318
7 Wave Transformation in Ice-Covered Water
cakes, will be considered. There is no general theory concerning this mechanism and it is researched mainly experimentally (Wadhams et al. 1986). For
the first time the simplest approach towards the solution of the problem was
suggested by Krylov (1948). He considers the case of a viscous layer simulating ice cakes in an ideal liquid surface. The following expression connecting
the frequency and the wave number is obtained as:
. L
2
w2
lWl/ k2 + 1 - ~ -
= 0 '
P2Y
w~
gk th[k(H- A)]
(7.10)
where v is the viscosity coefficient.
It is seen that (7.10) is a transcedental equation with an imaginary value.
Contrary to the non-damping oscillations considered above (7.4), this equation provides a finite solution of the wave number k, even for the case w = Wb·
Its solution consists of imaginary and real parts. The imaginary part of the
wave number k results in a damping coefficient. The value of this coefficient is
determined simply enough for every separate case of deep and shallow water.
So, for shallow water, a biquadratic equation can be obtained to determine
the wave number value. For the deep-water case, a cubic equation can be
obtained as well.
Supposing that the damping coefficient is small (p, « 1.0) for the low
frequencies ( w « Wb), the wave number can be written in the form k =
k0 (1 + iJ.l), where the value k0 is determined by (7.4) or (7.5). Neglecting
small values of second order, an approximate expression for the damping
coefficient is obtained. It can be written for deep water as follows:
vL
w 5
J.l = P2Y3 (1- w2 jw~)3 .
(7.11)
Similarly, for shallow water it can be presented as:
vL
w 5
J.l - - - -;-:---;:-;---~
- P2Hg2 (1-w2jw~)2.
(7.12)
It follows from these formulas that the wave damping becomes larger with
increase of ice cake thickness or wave frequency. Thus, if a wave perturbation
exists in a water area covered with ice cakes, the short-wave components,
corresponding to high frequencies, are observed only in the vicinity of the
perturbation area and they are damped outside it. These components are
also quickly damped in time after the perturbation is over. The undamped
long-wave components, moving with large velocity, propagate far away from
the perturbation area. The wave energy dissipation is infinitely increased,
as the wave frequency w approaches the ice oscillation frequency Wb· This
reveals some solution trends and shows the limitation of the applicability of
the approximation.
When applying the geometrical optics approximation to solve the problem
of spectrum evolution in a water area with spatially non-uniform ice cake
7 Wave Transformation in Ice-Covered Water
cakes, will be considered. There is no general theory concerning this mechanism and it is researched mainly experimentally (Wadhams et al. 1986). For
the first time the simplest approach towards the solution of the problem was
suggested by Krylov (1948). He considers the case of a viscous layer simulating ice cakes in an ideal liquid surface. The following expression connecting
the frequency and the wave number is obtained as:
. L
2
w2
lWl/ k2 + 1 - ~ -
= 0 '
P2Y
w~
gk th[k(H- A)]
(7.10)
where v is the viscosity coefficient.
It is seen that (7.10) is a transcedental equation with an imaginary value.
Contrary to the non-damping oscillations considered above (7.4), this equation provides a finite solution of the wave number k, even for the case w = Wb·
Its solution consists of imaginary and real parts. The imaginary part of the
wave number k results in a damping coefficient. The value of this coefficient is
determined simply enough for every separate case of deep and shallow water.
So, for shallow water, a biquadratic equation can be obtained to determine
the wave number value. For the deep-water case, a cubic equation can be
obtained as well.
Supposing that the damping coefficient is small (p, « 1.0) for the low
frequencies ( w « Wb), the wave number can be written in the form k =
k0 (1 + iJ.l), where the value k0 is determined by (7.4) or (7.5). Neglecting
small values of second order, an approximate expression for the damping
coefficient is obtained. It can be written for deep water as follows:
vL
w 5
J.l = P2Y3 (1- w2 jw~)3 .
(7.11)
Similarly, for shallow water it can be presented as:
vL
w 5
J.l - - - -;-:---;:-;---~
- P2Hg2 (1-w2jw~)2.
(7.12)
It follows from these formulas that the wave damping becomes larger with
increase of ice cake thickness or wave frequency. Thus, if a wave perturbation
exists in a water area covered with ice cakes, the short-wave components,
corresponding to high frequencies, are observed only in the vicinity of the
perturbation area and they are damped outside it. These components are
also quickly damped in time after the perturbation is over. The undamped
long-wave components, moving with large velocity, propagate far away from
the perturbation area. The wave energy dissipation is infinitely increased,
as the wave frequency w approaches the ice oscillation frequency Wb· This
reveals some solution trends and shows the limitation of the applicability of
the approximation.
When applying the geometrical optics approximation to solve the problem
of spectrum evolution in a water area with spatially non-uniform ice cake