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Of particular interest are one-mode densities which can
be conveniently obtained using a one-amplitude
generating function
where
is a real parameter. Then PDF of the wave
intensities
at each
can be written as a Laplace
transform,
![$\displaystyle P^{(a)}(s,t) = \langle \delta (\vert a_k\vert^2 -s) \rangle = {1 \over 2 \pi} \int_{0}^{\infty} Z(\lambda, t)e^{-s \lambda}d\lambda.$](img165.png) |
(26) |
For the one-point moments of the amplitude we have
![$\displaystyle M_k^{(p)} \equiv \langle
\vert a_k\vert^{2p}\rangle
=\langle
\vert a\vert^{2p}e^{\lambda \vert a\vert^2}\rangle\vert _{\lambda=0}=$](img166.png) |
|
|
|
![$\displaystyle Z_{\lambda \cdots \lambda}\vert _{\lambda=0}
= \int_{0}^{\infty} s^p P^{(a)}(s, t) \, ds,$](img167.png) |
|
|
(27) |
where
and subscript
means differentiation
with respect to
times.
The first of these moments,
, is the waveaction spectrum.
Higher moments
measure fluctuations of the waveaction
-space
distributions about their
mean values [22]. In particular the r.m.s. value of these fluctuations is
![$\displaystyle \sigma_k = \sqrt{ M_k^{(2)} - n_k^2}.$](img172.png) |
(28) |
Next: Separation of timescales: general
Up: Setting the stage II:
Previous: Generating functional.
Dr Yuri V Lvov
2007-01-23