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5 Wave Evolution in Non-uniform Currents in Deep Water
Large wave steepness values, occurring in a countercurrent, indicate the
necessity to take into account non-linear effects. Non-linearity can change the
current speed blocking waves.
The correction for the current blocking speed can be easily obtained using
the Stock wave expansion. Thus, the energy transfer speed can be estimated
as:
(5.59)
The value Cg is equal to zero at the blocking point. Then, the current
speed is V* = -0.32 c0 for the ultimate wave steepness ( ak ) 2 = 0.2, where c0
is the phase wave velocity of an infinitely small amplitude in the absence of the
current l-" = 0. Thus, it should be noted that the finite steepness wave value
shows an increased blocking current speed for steeper waves V* = -0.3 c0 as
compared with sloping ones V* = -0.25 c0 .
The non-linear wave effects around the caustic can be described by the
non-linear Schrodinger equation:
(5.60)
where b is a non-linearity parameter. It is shown that the local wave amplitude a around the caustic for the incident wave is stable (Smith, 1976). The
wave field increase is not practically influenced by non-linearity around the
caustic. The wave profile is asymmetric due to the amplitude modulation.
To some extent it is similar to waves in a water surface described by the
Airy function. Depending on the sign of the derivative amplitudes, the front
wave slope can be of greater steepness compared to the rear one, although
the reversed phenomenon can also be observed.
5. 7 Wind Wave Transformation
in Cross-Velocity Shear Current
Solution of the spectral equation.
The problem of wave spectrum
transformation will be considered in a shear horizontally non-uniform current. Its solution can be obtained in analytical form. The current velocity
V is assumed to be directed along the Oy axis with its changes along the x
coordinate: V = {0; Vy(x)}. The depth H = H(x) is also changed along this
direction (see Fig. 5.19). This situation can appear, for example, in a coastal
zone with an along shore current.
The equation (5.2) of wave packet propagation can be written in the
following form:
5 Wave Evolution in Non-uniform Currents in Deep Water
Large wave steepness values, occurring in a countercurrent, indicate the
necessity to take into account non-linear effects. Non-linearity can change the
current speed blocking waves.
The correction for the current blocking speed can be easily obtained using
the Stock wave expansion. Thus, the energy transfer speed can be estimated
as:
(5.59)
The value Cg is equal to zero at the blocking point. Then, the current
speed is V* = -0.32 c0 for the ultimate wave steepness ( ak ) 2 = 0.2, where c0
is the phase wave velocity of an infinitely small amplitude in the absence of the
current l-" = 0. Thus, it should be noted that the finite steepness wave value
shows an increased blocking current speed for steeper waves V* = -0.3 c0 as
compared with sloping ones V* = -0.25 c0 .
The non-linear wave effects around the caustic can be described by the
non-linear Schrodinger equation:
(5.60)
where b is a non-linearity parameter. It is shown that the local wave amplitude a around the caustic for the incident wave is stable (Smith, 1976). The
wave field increase is not practically influenced by non-linearity around the
caustic. The wave profile is asymmetric due to the amplitude modulation.
To some extent it is similar to waves in a water surface described by the
Airy function. Depending on the sign of the derivative amplitudes, the front
wave slope can be of greater steepness compared to the rear one, although
the reversed phenomenon can also be observed.
5. 7 Wind Wave Transformation
in Cross-Velocity Shear Current
Solution of the spectral equation.
The problem of wave spectrum
transformation will be considered in a shear horizontally non-uniform current. Its solution can be obtained in analytical form. The current velocity
V is assumed to be directed along the Oy axis with its changes along the x
coordinate: V = {0; Vy(x)}. The depth H = H(x) is also changed along this
direction (see Fig. 5.19). This situation can appear, for example, in a coastal
zone with an along shore current.
The equation (5.2) of wave packet propagation can be written in the
following form: