294
6 Wave Transformation in Shallow Water
obtain such a solution in the general case with the water depth and current
velocity being changed ad arbitrium. However, a correct estimation of wind
wave transformation in a certain natural basin is of the most practical interest. Estimation difficulties are connected with the problem formulation as it
is and its numerical implementation, including the availability of necessary
information about wind speed, current fields, morphometry, etc.
There have been numerous attempts to solve this problem. Some of them
have already been mentioned in the review of this monograph. As a rule, the
problem results in determining the wave elements for a certain coastal area, if
the initial wave values are known at open sea in deep water. The traditional
method of solving the problem according to GOST (the Russian State Building Standards and Rules, 1983) is based on defining the energy flow along
a ray tube originated from the initial boundary. However this can lead to
caustics and singularities in the problem solution. The method of developing
the characteristics for the spectral wave energy balance equation without resulting in such peculiarities is given in this monograph. This method obtains
rather an accurate solution, producing fine details of wave refraction, because it is not connected with numerical diffusion present in most numerical
schemes.
The problem is solved using the balance equation of the wave action
spectral density, written in the general form (1.84)-(1.89). One can pass from
the spectral density of the wave action N to the spectral density of the wave
energy S = N a, which is a function of the wave number k = I k I and the
angle (3. As for the source function G, the non-linear wave interaction in the
spectrum can be neglected in the case of wave transformation in a limited
local coastal area, as shown in Sect. 6.2. Taking into account the known
mechanism of wave energy dissipation Gds, connected with bottom friction,
the balance equation of the energy spectral density can be presented in the
following form:
dS = [~ j -1 (8H di? 8H dcp) j + 2 dkl S _ G
dt
2 H
81? dt + 8cp dt + k dt
ds '
(6.59)
where j = 1 + 2kH/sh(2kH).
Algorithm for problem solution. In order to obtain the energy wave
spectrum and its mean elements (heights, lengths, periods, etc.), it is necessary to integrate the set of equations (6.59) with the characteristics (1.86)(1.89). This integration should be fulfilled for a discrete set of wave numbers ki and directions (3j (see Fig. 6.16). However, the initial ray points
{ cpij, i?ij, ki, f3j} of phase space, making a contribution to the estimated spectrum and serving as initial values for the ray integration, are not known beforehand. That is why there are two stages to the numerical implementation
of the problem. In the first stage the "reverse problem" will be solved, i.e.
proceeding from the estimated point { cp 0 , 1? 0 , ki, (3j}, the rays and the corre-
6 Wave Transformation in Shallow Water
obtain such a solution in the general case with the water depth and current
velocity being changed ad arbitrium. However, a correct estimation of wind
wave transformation in a certain natural basin is of the most practical interest. Estimation difficulties are connected with the problem formulation as it
is and its numerical implementation, including the availability of necessary
information about wind speed, current fields, morphometry, etc.
There have been numerous attempts to solve this problem. Some of them
have already been mentioned in the review of this monograph. As a rule, the
problem results in determining the wave elements for a certain coastal area, if
the initial wave values are known at open sea in deep water. The traditional
method of solving the problem according to GOST (the Russian State Building Standards and Rules, 1983) is based on defining the energy flow along
a ray tube originated from the initial boundary. However this can lead to
caustics and singularities in the problem solution. The method of developing
the characteristics for the spectral wave energy balance equation without resulting in such peculiarities is given in this monograph. This method obtains
rather an accurate solution, producing fine details of wave refraction, because it is not connected with numerical diffusion present in most numerical
schemes.
The problem is solved using the balance equation of the wave action
spectral density, written in the general form (1.84)-(1.89). One can pass from
the spectral density of the wave action N to the spectral density of the wave
energy S = N a, which is a function of the wave number k = I k I and the
angle (3. As for the source function G, the non-linear wave interaction in the
spectrum can be neglected in the case of wave transformation in a limited
local coastal area, as shown in Sect. 6.2. Taking into account the known
mechanism of wave energy dissipation Gds, connected with bottom friction,
the balance equation of the energy spectral density can be presented in the
following form:
dS = [~ j -1 (8H di? 8H dcp) j + 2 dkl S _ G
dt
2 H
81? dt + 8cp dt + k dt
ds '
(6.59)
where j = 1 + 2kH/sh(2kH).
Algorithm for problem solution. In order to obtain the energy wave
spectrum and its mean elements (heights, lengths, periods, etc.), it is necessary to integrate the set of equations (6.59) with the characteristics (1.86)(1.89). This integration should be fulfilled for a discrete set of wave numbers ki and directions (3j (see Fig. 6.16). However, the initial ray points
{ cpij, i?ij, ki, f3j} of phase space, making a contribution to the estimated spectrum and serving as initial values for the ray integration, are not known beforehand. That is why there are two stages to the numerical implementation
of the problem. In the first stage the "reverse problem" will be solved, i.e.
proceeding from the estimated point { cp 0 , 1? 0 , ki, (3j}, the rays and the corre-