6.3 Bottom and a Non-Uniform Current Influence on Waves
277
If waves propagate in a countercurrent, an accurate determination of the
value of the parameters 0: and v makes some additional difficulties, because
the spectrum is different from (5.16) at the initial point due to the influence
of the current. It cuts off a high-frequency spectrum range, where components cannot propagate countercurrent. The expression (6.38) can be used
for estimation in the cases of no reverse waves, for example, as in a current not increasing along the Ox axis. If there are reverse waves (at the
current increasing along the Ox axis), then their spectral density can be
approximated by the equilibrium range due to wave breaking. In order to
estimate the values 0: and v, the parameter n = 4 will be taken instead of
n = 5.5. After that the values 0: = 3.17H0 j.\0 ; v ~ 1.78V0 /V'9To can be
estimated.
Results of mean wave element calculations.
Numerical calculations
of the obtained solution will be made. Firstly, the calculation results of the
expressions fr, h, h are presented for the cases when their values can be
compared with the known measurement data. A transformation of waves,
propagating without any current from deep water to the shallow coastal area,
is referred to such cases. The current velocity is assumed to be very small
at the initial boundary (v = 10- 5 ) and the non-dimensional initial depth is
equal to 0: = 10 (i.e. H 0 j.\0 ~ 2.5), corresponding to the deep-water case.
Variations of mean wave elements, with waves approaching a coast, are
shown in Fig. 6.10. A comparison of calculations and observations (Krylov
et al., 1976) is also presented. As the waves propagate, their heights are
smoothly decreased to h ~ 0.91 and then begin increasing. The wavelength
decreases and the mean period is changed. The wave height change coincides accurately enough with the known results of wave transformation in
a coastal area. It should be noted that the evolution of the mean wave elements calculated using the spectral solution is quantitatively different from
the monochromatic wave solution presented in Fig. 6.3 for different angles
of waves approaching a shore. The spectral approach slightly smoothes the
solution, averaging it over angle directions. The quantitative difference of two
solutions is determined by the angular distribution width for the spectrum
of waves directed to the shore.
The second control case of testing the general solution is wave evolution
in deep water for their propagation from a small current area against the
current increasing along its direction. This case has already been considered in
Sect. 5.5. Calculations of wave element transformation, made in this section,
coincide practically with those obtained earlier (see Fig. 5.15).
Not only quantitative, but also qualitative agreement of calculations and
observations given above indicates that the proposed mathematical model
could describe the wave parameter transformation both in the presence of
a horizontal non-uniform current and under basin depth variation. Now it
would be interesting to investigate the problem of the depth and current
velocity changing simultaneously.
277
If waves propagate in a countercurrent, an accurate determination of the
value of the parameters 0: and v makes some additional difficulties, because
the spectrum is different from (5.16) at the initial point due to the influence
of the current. It cuts off a high-frequency spectrum range, where components cannot propagate countercurrent. The expression (6.38) can be used
for estimation in the cases of no reverse waves, for example, as in a current not increasing along the Ox axis. If there are reverse waves (at the
current increasing along the Ox axis), then their spectral density can be
approximated by the equilibrium range due to wave breaking. In order to
estimate the values 0: and v, the parameter n = 4 will be taken instead of
n = 5.5. After that the values 0: = 3.17H0 j.\0 ; v ~ 1.78V0 /V'9To can be
estimated.
Results of mean wave element calculations.
Numerical calculations
of the obtained solution will be made. Firstly, the calculation results of the
expressions fr, h, h are presented for the cases when their values can be
compared with the known measurement data. A transformation of waves,
propagating without any current from deep water to the shallow coastal area,
is referred to such cases. The current velocity is assumed to be very small
at the initial boundary (v = 10- 5 ) and the non-dimensional initial depth is
equal to 0: = 10 (i.e. H 0 j.\0 ~ 2.5), corresponding to the deep-water case.
Variations of mean wave elements, with waves approaching a coast, are
shown in Fig. 6.10. A comparison of calculations and observations (Krylov
et al., 1976) is also presented. As the waves propagate, their heights are
smoothly decreased to h ~ 0.91 and then begin increasing. The wavelength
decreases and the mean period is changed. The wave height change coincides accurately enough with the known results of wave transformation in
a coastal area. It should be noted that the evolution of the mean wave elements calculated using the spectral solution is quantitatively different from
the monochromatic wave solution presented in Fig. 6.3 for different angles
of waves approaching a shore. The spectral approach slightly smoothes the
solution, averaging it over angle directions. The quantitative difference of two
solutions is determined by the angular distribution width for the spectrum
of waves directed to the shore.
The second control case of testing the general solution is wave evolution
in deep water for their propagation from a small current area against the
current increasing along its direction. This case has already been considered in
Sect. 5.5. Calculations of wave element transformation, made in this section,
coincide practically with those obtained earlier (see Fig. 5.15).
Not only quantitative, but also qualitative agreement of calculations and
observations given above indicates that the proposed mathematical model
could describe the wave parameter transformation both in the presence of
a horizontal non-uniform current and under basin depth variation. Now it
would be interesting to investigate the problem of the depth and current
velocity changing simultaneously.