6.3 Bottom and a Non-Uniform Current Influence on Waves
269
z
Fig. 6. 7. Scheme of wave transformation due to the joint effect of uneven bottom
and horizontal non-uniform current: H(x) is depth; V(x) is current velocity
where N0 and N 00 are the initial values of the wave action density and the
equilibrium range correspondingly, given at the initial time to. The arguments
N0 and N 00 depend on the wave vector k considered at time t and point r.
The latter dependencies are derived using the set of the equations (5.20)(5.22).
Using the spectral density of the wave action N, the energy spectral density F, depending on the wave number I k I and the angle between its components f3 =arctan (ky/kx), can easily be found:
F (k, (3, r, t) =min { a:~o Fo (ko, f3o, ro, t) ; F 00 (k, (3)} , (6.27b)
where the least value of the two functions in braces is assigned to the function F; F0 is the initial spectral energy density; F 00 is the spectral density of
the equilibrium range; and k0 , (30 , r 0 are derived with the help of the equation
set (5.20)-(5.22). Using the dispersion ratio a 2 = gkth (kH), the expression
I
.
k 3 th(kH)
ak a0 k0 can be wntten as:
k& th(koHo) ·
The spectral energy density, prescribed for the initial boundary at x = 0,
is assumed to be the uniform and stationary function: F0 (k0 , (30 ), determined
using the spectrum (5.16).
Wave kinematic investigation in finite depth water. In order to determine the unknown values k0 , (30 , r 0 , t0 in (6.27), it is not necessary to solve
directly the set the equations (5.20)-(5.22) numerically, but an attempt is undertaken instead to derive their solution analytically. The motion integrals of
wave packet propagation, following from this set, will be used. They are: the
preservation of the wave vector component ky and the frequency w, as soon
269
z
Fig. 6. 7. Scheme of wave transformation due to the joint effect of uneven bottom
and horizontal non-uniform current: H(x) is depth; V(x) is current velocity
where N0 and N 00 are the initial values of the wave action density and the
equilibrium range correspondingly, given at the initial time to. The arguments
N0 and N 00 depend on the wave vector k considered at time t and point r.
The latter dependencies are derived using the set of the equations (5.20)(5.22).
Using the spectral density of the wave action N, the energy spectral density F, depending on the wave number I k I and the angle between its components f3 =arctan (ky/kx), can easily be found:
F (k, (3, r, t) =min { a:~o Fo (ko, f3o, ro, t) ; F 00 (k, (3)} , (6.27b)
where the least value of the two functions in braces is assigned to the function F; F0 is the initial spectral energy density; F 00 is the spectral density of
the equilibrium range; and k0 , (30 , r 0 are derived with the help of the equation
set (5.20)-(5.22). Using the dispersion ratio a 2 = gkth (kH), the expression
I
.
k 3 th(kH)
ak a0 k0 can be wntten as:
k& th(koHo) ·
The spectral energy density, prescribed for the initial boundary at x = 0,
is assumed to be the uniform and stationary function: F0 (k0 , (30 ), determined
using the spectrum (5.16).
Wave kinematic investigation in finite depth water. In order to determine the unknown values k0 , (30 , r 0 , t0 in (6.27), it is not necessary to solve
directly the set the equations (5.20)-(5.22) numerically, but an attempt is undertaken instead to derive their solution analytically. The motion integrals of
wave packet propagation, following from this set, will be used. They are: the
preservation of the wave vector component ky and the frequency w, as soon