5.3 Spectral Model of Rips
169
speed. Both straight and reverse waves can exist in the area (x < Xm) for
V* < Vmax· There are no reverse waves for V* > Vmax· Only straight waves,
passing through the "barrier", can exist in the other area (x > Xm)· Thus, in
the area (x < xm), the frequency spectrum S(a) can be written in the form:
(5.25)
where 'Y = V/Vmax; fJ = V a/ 9 is a non-dimensional frequency; and 8(fl) is
the Heaviside function.
A similar expression can be written for waves passing through the point
with maximum current speed ( x > Xm):
S2(a) =So (a(1- fl)) { 1 + ~ [aF:~~~~~ 1 -!~)- 5 r [(1- y)- 7 q -1]} -
1 /q
x ( 1 ~ fl) 2 { e [ 4 ( 1 ~ fl) - fJ J e ( ~ - fJ)} .
(5.26)
As can be seen, the expression (5.25) is transformed into (5.26) at the
point x = Xm·
Now the wave spectrum described by the obtained solution will be investigated. In order to do this, it is necessary to prescribe the values of the
parameters in (5.25) and (5.26). The parameter q is assumed to be equal to 2.
It corresponds to the value which is frequently assumed by different authors
in wind wave calculations (Ocean Wave Modeling, 1985).
The spectra (5.25) and (5.26) are written in a more general form by normalizing the spectra 8 1 and 8 2 by the maximum value of the initial spectrum
So(crmax) = (n + 1)mo exp (- n! 1 ) /a max· The normalized spectra fh and S2
are written as a function of the non-dimensional frequency fl. The following
non-dimensional parameters appear is these spectra: v = V CTmax/ 9 is a nondimensional current speed and Vm = VmaxCTmax/ 9 is the effectiveness of the
influence of the selected current profile on the major energy spectrum components. It can be shown that the current influence on the wave spectrum
is negligible for Vm < 0.1. On the other hand, the wave spectrum is blocked
completely by the current (S2 ~ 0) for Vm > 0.5. That is why there is no
sense in considering the case with Vm being greater than the indicated value.
Firstly, the case of wave blocking at Vm = 0.5 is considered. It is necessary
to define the dispersion m 0 to determine the initial spectrum value. The
solution (5.25) and (5.26) includes the dispersion m 0 as a non-dimensional
product of mo(n + 1)a~axjap9 2 = 8o, which can be connected with the
mean initial wave steepness h0 j>.0 = 2.722 x w- 2 J(50 (where h0 and >.0 are
the initial mean wave height and length, respectively).
169
speed. Both straight and reverse waves can exist in the area (x < Xm) for
V* < Vmax· There are no reverse waves for V* > Vmax· Only straight waves,
passing through the "barrier", can exist in the other area (x > Xm)· Thus, in
the area (x < xm), the frequency spectrum S(a) can be written in the form:
(5.25)
where 'Y = V/Vmax; fJ = V a/ 9 is a non-dimensional frequency; and 8(fl) is
the Heaviside function.
A similar expression can be written for waves passing through the point
with maximum current speed ( x > Xm):
S2(a) =So (a(1- fl)) { 1 + ~ [aF:~~~~~ 1 -!~)- 5 r [(1- y)- 7 q -1]} -
1 /q
x ( 1 ~ fl) 2 { e [ 4 ( 1 ~ fl) - fJ J e ( ~ - fJ)} .
(5.26)
As can be seen, the expression (5.25) is transformed into (5.26) at the
point x = Xm·
Now the wave spectrum described by the obtained solution will be investigated. In order to do this, it is necessary to prescribe the values of the
parameters in (5.25) and (5.26). The parameter q is assumed to be equal to 2.
It corresponds to the value which is frequently assumed by different authors
in wind wave calculations (Ocean Wave Modeling, 1985).
The spectra (5.25) and (5.26) are written in a more general form by normalizing the spectra 8 1 and 8 2 by the maximum value of the initial spectrum
So(crmax) = (n + 1)mo exp (- n! 1 ) /a max· The normalized spectra fh and S2
are written as a function of the non-dimensional frequency fl. The following
non-dimensional parameters appear is these spectra: v = V CTmax/ 9 is a nondimensional current speed and Vm = VmaxCTmax/ 9 is the effectiveness of the
influence of the selected current profile on the major energy spectrum components. It can be shown that the current influence on the wave spectrum
is negligible for Vm < 0.1. On the other hand, the wave spectrum is blocked
completely by the current (S2 ~ 0) for Vm > 0.5. That is why there is no
sense in considering the case with Vm being greater than the indicated value.
Firstly, the case of wave blocking at Vm = 0.5 is considered. It is necessary
to define the dispersion m 0 to determine the initial spectrum value. The
solution (5.25) and (5.26) includes the dispersion m 0 as a non-dimensional
product of mo(n + 1)a~axjap9 2 = 8o, which can be connected with the
mean initial wave steepness h0 j>.0 = 2.722 x w- 2 J(50 (where h0 and >.0 are
the initial mean wave height and length, respectively).