154
5 Wave Evolution in Non-uniform Currents in Deep Water
dr
aw dk
dt
8k '
dt
aw dw aw
- ar ' dt at '
(5.2)
where w is the wave frequency measured in an immovable coordinate system.
The frequency w can be different from the wave frequency a measured
in a moving coordinate system connected with the current a 2 = (w- kV) 2 ,
where V = V(r, t) is the current velocity. For surface gravity waves, the
dispersion relation can be written as a 2 = gkth(kH), where H = H(r) is the
water depth. These equations are valid for the case of the current velocity,
which does not vary vertically. The cases of vertical non-uniform current
influence on waves are considered in Sect. 5.9.
In order to solve the problem of determining the wave action spectral
density N(k, r, t), it is necessary to define the current velocities V(r, t) and
the water depth H(r) and solve (5.2) and (5.1).
It should be noted that the solution is delivered in the phase space
{ k, r, t}. 1 Only one phase trajectory can pass through each space point, i.e.
the phase trajectories do not intersect. In fact, this quality is a consequence
of the uniqueness theorem for a set of ordinary differential equations with
given initial conditions. There are some remarkable properties of propagating
wave packet trajectories (Maslov & Fedoryuk, 1976; Vaynberg, 1982; Arnold,
1989).
It follows from (5.1) that when the source function is equal to zero (at
G = 0): dN(x, y, kx, ky, t)/ dt = 0. This means that the spectral density of
the energy wave action is preserved along the ray:
(5.3)
The Jacobian of the transition from the initial to current values is absent
in this case, unlike the wave description in physical space (see Sect. 1.3). This
can be explained with the help of the wave description in phase space, dealing
with canonical variables. The preservation of the wave action value in unit
phase volume can be written as N tlxtlyllkxllky = N 0 tlxotly0 llkxollkyo·
When the system motion in phase space is described by the Hamiltonian equations, the preservation of the phase space element volume follows from the Liouville theorem (Landau & Lifshits, 1973). The Jacobian
of the transition from the initial phase volume element to the current one
is 8(xo, yo, kxo, kyo)/8(x, y, kx, ky) = 1. Thus, the relation (5.3) is proved.
There are no caustic peculiarities associated with the vanishing Jacobian, as
shown in the case of wave behaviour in physical space (see Sect. 1.3).
It follows from the condition (5.3) that the preservation of the wave action spectral density occurs along the trajectory of propagating wave packets
without any sources or wave dissipation effects. It should be noted that this
1 This space is also called the coordinate-impulse or simplex, and some sections
of modern geometry (Arnold, 1989) are devoted to research of the behaviour of
Hamiltonian systems in such space.
5 Wave Evolution in Non-uniform Currents in Deep Water
dr
aw dk
dt
8k '
dt
aw dw aw
- ar ' dt at '
(5.2)
where w is the wave frequency measured in an immovable coordinate system.
The frequency w can be different from the wave frequency a measured
in a moving coordinate system connected with the current a 2 = (w- kV) 2 ,
where V = V(r, t) is the current velocity. For surface gravity waves, the
dispersion relation can be written as a 2 = gkth(kH), where H = H(r) is the
water depth. These equations are valid for the case of the current velocity,
which does not vary vertically. The cases of vertical non-uniform current
influence on waves are considered in Sect. 5.9.
In order to solve the problem of determining the wave action spectral
density N(k, r, t), it is necessary to define the current velocities V(r, t) and
the water depth H(r) and solve (5.2) and (5.1).
It should be noted that the solution is delivered in the phase space
{ k, r, t}. 1 Only one phase trajectory can pass through each space point, i.e.
the phase trajectories do not intersect. In fact, this quality is a consequence
of the uniqueness theorem for a set of ordinary differential equations with
given initial conditions. There are some remarkable properties of propagating
wave packet trajectories (Maslov & Fedoryuk, 1976; Vaynberg, 1982; Arnold,
1989).
It follows from (5.1) that when the source function is equal to zero (at
G = 0): dN(x, y, kx, ky, t)/ dt = 0. This means that the spectral density of
the energy wave action is preserved along the ray:
(5.3)
The Jacobian of the transition from the initial to current values is absent
in this case, unlike the wave description in physical space (see Sect. 1.3). This
can be explained with the help of the wave description in phase space, dealing
with canonical variables. The preservation of the wave action value in unit
phase volume can be written as N tlxtlyllkxllky = N 0 tlxotly0 llkxollkyo·
When the system motion in phase space is described by the Hamiltonian equations, the preservation of the phase space element volume follows from the Liouville theorem (Landau & Lifshits, 1973). The Jacobian
of the transition from the initial phase volume element to the current one
is 8(xo, yo, kxo, kyo)/8(x, y, kx, ky) = 1. Thus, the relation (5.3) is proved.
There are no caustic peculiarities associated with the vanishing Jacobian, as
shown in the case of wave behaviour in physical space (see Sect. 1.3).
It follows from the condition (5.3) that the preservation of the wave action spectral density occurs along the trajectory of propagating wave packets
without any sources or wave dissipation effects. It should be noted that this
1 This space is also called the coordinate-impulse or simplex, and some sections
of modern geometry (Arnold, 1989) are devoted to research of the behaviour of
Hamiltonian systems in such space.