276
6 Wave Transformation in Shallow Water
distribution within the equilibrium range; O:F is the Phillips constant; and /38
is the initial general direction of wave propagation.
The constructed spectrum is a function of the arguments y and /3, whereas
its defining parameters /38, a, v, "' (, J. L fix the initial spectrum state: the initial
depth Ho and the current velocity Vo, as well as the maximum current velocity
value Vmax· Using the given initial values of these parameters, the spectral
evolution at different basin points is determined only by the parameter 1.
The mean values for wave elements changing along the current are obtained with the help of the spectral expression (6.35). By integrating the
spectrum (6.35) over y and /3, similar to (5.40)-(5.42), the following values
can be found:
the ratio of the mean height li to the mean height of the initial wave li0:
the ratio of mean wavelengths:
and the ratio of mean periods:
r = r/ro = h (/38,a,v,"f,J.L).
The values !1, h, h are functions dependent on the non-dimensional
parameters /38, a, v, "f, J.L· This makes the obtained formulas more universal.
The ratios !1, h, !J, presented as integrals, are numerically calculated
using the Chebyshev quadrature method.
In order to simplify the physical interpretation of the results, the parameters a and v are expressed through the mean values of the initial wave
elements. With the help of the initial spectrum (5.16), the parameter a can
be expressed using the mean length >.0 :
(6.36)
Analogously, the parameter v can be written as:
(6.37)
It should be noted that the ratios (6.36) and (6.37) are accurate for wave
propagation cases in a fair current, when the reverse waves are absent and all
spectral components propagate from the initial boundary to the calculated
point. In such a case the following is obtained for n = 5.5:
a= 4.19Ho/Xo ; v = 2.05Vo/ ~ .
(6.38)
6 Wave Transformation in Shallow Water
distribution within the equilibrium range; O:F is the Phillips constant; and /38
is the initial general direction of wave propagation.
The constructed spectrum is a function of the arguments y and /3, whereas
its defining parameters /38, a, v, "' (, J. L fix the initial spectrum state: the initial
depth Ho and the current velocity Vo, as well as the maximum current velocity
value Vmax· Using the given initial values of these parameters, the spectral
evolution at different basin points is determined only by the parameter 1.
The mean values for wave elements changing along the current are obtained with the help of the spectral expression (6.35). By integrating the
spectrum (6.35) over y and /3, similar to (5.40)-(5.42), the following values
can be found:
the ratio of the mean height li to the mean height of the initial wave li0:
the ratio of mean wavelengths:
and the ratio of mean periods:
r = r/ro = h (/38,a,v,"f,J.L).
The values !1, h, h are functions dependent on the non-dimensional
parameters /38, a, v, "f, J.L· This makes the obtained formulas more universal.
The ratios !1, h, !J, presented as integrals, are numerically calculated
using the Chebyshev quadrature method.
In order to simplify the physical interpretation of the results, the parameters a and v are expressed through the mean values of the initial wave
elements. With the help of the initial spectrum (5.16), the parameter a can
be expressed using the mean length >.0 :
(6.36)
Analogously, the parameter v can be written as:
(6.37)
It should be noted that the ratios (6.36) and (6.37) are accurate for wave
propagation cases in a fair current, when the reverse waves are absent and all
spectral components propagate from the initial boundary to the calculated
point. In such a case the following is obtained for n = 5.5:
a= 4.19Ho/Xo ; v = 2.05Vo/ ~ .
(6.38)