One-mode statistics

Of particular interest are one-mode densities which can be conveniently obtained using a one-amplitude generating function

$$Z(t,\lambda)=\langle e^{\lambda \vert a_k\vert^2}\rangle,$$

where $\lambda$ is a real parameter. Then PDF of the wave intensities $s= \vert a_k\vert^2$ at each ${\bf k}$ can be written as a Laplace transform,

$$P^{(a)}(s,t) = \langle \delta (\vert a_k\vert^2 -s) \rangle = {1 \over 2 \pi} \int_{0}^{\infty} Z(\lambda, t)e^{-s \lambda}d\lambda.$$ (26)

For the one-point moments of the amplitude we have

$$M_k^{(p)} \equiv \langle \vert a_k\vert^{2p}\rangle =\langle \vert a\vert^{2p}e^{\lambda \vert a\vert^2}\rangle\vert _{\lambda=0}=$$

$$Z_{\lambda \cdots \lambda}\vert _{\lambda=0} = \int_{0}^{\infty} s^p P^{(a)}(s, t) \, ds,$$ (27)

where $p \in {\cal N}$ and subscript $\lambda$ means differentiation with respect to $\lambda$ $p$ times.

The first of these moments, $n_k = M_k^{(1)}$, is the waveaction spectrum. Higher moments $M_k^{(p)}$ measure fluctuations of the waveaction $k$-space distributions about their mean values [22]. In particular the r.m.s. value of these fluctuations is

$$\sigma_k = \sqrt{ M_k^{(2)} - n_k^2}.$$ (28)