1.4 Statistical Wind Wave Description
23
y
Fig. 1.1. Wave rays in the background of a non-uniform current
the wave field can be significantly different from a local plane one. Thus, if
the Jacobian becomes zero, J 1 = 0, in the solution (1.46), then a singularity
(caustic) occurs due to the ray tube width being decreased to zero. An infinite
narrowing of the ray tube width occurs due to ray intersection in the ratio
(1.47). In this case the estimate of the wave height becomes unrealistically
large in the vicinity of the caustic. These are considered below when the use
of diffraction approaches is necessary.
An alternative solution method can be developed using the spectral approach developed in this monograph. The advantage of this method is that
the ray set r = r(~, (, T) in physical space can have a sufficiently complicated
form. This results in complexity of constructing the smooth solution for the
entire space. However, there can be only one phase trajectory, crossing a given
point in the phase space { k, r}, i.e. the phase trajectories do not intersect
each other in the spectral solution. In fact, this quality is a consequence of
the uniqueness of solutions for ordinary differential equation systems with
given initial conditions.
1.4 Statistical Wind Wave Description
An obvious feature of wind waves is their random character. As wind waves
present a non-stationary probabilistic hydrodynamic process, the ideas and
methods of random process theory are widely used for their theoretical and
experimental investigation. The displacement of the water-air interface ry(r, t)
is the main experimentally observed characteristic of wind waves. That is why
it is necessary to consider ry(r, t) as a random moving surface in the probabilistic description of wind waves. Probabilistic distributions of the values rJ in
23
y
Fig. 1.1. Wave rays in the background of a non-uniform current
the wave field can be significantly different from a local plane one. Thus, if
the Jacobian becomes zero, J 1 = 0, in the solution (1.46), then a singularity
(caustic) occurs due to the ray tube width being decreased to zero. An infinite
narrowing of the ray tube width occurs due to ray intersection in the ratio
(1.47). In this case the estimate of the wave height becomes unrealistically
large in the vicinity of the caustic. These are considered below when the use
of diffraction approaches is necessary.
An alternative solution method can be developed using the spectral approach developed in this monograph. The advantage of this method is that
the ray set r = r(~, (, T) in physical space can have a sufficiently complicated
form. This results in complexity of constructing the smooth solution for the
entire space. However, there can be only one phase trajectory, crossing a given
point in the phase space { k, r}, i.e. the phase trajectories do not intersect
each other in the spectral solution. In fact, this quality is a consequence of
the uniqueness of solutions for ordinary differential equation systems with
given initial conditions.
1.4 Statistical Wind Wave Description
An obvious feature of wind waves is their random character. As wind waves
present a non-stationary probabilistic hydrodynamic process, the ideas and
methods of random process theory are widely used for their theoretical and
experimental investigation. The displacement of the water-air interface ry(r, t)
is the main experimentally observed characteristic of wind waves. That is why
it is necessary to consider ry(r, t) as a random moving surface in the probabilistic description of wind waves. Probabilistic distributions of the values rJ in