244
5 Wave Evolution in Non-uniform Currents in Deep Water
The mean period can be determined differently depending on the chosen
coordinate system. It can be calculated with the help of formulas similar to
(5.150), (5.151) in the coordinate system moving with the current:
am(x) (n~l)~ Jr(1-2/n)Jmo(x) -
- - - - - - - - - - ' - - . . . . , - 1 - F l (X, i)
[l£ks(k,(1,X,i) dkdf1r
= F3 (x,-r) .
(5.152)
Determining the mean period in the immovable coordinate system 72 ,
it is necessary to take into account the Doppler shift connected with the
current. The frequency w used to determine the mean period 72 can attain
both positive and negative values (Peregrine, 1976). The latter holds for waves
with their own phase velocity c = ak / k 2 , directed opposite to the current and
being less in value. In order to calculate the mean period 7'2 , the expression of
the mean value should be used rather than for the RMS value, as usually used
in (5.150)-(5.152). Otherwise, the contribution of harmonics with a negative
frequency to the mean period 7' 2 could be taken into account with the wrong
sign. The expression for the mean wave period in the immovable coordinate
system can be written as follows:
am (x) (n~ 1 )~ T(1-4/n)m0 (x)
2 ( _ -)
------'---'------------'--'------,, F 1 X,"!
[ l [ ( V k - i k cos ((1)) s ( k, !1, x, i) dk df1]
2
= F4 (x,-r) .
(5.153)
Thus, the obtained integral formulas (5.150)-(5.153), describing the change
of the mean wave elements in a current, are functions of the non-dimensional
fetch X and the relative current velocity i- The non-dimensional form of
the determining parameters X and i makes the obtained expressions more
universal.
The obtained integral expressions F1 , F2 , F3 , F 4 are calculated numerically. The integrands are presented in a discrete way as eight angular directions f'i (within the range from n/2 to n) and 50 components by the wave
numbers kj, taken in geometrical progression. Thus, it provides accuracy of
integral calculation with not more than 1-2 per cent numerical error.
The non-dimensional fetch X is varied from 10 to 10 5 , and the parameter
of the relative velocity i is assumed to be within the range from -0.5 to
+0.5. It covers a reasonable range of these parameter variations under fullscale conditions.
5 Wave Evolution in Non-uniform Currents in Deep Water
The mean period can be determined differently depending on the chosen
coordinate system. It can be calculated with the help of formulas similar to
(5.150), (5.151) in the coordinate system moving with the current:
am(x) (n~l)~ Jr(1-2/n)Jmo(x) -
- - - - - - - - - - ' - - . . . . , - 1 - F l (X, i)
[l£ks(k,(1,X,i) dkdf1r
= F3 (x,-r) .
(5.152)
Determining the mean period in the immovable coordinate system 72 ,
it is necessary to take into account the Doppler shift connected with the
current. The frequency w used to determine the mean period 72 can attain
both positive and negative values (Peregrine, 1976). The latter holds for waves
with their own phase velocity c = ak / k 2 , directed opposite to the current and
being less in value. In order to calculate the mean period 7'2 , the expression of
the mean value should be used rather than for the RMS value, as usually used
in (5.150)-(5.152). Otherwise, the contribution of harmonics with a negative
frequency to the mean period 7' 2 could be taken into account with the wrong
sign. The expression for the mean wave period in the immovable coordinate
system can be written as follows:
am (x) (n~ 1 )~ T(1-4/n)m0 (x)
2 ( _ -)
------'---'------------'--'------,, F 1 X,"!
[ l [ ( V k - i k cos ((1)) s ( k, !1, x, i) dk df1]
2
= F4 (x,-r) .
(5.153)
Thus, the obtained integral formulas (5.150)-(5.153), describing the change
of the mean wave elements in a current, are functions of the non-dimensional
fetch X and the relative current velocity i- The non-dimensional form of
the determining parameters X and i makes the obtained expressions more
universal.
The obtained integral expressions F1 , F2 , F3 , F 4 are calculated numerically. The integrands are presented in a discrete way as eight angular directions f'i (within the range from n/2 to n) and 50 components by the wave
numbers kj, taken in geometrical progression. Thus, it provides accuracy of
integral calculation with not more than 1-2 per cent numerical error.
The non-dimensional fetch X is varied from 10 to 10 5 , and the parameter
of the relative velocity i is assumed to be within the range from -0.5 to
+0.5. It covers a reasonable range of these parameter variations under fullscale conditions.