Time-dependent response of floating ice to a steadily moving load

Abstract

When a steadily moving load is applied to a floating ice plate, the disturbance will generally approach a steady state (relative to the load) as time t → ∞. However, for certain ‘critical’ load speeds the disturbance may grow continuously with time, which represents some danger to vehicles driving on ice. To understand this phenomenon and the overall time development of the ice response, this paper analyses the problem of an impulsively applied, concentrated line load on a perfectly elastic homogeneous floating ice plate. An exact expression for the ice deflection is derived, and then interpreted by means of asymptotic expansions for large t in the vicinity of the source. The spatial development of the disturbance is analysed by considering asymptotic expansions as t → ∞ near an observer moving away from the load. Theoretical results are compared with field measurements, and some hitherto unexplained features can be understood.

The response of a floating ice sheet to an accelerating line load

“…Figure 1(a) shows the wave pattern created by a load impulsively accelerated to a subcritical speed v < 1.0. which then continues at constant speed. This is similar to figure 7 of Schulkes & Sneyd (1988). Shorter, faster, waves propagate before the source, and slower, longer, waves behind.…”

“…Figure 1(b) is similar, but now the speed is supercritical with v > 1.0. Here the difference in wavelength between the forward and backward waves is more pronounced, as in figure 8 of Schulkes & Sneyd (1988).…”

“…Figures 1(a)-1(d) show the time development of the wave system due to line loads moving with constant (or zero) acceleration. As in Schulkes & Sneyd (1988) we present the results in the form of mesh plots of the surface elevation in the (x, t)-plane. The x-axis is placed in the immediate foreground, and the second horizontal axis represents time.…”

“…We find no noticeable effect as a load accelerates through the critical speed c cmin . This is to be expected since the displacement due to a load travelling at constant speed c cmin grows linearly with time (Schulkes & Sneyd 1988). For loads moving with nonconstant acceleration special methods may be necessary to evaluate rapidly oscillating integrals.…”

Three-dimensional waves beneath an ice sheet due to a steadily moving pressure

“…In two dimensions, its effect was discussed by Schulkes & Sneyd [23]. We expect that it will play an important role in the pattern of solutions, so a boundary integral method based on a Green function that incorporates the boundary condition at the bottom of the fluid needs to be developed [31].…”