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7 Wave Transformation in Ice-Covered Water
The movement equations for wave packets can be written in the following
form:
dx
dt = C 9 x = C 9 cos((J) ;
dy
.
dt = C9y = C9 sm((J) ;
(7.17)
dkx _ 1 ( w )
2
a L .
----w -
-
dt
2
Wb
ax
1
dky
dt =0.
(7.18)
An attempt will be undertaken to find movement integrals for the equation
set (7.17), (7.18). It can be seen that the coordinate y is cyclic and the
component ky of the wave vector k remains constant along the trajectory of
wave packet propagation. So it can be written as:
ky = k sin((J) = k0 sin(f3o) .
(7.19)
Preservation of the frequency w is the second integral of movement. It can
be presented as follows:
The ratios (7.19) and (7.20) turn out to be sufficient to determine the
wave number k and the angle (3 along the trajectory depending on variations
of the ice thickness L( x). If the ice thickness is monotonically increased along
the positive direction of the Ox axis from zero to some definite values, the
wave number is increased. According to (7.19) the wave angle propagation (3
is decreased, i.e. it is turned to the positive direction of the Ox axis. The
waves are directed along the Ox axis accurately at the point where the wave
frequency coincides with the float oscillation frequency.
Similar wave behaviour is observed in shallow water, with waves propagating to the shore. However, there are some differences. In shallow water, the
shore line (i.e. depth equal to zero) is a special point for every wave, where
the group velocity is equal to zero. As waves propagate in this situation, the
eigenfrequency of ice oscillation is varied along the Ox axis, i.e., the position
of the special point is different for different waves.
Using (7.14), the solution of frequency angular spectrum evolution along
the wave propagation trajectory can be written as:
kC 0
S(w, (3, x) = koJg So(w, f3o, xo) exp [E(w, (30 , x)]
5
= (1- :;) -
2 So(w,(Jo,xo) exp[E(w,fJo,x)],
(7.21)
where E(w,(30 ,x) is the integral value of the dissipative function along the
characteristic in (7.16).
7 Wave Transformation in Ice-Covered Water
The movement equations for wave packets can be written in the following
form:
dx
dt = C 9 x = C 9 cos((J) ;
dy
.
dt = C9y = C9 sm((J) ;
(7.17)
dkx _ 1 ( w )
2
a L .
----w -
-
dt
2
Wb
ax
1
dky
dt =0.
(7.18)
An attempt will be undertaken to find movement integrals for the equation
set (7.17), (7.18). It can be seen that the coordinate y is cyclic and the
component ky of the wave vector k remains constant along the trajectory of
wave packet propagation. So it can be written as:
ky = k sin((J) = k0 sin(f3o) .
(7.19)
Preservation of the frequency w is the second integral of movement. It can
be presented as follows:
The ratios (7.19) and (7.20) turn out to be sufficient to determine the
wave number k and the angle (3 along the trajectory depending on variations
of the ice thickness L( x). If the ice thickness is monotonically increased along
the positive direction of the Ox axis from zero to some definite values, the
wave number is increased. According to (7.19) the wave angle propagation (3
is decreased, i.e. it is turned to the positive direction of the Ox axis. The
waves are directed along the Ox axis accurately at the point where the wave
frequency coincides with the float oscillation frequency.
Similar wave behaviour is observed in shallow water, with waves propagating to the shore. However, there are some differences. In shallow water, the
shore line (i.e. depth equal to zero) is a special point for every wave, where
the group velocity is equal to zero. As waves propagate in this situation, the
eigenfrequency of ice oscillation is varied along the Ox axis, i.e., the position
of the special point is different for different waves.
Using (7.14), the solution of frequency angular spectrum evolution along
the wave propagation trajectory can be written as:
kC 0
S(w, (3, x) = koJg So(w, f3o, xo) exp [E(w, (30 , x)]
5
= (1- :;) -
2 So(w,(Jo,xo) exp[E(w,fJo,x)],
(7.21)
where E(w,(30 ,x) is the integral value of the dissipative function along the
characteristic in (7.16).