18
1 General Problem Formulation
where the Hamiltonian function F 1 is connected with the function F (1.22)
by the ratio:
The group velocity Cg in the moving coordinate system is expressed by
the stationary coordinate system with ratio cg = C g - V.
Thus, it is sufficient to use the aforesaid formulae, to pass from the moving
coordinate system to the stationary one and vice versa.
1.3 Wave Action Conservation Law
The kinematic equations (derived in the previous section on the basis of the
geometric optics method) determine a non-vortical field of wave vector k in
space and time. In order to obtain the distribution of the dynamic wave characteristics, such as the energy density, it is necessary to get information about
the wave dynamics and wave interactions with the medium. If the wavelength
and period are assumed to be small in comparison with the medium parameter variations, the amplitude evolution of gravity wave propagation over the
oceanic surface in the presence of spatial-non-uniform currents and an uneven
bottom can be considered using the geometric optics approximation.
The ocean is assumed to be an incompressible heavy homogeneous fluid.
Its hydrodynamic equations are written in the form (1.1)-(1.3), whereas the
Coriolis force for wind waves is not essential. The velocity vector U is assumed
to consist of the horizontal V and the vertical W components.
The boundary conditions at the free surface at z = ry(r, t) can be written
in the form:
P=Pa =0;
8ry
W = 8t + (VV')ry ,
(1.28)
where Pa is the atmospheric pressure.
At z = H(r, t) the bottom condition is written as:
W + (VV')H = 0 .
(1.29)
The small parameter c introduced above is assumed to characterize the
slowness of the main motion change along the horizontal coordinates and in
time. No such slowness of variation along the vertical coordinate is assumed.
All hydrodynamic fields are presented in the equations in the following form:
where
1 General Problem Formulation
where the Hamiltonian function F 1 is connected with the function F (1.22)
by the ratio:
The group velocity Cg in the moving coordinate system is expressed by
the stationary coordinate system with ratio cg = C g - V.
Thus, it is sufficient to use the aforesaid formulae, to pass from the moving
coordinate system to the stationary one and vice versa.
1.3 Wave Action Conservation Law
The kinematic equations (derived in the previous section on the basis of the
geometric optics method) determine a non-vortical field of wave vector k in
space and time. In order to obtain the distribution of the dynamic wave characteristics, such as the energy density, it is necessary to get information about
the wave dynamics and wave interactions with the medium. If the wavelength
and period are assumed to be small in comparison with the medium parameter variations, the amplitude evolution of gravity wave propagation over the
oceanic surface in the presence of spatial-non-uniform currents and an uneven
bottom can be considered using the geometric optics approximation.
The ocean is assumed to be an incompressible heavy homogeneous fluid.
Its hydrodynamic equations are written in the form (1.1)-(1.3), whereas the
Coriolis force for wind waves is not essential. The velocity vector U is assumed
to consist of the horizontal V and the vertical W components.
The boundary conditions at the free surface at z = ry(r, t) can be written
in the form:
P=Pa =0;
8ry
W = 8t + (VV')ry ,
(1.28)
where Pa is the atmospheric pressure.
At z = H(r, t) the bottom condition is written as:
W + (VV')H = 0 .
(1.29)
The small parameter c introduced above is assumed to characterize the
slowness of the main motion change along the horizontal coordinates and in
time. No such slowness of variation along the vertical coordinate is assumed.
All hydrodynamic fields are presented in the equations in the following form:
where