1.3 Wave Action Conservation Law
21
relation u 2 = gk th(kH) can be easily obtained. The adiabatic invariant
transfer rate C g is equal to:
au
1 k (g th(kH)) 112 ( 2kH )
Cg = Vo + ak = Vo + 2k
k
1 + sh(2kH)
Using the expressions (1.41 )-(1.44) it follows that:
A-E
-
'
(j
where E is the wave energy density.
(1.44)
(1.45)
The expression (1.45) is widely known in science the wave action density. The law of wave action density preservation (1.41) with (1.44) is the
simplest and the most common expression in wave dynamics. It was established on the basis of the variation principle by B. Whitham (1965) and developed by F. Bretherton and C. Garret (1968) and F. Bretherton (1971), and
A. Voronovich and V. Goncharov (1982). It should be noted that the equation
of adiabatic invariant preservation law (1.41)-(1.43) represents a more general law as compared to the principle of wave action conservation, because it
takes into account the vertical non-uniformity of the mean current velocity.
The equation (1.41) indicates that the local velocity of wave action change
is balanced by the action flow divergence transferred with the group velocity C g relative to moving medium. If the mean velocity V does not remain
constant, then the wave vector k and the frequency u can be varied in space
and time according to (1.24). In this case the wave energy density is not conserved with preservation of the wave action A. There is some energy exchange
between the waves and the current.
An important consequence of solving the problem is that the characteristics of (1.41) coincide with (1.24), being in turn characteristics of the phase
equation (1.23).
Now the problem with initial conditions should be considered. It is necessary to determine the initial surface S, where the starting values are prescribed. The equations of the surface S are written in the parametric form
r = r0 (~, (), where ~ and ( are curvilinear coordinates in the surface S. Let
the wave field rl(~, () be determined by the initial value of the wave phase
'lj!ls = '1/:_ 0 (~, () and let the amplitude al.s = a 0 (~, () be prescribed in the
surface S at r = 0 ( r is a parameter varying along a ray, for example, time,
i.e. r = t). If a wave propagates along a ray, then the existing ray point
r( r0 ) = r0 (~, () on the surfaceS is a natural initial condition for the ray trajectory r = r(r). The solutions of differential equations for rays (1.24) corresponding to the initial conditions can be given in the form of r = r( ~, (, T),
k = k(~, (, r). In this case the parameters~ and (indicate rays leaving the
surface S, with the parameter T pointing at a fixed ray position. The value
combination ~' (, r is called the ray coordinates. In the general case these
coordinates are not orthogonal.
21
relation u 2 = gk th(kH) can be easily obtained. The adiabatic invariant
transfer rate C g is equal to:
au
1 k (g th(kH)) 112 ( 2kH )
Cg = Vo + ak = Vo + 2k
k
1 + sh(2kH)
Using the expressions (1.41 )-(1.44) it follows that:
A-E
-
'
(j
where E is the wave energy density.
(1.44)
(1.45)
The expression (1.45) is widely known in science the wave action density. The law of wave action density preservation (1.41) with (1.44) is the
simplest and the most common expression in wave dynamics. It was established on the basis of the variation principle by B. Whitham (1965) and developed by F. Bretherton and C. Garret (1968) and F. Bretherton (1971), and
A. Voronovich and V. Goncharov (1982). It should be noted that the equation
of adiabatic invariant preservation law (1.41)-(1.43) represents a more general law as compared to the principle of wave action conservation, because it
takes into account the vertical non-uniformity of the mean current velocity.
The equation (1.41) indicates that the local velocity of wave action change
is balanced by the action flow divergence transferred with the group velocity C g relative to moving medium. If the mean velocity V does not remain
constant, then the wave vector k and the frequency u can be varied in space
and time according to (1.24). In this case the wave energy density is not conserved with preservation of the wave action A. There is some energy exchange
between the waves and the current.
An important consequence of solving the problem is that the characteristics of (1.41) coincide with (1.24), being in turn characteristics of the phase
equation (1.23).
Now the problem with initial conditions should be considered. It is necessary to determine the initial surface S, where the starting values are prescribed. The equations of the surface S are written in the parametric form
r = r0 (~, (), where ~ and ( are curvilinear coordinates in the surface S. Let
the wave field rl(~, () be determined by the initial value of the wave phase
'lj!ls = '1/:_ 0 (~, () and let the amplitude al.s = a 0 (~, () be prescribed in the
surface S at r = 0 ( r is a parameter varying along a ray, for example, time,
i.e. r = t). If a wave propagates along a ray, then the existing ray point
r( r0 ) = r0 (~, () on the surfaceS is a natural initial condition for the ray trajectory r = r(r). The solutions of differential equations for rays (1.24) corresponding to the initial conditions can be given in the form of r = r( ~, (, T),
k = k(~, (, r). In this case the parameters~ and (indicate rays leaving the
surface S, with the parameter T pointing at a fixed ray position. The value
combination ~' (, r is called the ray coordinates. In the general case these
coordinates are not orthogonal.