24
1 General Problem Formulation
finite space and time sets { rn, tn} (n = 1, 2) become the objects of study. As
the measurement data indicate, the distribution of probabilities for a value "1
at a fixed point is approximated by the Gaussian distribution.
A theoretical description of wind waves in the terms of finite-dimensional
densities is associated with significant difficulties. It allows investigating only
the simplest statistical characteristics of the value ' TJ, with the second moment
or the correlation function being the most important:
K(~r, ~t) = (ry(r, t)rJ(r + ~r, t + ~t)) ,
(1.48)
where angle brackets denote averaging by a statistical ensemble.
The spatial-temporal correlation function K(~r, ~t) is connected with
the spectrum S(k,w) of a random process by the Fourier transformation:
S(k, w) = ( 2 ~)3 J K(~r, ~t) exp[-i(k~r- w~t)] d~r d~t. (1.49)
The dispersion of surface elevation (ry 2 ) is calculated by integrating the
spectrum S(k,w) over the two-dimensional wave vector k and the frequency w.
The two-dimensional spatial wave spectrum S(k) is determined by (1.49)
according to the formula:
S(k) = J S(k,w) dw = ( 2 ~)2 J K(~r,O) exp(-ik~r) d~r (1.50)
and the frequency spectrum S(w) is determined as:
S(w) = J S(k,w) dk = 2 ~ J K(O,~t) eiwat d~t. (1.51)
The second moments or their corresponding spectra are known to give
complete statistical information about a random field, in the case of a Gaussian process (Davidan et al., 1978). This determines the importance of the
spectral wave characteristic information, as soon as the experimental data
of the distribution function allow us to consider a random field of the level
disturbances rJ as being approximately Gaussian. For the given spectrum the
Gaussian surface model can be a basis for obtaining statistical information
about the geometric characteristics of a moving random surface: the mean
number of stationary points (maxima, minima, hyperbolic points, etc.) per
unit surface area, statistical distributions of maximum and minimum heights,
etc. Numerous results have been obtained by W. Pierson, Yu. Krylov, and
by others. The methods of statistical geometry for random surfaces were
most consistently developed by M. Longuet-Higgins in the 1960s and later by
V. Rozhkov and Yu. Trapeznikov (1990).
The first empirical estimates for wind waves were based on the relation
between their simplest characteristics and wind speed. The main purpose of
1 General Problem Formulation
finite space and time sets { rn, tn} (n = 1, 2) become the objects of study. As
the measurement data indicate, the distribution of probabilities for a value "1
at a fixed point is approximated by the Gaussian distribution.
A theoretical description of wind waves in the terms of finite-dimensional
densities is associated with significant difficulties. It allows investigating only
the simplest statistical characteristics of the value ' TJ, with the second moment
or the correlation function being the most important:
K(~r, ~t) = (ry(r, t)rJ(r + ~r, t + ~t)) ,
(1.48)
where angle brackets denote averaging by a statistical ensemble.
The spatial-temporal correlation function K(~r, ~t) is connected with
the spectrum S(k,w) of a random process by the Fourier transformation:
S(k, w) = ( 2 ~)3 J K(~r, ~t) exp[-i(k~r- w~t)] d~r d~t. (1.49)
The dispersion of surface elevation (ry 2 ) is calculated by integrating the
spectrum S(k,w) over the two-dimensional wave vector k and the frequency w.
The two-dimensional spatial wave spectrum S(k) is determined by (1.49)
according to the formula:
S(k) = J S(k,w) dw = ( 2 ~)2 J K(~r,O) exp(-ik~r) d~r (1.50)
and the frequency spectrum S(w) is determined as:
S(w) = J S(k,w) dk = 2 ~ J K(O,~t) eiwat d~t. (1.51)
The second moments or their corresponding spectra are known to give
complete statistical information about a random field, in the case of a Gaussian process (Davidan et al., 1978). This determines the importance of the
spectral wave characteristic information, as soon as the experimental data
of the distribution function allow us to consider a random field of the level
disturbances rJ as being approximately Gaussian. For the given spectrum the
Gaussian surface model can be a basis for obtaining statistical information
about the geometric characteristics of a moving random surface: the mean
number of stationary points (maxima, minima, hyperbolic points, etc.) per
unit surface area, statistical distributions of maximum and minimum heights,
etc. Numerous results have been obtained by W. Pierson, Yu. Krylov, and
by others. The methods of statistical geometry for random surfaces were
most consistently developed by M. Longuet-Higgins in the 1960s and later by
V. Rozhkov and Yu. Trapeznikov (1990).
The first empirical estimates for wind waves were based on the relation
between their simplest characteristics and wind speed. The main purpose of