262
6 Wave Transformation in Shallow Water
The wave frequency w is assumed to be connected with the wave number k
using the dispersion ratio w 2 = gk th ( kH).
A numerical solution of (6.17) for finite depth water is quite a difficult
problem demanding much computing time. The recommendations of Herterich & Hasselmann (1980) can simplify the solution. The results of numerical
calculation of non-linear energy transfer for the frequency-angular spectrum
S ( w, {3) in the case of finite depth are shown to be similar to those obtained
for infinitely deep water (w 2 = gk) and differ only by the multiplier R:
GNL (w, {3) lw2=gk th(kH) = R (kH) Gnl (w, {3) lw2=gk ,
(6.18)
where GNL is a function of the non-linear energy transfer for the wave spectrum in finite depth water and Gnl is a similar function for the infinitely
deep water case. The expression for the function R can determined as follows
(Komen et al. 1988, 1994):
R(z) = 1 + 5 ~ 5 (1- ~z) exp ( -~z)
(6.19)
where the valueR is a function of the parameter z = kH.
Thus, the solution of the spectrum evolution ( 6.17) is reduced to solving
the equation where the right-hand side is calculated using the formula for
infinitely deep water multiplying it by the function R.
Algorithm for solving the evolution problem. In order to solve numerically the evolution problem, the collision integral is calculated using the
method described in Chap. 4.
It should be taken into account that the time required for calculation of
the integral is sufficiently large. It has to be calculated numerically many
times, i.e. at every step of the spectrum evolution. To save calculation
time, the spectrum and the collision integral can be presented in the nondimensional form:
S (w, {3) = SmaxS (w, {3)
Gnl (w,{3) = S!axw;;axanl (w,{3) /l,
(6.20)
(6.21)
where Smax is the maximum spectrum value; S is its non-dimensional value;
and Gn1 is the non-dimensional collision integral value. In a similar way, the
core function Tin ( 4.4) of the integrand ( 4.3), not depending on the spectrum,
is reduced to non-dimensional form.
In order to calculate the non-dimensional values, a grid with the relative
frequencies Wi (i = 1.30) and directions {3j (j = 1.36) is chosen. A set of
core values is calculated for these grid points. The numerical solution of the
equation is obtained by stages. At the first stage, using the given spectrum
S (w,{3, tn), the spectral maximum Smax and the corresponding frequency
6 Wave Transformation in Shallow Water
The wave frequency w is assumed to be connected with the wave number k
using the dispersion ratio w 2 = gk th ( kH).
A numerical solution of (6.17) for finite depth water is quite a difficult
problem demanding much computing time. The recommendations of Herterich & Hasselmann (1980) can simplify the solution. The results of numerical
calculation of non-linear energy transfer for the frequency-angular spectrum
S ( w, {3) in the case of finite depth are shown to be similar to those obtained
for infinitely deep water (w 2 = gk) and differ only by the multiplier R:
GNL (w, {3) lw2=gk th(kH) = R (kH) Gnl (w, {3) lw2=gk ,
(6.18)
where GNL is a function of the non-linear energy transfer for the wave spectrum in finite depth water and Gnl is a similar function for the infinitely
deep water case. The expression for the function R can determined as follows
(Komen et al. 1988, 1994):
R(z) = 1 + 5 ~ 5 (1- ~z) exp ( -~z)
(6.19)
where the valueR is a function of the parameter z = kH.
Thus, the solution of the spectrum evolution ( 6.17) is reduced to solving
the equation where the right-hand side is calculated using the formula for
infinitely deep water multiplying it by the function R.
Algorithm for solving the evolution problem. In order to solve numerically the evolution problem, the collision integral is calculated using the
method described in Chap. 4.
It should be taken into account that the time required for calculation of
the integral is sufficiently large. It has to be calculated numerically many
times, i.e. at every step of the spectrum evolution. To save calculation
time, the spectrum and the collision integral can be presented in the nondimensional form:
S (w, {3) = SmaxS (w, {3)
Gnl (w,{3) = S!axw;;axanl (w,{3) /l,
(6.20)
(6.21)
where Smax is the maximum spectrum value; S is its non-dimensional value;
and Gn1 is the non-dimensional collision integral value. In a similar way, the
core function Tin ( 4.4) of the integrand ( 4.3), not depending on the spectrum,
is reduced to non-dimensional form.
In order to calculate the non-dimensional values, a grid with the relative
frequencies Wi (i = 1.30) and directions {3j (j = 1.36) is chosen. A set of
core values is calculated for these grid points. The numerical solution of the
equation is obtained by stages. At the first stage, using the given spectrum
S (w,{3, tn), the spectral maximum Smax and the corresponding frequency