Mathematical Analysis
Liu and Mollo-Christensen (1988)
developed an analytical approach to the problem of wave propagation under an
ice pack. They began with the usual
expected solution of the form:
n(x,t) =
a sin (kx - wt)
wherein n = vertical
displacement of the ice-water interface, a = amplitude of the
wave, k = wave number, x = distance covered by the wave, w =
frequency of the wave and t = time. They
considered the ice to be a thin and plastic surface of thickness = h on the
sea, using the work of previous investigators to quantify the dispersion
relation of waves moving under the ice pack. The resulting dispersion relation was:
w2 (t)
= ( gk + Bk5 - Qk3) / ( 1 +
kM )
when k5 = the 5th power of k, B
= the effect on frequency of bending, Q = the effect
of compression and M = the effect of the inertia of the ice. These terms are, in turn, expressed:
B = Eh3
/ 12 (1 - s2) rw,
Q
= Ph / rw,
M = ri
h / rw
with E = Young's
modulus for ice elasticity, ~ 6 x 10e9 , rw = density of water
= 1025 kg/m3, ri = density of ice = .9 rw and
10e9 = 10 to the 9th power. The
important aspect of their findings is that frequency is proportional to several
powers of h (ice thickness.) Thus, as
ice thickens, the higher the frequency of the waves beneath. Given a constant frequency, wave length would
increase in open water, all other conditions being roughly equal.
Liu and Mollo-Christensen's
analysis suggested that long waves (with a wavelength of 500 m or more) didn't
"feel" the ice pack on the surface; rather, the ice floats as though it were a
field of small, thin rigid plates. Shorter
wave length seas cause the ice to flex, absorbing energy from the sea and
swell. In the case of low amplitude,
long wavelength seas, the pack ice can be regarded as a thin film with very
limited strength, afloat on and at the mercy of a restless ocean.
G. de Q. Robin (1963)
investigated movement of waves under pack ice in the