5.5 Wave Element Transformations in Current, Varying Along its Direction
185
curve J 1 . It takes an infinitely large value (singularity) at the special point
(Vfc0 = -1/4). The straight wave length is decreased in a countercurrent,
being four times less at the special point than the initial value >.0 . There is
also a change of the wave parameters, corresponding to the (-) sign in the
formula (5.38), i.e. for a reverse wave. It is described by the curve J2, which
is also of infinitely large value at the point V /eo = -1/4.
The second singularity appears in the ratio h/ho for a reverse wave with
decreasing countercurrent velocity. Its wavelength tends to zero (although,
in reality, the capillarity does not allow the wave number to increase up to
infinity as V-+ 0). The wave height h/ho is decreased in a fair current, while
the wavelength .A/ >.0 is increased.
It is necessary to note that for a long time different researchers used
the relation (5.38) only with the ( +) sign. In this case the reverse waves
were not taken into consideration in the countercurrent. The ambiguity of
this expression was considered to be purely formal, whereas the (-) sign
was supposed to have no physical sense. However, reverse waves really do
exist, as shown by the experimental (Pokazeyev & Rozenberg, 1983) and
theoretical (Basovich & Bakhanov, 1979; Lavrenov, 1986, 1998; Peregrine,
1976) studies. These are waves reflected from a non-uniform current and
carried away downstream. The relation (5.37), applied near a special point
(caustic), admittedly leads to a wrong result, as soon as there is a violation
of the Longuet-Higgins and Stewart theory.
In order to solve the problem the idea of using a spectral approach was
proposed by Lavrenov (1986, 1988b). The approach eliminates the aforementioned singularity and produces an adequate wave field description.
A regular monochromatic wave discussed above is an idealization that
does not occur in natural conditions. Usual wind waves consist of different
spectral components, corresponding to different wave vectors k = {kx, ky}·
That is why there can exit a separate caustic for every component k =
{ kx, ky}. Thus, the entire horizontal plane r = { x, y} could be covered with
caustics. The statistical averaging of different waves results in a smoothing
caustic. Thus, there is no need to introduce any corrections in the geometrical
optics approximation for accurate wave field estimation (Krasitskii, 1974).
The kinetic equation (5.1) in the form (5.19) will be used to describe the
wave evolution in a current. Unlike in Sect. 5.2, the wave number k (or the
frequency a) and the angle {3 will be used instead of the variables w and {3,
because the dependence of the wave number k (or frequency a) on them
is ambiguous. Implementation of these variables avoids the aforementioned
shortcoming. There is no singularity in the spectral energy density S(k, {3),
occurring in (5.9). Furthermore, the variables k and {3 make it possible to
use easily in the estimations the idea of Phillips about the invariance of the
equilibrium interval (Kitaigorodskii et al., 1975).
The propagation of waves in deep water from an area where the current is
absent, to a non-uniform stationary current will be considered with the help
of (5.19). The current velocity is considered to be directed along the Ox axis
185
curve J 1 . It takes an infinitely large value (singularity) at the special point
(Vfc0 = -1/4). The straight wave length is decreased in a countercurrent,
being four times less at the special point than the initial value >.0 . There is
also a change of the wave parameters, corresponding to the (-) sign in the
formula (5.38), i.e. for a reverse wave. It is described by the curve J2, which
is also of infinitely large value at the point V /eo = -1/4.
The second singularity appears in the ratio h/ho for a reverse wave with
decreasing countercurrent velocity. Its wavelength tends to zero (although,
in reality, the capillarity does not allow the wave number to increase up to
infinity as V-+ 0). The wave height h/ho is decreased in a fair current, while
the wavelength .A/ >.0 is increased.
It is necessary to note that for a long time different researchers used
the relation (5.38) only with the ( +) sign. In this case the reverse waves
were not taken into consideration in the countercurrent. The ambiguity of
this expression was considered to be purely formal, whereas the (-) sign
was supposed to have no physical sense. However, reverse waves really do
exist, as shown by the experimental (Pokazeyev & Rozenberg, 1983) and
theoretical (Basovich & Bakhanov, 1979; Lavrenov, 1986, 1998; Peregrine,
1976) studies. These are waves reflected from a non-uniform current and
carried away downstream. The relation (5.37), applied near a special point
(caustic), admittedly leads to a wrong result, as soon as there is a violation
of the Longuet-Higgins and Stewart theory.
In order to solve the problem the idea of using a spectral approach was
proposed by Lavrenov (1986, 1988b). The approach eliminates the aforementioned singularity and produces an adequate wave field description.
A regular monochromatic wave discussed above is an idealization that
does not occur in natural conditions. Usual wind waves consist of different
spectral components, corresponding to different wave vectors k = {kx, ky}·
That is why there can exit a separate caustic for every component k =
{ kx, ky}. Thus, the entire horizontal plane r = { x, y} could be covered with
caustics. The statistical averaging of different waves results in a smoothing
caustic. Thus, there is no need to introduce any corrections in the geometrical
optics approximation for accurate wave field estimation (Krasitskii, 1974).
The kinetic equation (5.1) in the form (5.19) will be used to describe the
wave evolution in a current. Unlike in Sect. 5.2, the wave number k (or the
frequency a) and the angle {3 will be used instead of the variables w and {3,
because the dependence of the wave number k (or frequency a) on them
is ambiguous. Implementation of these variables avoids the aforementioned
shortcoming. There is no singularity in the spectral energy density S(k, {3),
occurring in (5.9). Furthermore, the variables k and {3 make it possible to
use easily in the estimations the idea of Phillips about the invariance of the
equilibrium interval (Kitaigorodskii et al., 1975).
The propagation of waves in deep water from an area where the current is
absent, to a non-uniform stationary current will be considered with the help
of (5.19). The current velocity is considered to be directed along the Ox axis