296
6 Wave Transformation in Shallow Water
( -130 x 10 3 m < y < 130 x 10 3 m) for wave transformation at the point under
consideration.
The initial wave spectrum in deep water (for x = 0) is accepted according to (5.16), with the angular distribution taken in the form of the specified approximation (4.14) and (4.15). The frequency O"max of the maximum
spectrum is accepted to be equal to 1.0 rads- 1 , and the wave number is presented in discrete form (i.e. the set of 17 values is varied within the range
0.06 < ki < 1.26m- 1 ). The set of angle directions /3j consists of 30 values
within the range -0.96n/2 < /3j < 0.96nj2.
Applying this algorithm, the equation set (1.86)-(1.89) is integrated "forward" and after that "backward" jointly with the wave energy balance equation, using the Runge-Kutta method of fourth-order accuracy with automatic
choice of integration step.
As a result of numerical calculations made for a general wave direction coinciding with the Ox axis the following values: spectrum S(k,/3), frequency angular spectrum S ( O", (3) = S ( k, (3) dk j dO", spectrum of wave numbers S(k) = J S(k,/3) d/3 and frequency spectrum S(O") = J S(0",/3) df3 are
obtained for different points along the Ox axis (for y = 0).
The frequency angular spectrum S ( O", (3) appears to reveal a tendency to
increase at low frequencies and decrease somewhat at high frequencies (for
f3 ~ 0). The spectral density decrease takes place at every frequency for the
angles f3, being sufficiently different from the general directions. The spectral
density is decreased to zero at low frequencies for large angles (/3 > 30°). This
is explained by the absence of proper spectral components and by narrowing
the angular distribution of the wave energy.
The behaviour of the spectrum S ( O", (3) is determined by diminishing the
depth with the group velocity being decreased at first and then increased,
whereas the wave number is monotonically increased, and the propagation
rays of the spectral components are turned perpendicular to the shore line.
The frequency spectrum S(O") is not essentially changed (at least up to 5 m
depth) and the frequency of the spectral maximum actually remains the same.
The spectrum S(k) is shifted to the area of large wave numbers, becoming
wider with its maximum being slightly decreased. Such spectrum behaviour
is proved by field observations (Krylov et al., 1976).
After obtaining the spectrum value at the point under consideration, it
is integrated over angles and wave numbers using the Simpson cubature
method. Its statistical moments and the wave element estimations, such as
the height h, length >.. and period T, are obtained. Relative values of the mean
height h = hjh0 , length ~ = 5..j5..0 and period i = f /fa, presented as functions of the relative depth Hj5..0 , practically coincide with accurate spectral
solutions (see Fig. 6.10, Sect. 6.3). In particular for a straight isobaths, the
relative height hjh0 is decreased at first to the value ~ 0.9, and then it is
increased. The mean wave period is weakly decreased (up to 3 per cent), then
slowly increased. The wavelengths are monotonically decreased.
6 Wave Transformation in Shallow Water
( -130 x 10 3 m < y < 130 x 10 3 m) for wave transformation at the point under
consideration.
The initial wave spectrum in deep water (for x = 0) is accepted according to (5.16), with the angular distribution taken in the form of the specified approximation (4.14) and (4.15). The frequency O"max of the maximum
spectrum is accepted to be equal to 1.0 rads- 1 , and the wave number is presented in discrete form (i.e. the set of 17 values is varied within the range
0.06 < ki < 1.26m- 1 ). The set of angle directions /3j consists of 30 values
within the range -0.96n/2 < /3j < 0.96nj2.
Applying this algorithm, the equation set (1.86)-(1.89) is integrated "forward" and after that "backward" jointly with the wave energy balance equation, using the Runge-Kutta method of fourth-order accuracy with automatic
choice of integration step.
As a result of numerical calculations made for a general wave direction coinciding with the Ox axis the following values: spectrum S(k,/3), frequency angular spectrum S ( O", (3) = S ( k, (3) dk j dO", spectrum of wave numbers S(k) = J S(k,/3) d/3 and frequency spectrum S(O") = J S(0",/3) df3 are
obtained for different points along the Ox axis (for y = 0).
The frequency angular spectrum S ( O", (3) appears to reveal a tendency to
increase at low frequencies and decrease somewhat at high frequencies (for
f3 ~ 0). The spectral density decrease takes place at every frequency for the
angles f3, being sufficiently different from the general directions. The spectral
density is decreased to zero at low frequencies for large angles (/3 > 30°). This
is explained by the absence of proper spectral components and by narrowing
the angular distribution of the wave energy.
The behaviour of the spectrum S ( O", (3) is determined by diminishing the
depth with the group velocity being decreased at first and then increased,
whereas the wave number is monotonically increased, and the propagation
rays of the spectral components are turned perpendicular to the shore line.
The frequency spectrum S(O") is not essentially changed (at least up to 5 m
depth) and the frequency of the spectral maximum actually remains the same.
The spectrum S(k) is shifted to the area of large wave numbers, becoming
wider with its maximum being slightly decreased. Such spectrum behaviour
is proved by field observations (Krylov et al., 1976).
After obtaining the spectrum value at the point under consideration, it
is integrated over angles and wave numbers using the Simpson cubature
method. Its statistical moments and the wave element estimations, such as
the height h, length >.. and period T, are obtained. Relative values of the mean
height h = hjh0 , length ~ = 5..j5..0 and period i = f /fa, presented as functions of the relative depth Hj5..0 , practically coincide with accurate spectral
solutions (see Fig. 6.10, Sect. 6.3). In particular for a straight isobaths, the
relative height hjh0 is decreased at first to the value ~ 0.9, and then it is
increased. The mean wave period is weakly decreased (up to 3 per cent), then
slowly increased. The wavelengths are monotonically decreased.