Eq.) predicts wave moments of the form

$$E[\eta^n] = \int_{0}^{\infty}\!\!\int_{0}^{2\pi} d\alpha d\theta \eta^n(\alpha,\theta) F(\alpha,\theta)$$ (32)

where $F(alpha,theta)$ is the joint probability density of $alpha$ and $theta$. For example, in lowest significant order, the skewness becomes

$$\mu_3 = \frac{\langle \eta^3 \rangle}{\langle \eta^2 \rangle^{3/2}}= 3\epsilon,$$ (33)

where $epsilon=k_0m_0^{1/2}$ is a measure of spectral steepness, while the kurtosis becomes

$$\kappa_{40} = 24\epsilon^2.$$ (34)

Similar results were obtained by (). Comparing Eq.() with Eq.() it is evident that for a narrow-band wave train the contribution of the (non)-resonant waves dominates the one from the bound waves when the relative width $Delta$ satisfies the inequality $Delta^2 < pi/12sqrt{3}simeq 0.15$. In practice this condition is easily achieved.