150
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
the non-linear energy transfer appear in the wave spectrum. There is a cubic proportion between the non-linear transfer and the spectral density. Only
a double spectral density enlargement within its frequency maximum range is
able to increase the non-linear transfer intensity almost by an order of magnitude. The periodic changes of the non-linear energy transfer occur within the
quasi-cyclic alterations of its peakness. Should the peakness within the periods become larger compared with its average value, then the intensity of the
non-linear energy transfer is increased. The non-linear transfer is decreased
in the case of small peakness values.
Now an average analytical estimate of the non-linear energy transfer in
the cyclically changed peakness will be obtained. According to ( 4.1), the
first estimation of the non-linear transfer value is evaluated as Gni(S) "'
w~axS!ax/ g 4 . It should be noted that more accurate numerical simulation
results (see Fig. 4.24) carried out for the spectrum (4.56) reveal that Gn1
is proportional to s;;,ax rather than to S!ax. Assuming that the maximum
frequency Wmax is not changed within one cycle and the spectral peak value
is changed as Smax(t) rv "(p,sin( 2 nt/Tl, integration of the non-linear transfer
value within one quasi-oscillation period produces the average as follows:
T
1 J
-
(Gn,) = -:;.
Gn,(S(t)) dt = Gn, Io(p) ,
( 4.75)
0
where Gn1 is a non-linear transfer without quasi-oscillations (f.L = 0), I 0 is
the modified first-order Bessel function, and the parameter p is determined
asp= 2Jl,ln("f). Assuming that f.L = 1.0 and"(= 3.3, it is possible to derive
p ~ 2.39 (i.e. p > 1). The Bessel function ( 4. 75) is estimated by the following
asymptotic formula:
eP
(
12
12 . 32
)
Io(P) ~ VJiiij 1 + 1! 8p + 2! (8p)2 + ... .
(4.76)
For the specific parameter values it can be found that I 0 ~ 3.05.
It can be pointed out that the spectrum mean over the period of quasioscillations determined as ( 4. 70) differs from its corresponding value computed for the case of "no oscillation" (f.L = 0). The mean spectrum for the
same period is equal to:
(S) = S · Io(p/2),
(4. 77)
where Sis the spectrum value without quasi-oscillations (/1 = 0).
The factor is equal to I 0 (p/2) ~ 1.38 in (4.77). If the non-linear energy
transfer for the mean spectrum is recalculated, the value is 1.92 times greater,
i.e. 1.59 times less than the non-linear energy transfer with quasi-oscillations.
Thus, it can be concluded that the averaged non-linear energy transfer
within the quasi-oscillation period is essentially greater than the average spectrum value for the same period of time. An intensive energy flux towards the
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
the non-linear energy transfer appear in the wave spectrum. There is a cubic proportion between the non-linear transfer and the spectral density. Only
a double spectral density enlargement within its frequency maximum range is
able to increase the non-linear transfer intensity almost by an order of magnitude. The periodic changes of the non-linear energy transfer occur within the
quasi-cyclic alterations of its peakness. Should the peakness within the periods become larger compared with its average value, then the intensity of the
non-linear energy transfer is increased. The non-linear transfer is decreased
in the case of small peakness values.
Now an average analytical estimate of the non-linear energy transfer in
the cyclically changed peakness will be obtained. According to ( 4.1), the
first estimation of the non-linear transfer value is evaluated as Gni(S) "'
w~axS!ax/ g 4 . It should be noted that more accurate numerical simulation
results (see Fig. 4.24) carried out for the spectrum (4.56) reveal that Gn1
is proportional to s;;,ax rather than to S!ax. Assuming that the maximum
frequency Wmax is not changed within one cycle and the spectral peak value
is changed as Smax(t) rv "(p,sin( 2 nt/Tl, integration of the non-linear transfer
value within one quasi-oscillation period produces the average as follows:
T
1 J
-
(Gn,) = -:;.
Gn,(S(t)) dt = Gn, Io(p) ,
( 4.75)
0
where Gn1 is a non-linear transfer without quasi-oscillations (f.L = 0), I 0 is
the modified first-order Bessel function, and the parameter p is determined
asp= 2Jl,ln("f). Assuming that f.L = 1.0 and"(= 3.3, it is possible to derive
p ~ 2.39 (i.e. p > 1). The Bessel function ( 4. 75) is estimated by the following
asymptotic formula:
eP
(
12
12 . 32
)
Io(P) ~ VJiiij 1 + 1! 8p + 2! (8p)2 + ... .
(4.76)
For the specific parameter values it can be found that I 0 ~ 3.05.
It can be pointed out that the spectrum mean over the period of quasioscillations determined as ( 4. 70) differs from its corresponding value computed for the case of "no oscillation" (f.L = 0). The mean spectrum for the
same period is equal to:
(S) = S · Io(p/2),
(4. 77)
where Sis the spectrum value without quasi-oscillations (/1 = 0).
The factor is equal to I 0 (p/2) ~ 1.38 in (4.77). If the non-linear energy
transfer for the mean spectrum is recalculated, the value is 1.92 times greater,
i.e. 1.59 times less than the non-linear energy transfer with quasi-oscillations.
Thus, it can be concluded that the averaged non-linear energy transfer
within the quasi-oscillation period is essentially greater than the average spectrum value for the same period of time. An intensive energy flux towards the