Ice as a Thin Elastic Plate

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An extensive amount of literature exists to support the modelling of an ice-sheet as a Floating Elastic Plate, (also cf. Mindlin 1951; Fung 1965). Greenhill 1887 was the first to propose modelling a floating ice sheet by a thin elastic beam on a fluid foundation, suggesting a dispersion relation based on the Euler-Bernoulli beam theory. He also wrote a further paper, humourously entitled "Skating on thin ice" Greenhill 1916, in which he elaborated further on the dispersion of waves in ice, as well as touching on other related problems. (Another noteworthy comment that was made in that paper is that ice was the first material for which an experimental value of Young's modulus was obtained.)

This elastic model for ice has since been corroborated experimentally by numerous researchers, with slight improvements to Greenhill's theory being made with the passage of time. For example, during a series of seismic experiments carried out by Ewing et al. 1934, the generation of flexural waves in thin lake/canal ice was observed in addition to the longitudinal and transverse waves being investigated at the time. Extending the theoretical component of Greenhill's work to include the compressibility of the water beneath the ice sheet and to model the ice as a thin plate rather than as a beam (i.e. as a three-dimensional structure rather than as a two-dimensional one), the measured group velocities of the flexural waves corresponded quite well to those predicted by their theory (Ewing and Crary 1934).

A more extensive set of seismic experiments on lake ice to create ice-coupled waves artificially was done by Press et al. 1951, who also observed air-coupled flexural waves travelling at the speed of sound (discussed in further detail by Press and Ewing 1951). Again, the measured flexural wave dispersion agreed quite well with the thin plate dispersion theory.

Press and Ewing 1951b developed the elastic model of ice further by allowing for the horizontal and vertical displacements to vary arbitrarily in the vertical direction, although they neglect the effect of gravity. In the thin plate model, the horizontal displacements are neglected, while the vertical displacements are taken to be linear in the vertical coordinate. The latter approximations are valid when the thickness of the ice is small in comparison to the wavelength. Indeed, in the large wavelength limit, the dispersion relation of Press and Ewing 1951b explicitly converges to the thin plate relation (allowing for the fact that the gravity term also becomes negligible for large wavelengths).

In the short wave limit, where gravity does have more of an effect, their dispersion relation reproduces the relation for an infinitely thick ice sheet with unattenuated Rayleigh waves travelling along the ice-air surface and with attenuated Rayleigh waves travelling along the ice-water interface. The disadvantage of the theory of Press and Ewing 1951b is that they were only able to solve their dispersion relation in the above long and short wave limits.

Oliver et al. 1954 did a similar seismic investigation to Press et al. 1951 in the Beaufort Sea and the Arctic Ocean amongst Arctic sea ice, and applied the theory of Press and Ewing 1951b in their data analysis. In particular, studies of flexural wave velocities in shore-fast ice off Barter Island showed that for smaller wavelengths dispersion results were in very good agreement with theoretical predictions. Experimental results for longer waves were still quite good, although not as good as for shorter waves. The deviation was attributed to sea-bottom effects due to the relative shallowness of the water (3m). Oliver et al. 1954 also compared measured ice thicknesses in different areas with those deduced from the group velocities of flexural waves or from the frequency at which the air-coupled waves occurred. Probably due to ice inhomogeneity, the flexural wave dispersion was not as reliable a guide to thickness as the frequency of the air-coupled wave. In general, however, the air-coupled frequency still significantly underpredicted the ice thickness (by around 11% to 15%).

Clearly, in seismological studies, the effect of water compression will become more important, especially if the charge is exploded in the liquid. However, in less destructive studies of ice-ocean interaction, such as the one carried out in Notre Dame Bay, Newfoundland by Squire and Allan 1980 involving strain gauges, neglecting that effect is less consequential. Moreover, in the analysis of Press et al. 1951, carried out before the work of Press and Ewing 1951b, the success of the thin plate model in describing the observed dispersion seems to suggest that the thin plate assumptions about the horizontal and vertical motion are not too significant. In any event, Squire and Allan 1980 were able to verify that the thin plate dispersion relation predicted wavelengths that agreed with measured ones to within experimental error. In addition, by considering the problem of a semi-infinite floating ice sheet, they were able to predict the variation of strain in the ice with distance from the ice edge, given the spectrum of the incident waves. Although no such data were available, they were able to estimate the incident wave spectrum by calculating the ratio of the measured strains to the theoretical strains given a uniform spectrum of unit amplitude. If the calculated incident wave spectrum was independent of the position of the strain-meter producing the experimental results, it would serve as a good check of the theory, although it would not confirm it absolutely. Results for three different meters were qualitatively quite similar for three different locations, and differences could possibly have been able to be attributed to the theoretical strains being based on an incomplete solution Wadhams 1973.

Mindlin 1951 also develops a thick plate model which allows for rotational and shear effects inside the plate. For a semi-infinite ice sheet, theoretical predictions (e.g. Fox and Squire 1990; Balmforth and Craster 1999) show that those effects make little difference, although they might be more important near a structure of smaller width such as a pressure ridge.