# Waves generated by a load moving on an ice sheet over water

## Date

2018-07-25

## Authors

Nugroho, Widijanto Satyo

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## Abstract

A load moving on a floating ice sheet produces a deflection of the ice sheet. In this Dissertation, three problems associated with mathematical models of the ice water system are examined.
A mathematical model involving a steadily moving rectangular load on an ice sheet where the supporting fluid is of infinite depth is analyzed. The solution is written as a Fourier integral and is estimated using an asymptotic method. The results show that the amplitude of the ice deflection is similar to the case where the supporting fluid is of finite depth. The only significant difference is that, in contrast to the case where the supporting fluid is of finite depth where a quiescent zone appears behind the load when its speed exceeds the speed of gravity waves on shallow water, waves appear behind the load for all supercritical load speeds.
A mathematical model of an ice plate that takes into account the thickness of the ice is derived by assuming that the vertical shearing forces vary linearly through the ice plate. The equations obtained are similar to those used to describe a mathematical model using a thin plate approximation subjected to in-plane forces. A comparison of the dispersion relation is carried out between the mathematical model of an ice plate that takes into account the plate thickness, the mathematical model of an ice plate using the thin plate approximation, and the mathematical model of an ice plate using the thin plate approximation subjected to in-plane forces. The results show that taking the ice thickness into consideration decreases the minimum phase speed. However, this effect is small.
The major contribution of this Dissertation is the determination of the large time response of the deflection of an ice sheet caused by the steady motion of an impulsively-started point load. The results obtained are new. The solution of the ice deflection is written as a Fourier integral and asymptotic methods are used to estimate the large time behaviour of the rate of change of the ice deflection with respect to time. The large time behaviour of the ice deflection itself is inferred from this estimate. This is done for the full range of load speeds and the results are verified numerically using the Fast Fourier Transform. The results in this Dissertation show that the minimum of the phase speed is the only critical speed, in the sense that no finite steady-state is attainable. At this speed the ice deflection grows logarithmically with time. This is in contrast with the case of a line load where there are two critical speeds: the minimum of the phase speed at which the ice deflection grows as the square-root of time, and the speed of gravity waves in shallow water at which the ice deflection grows as the cube-root of time. For a point load, it is found that the transient part of the ice deflection decays as the cube-root of time when the load speed is the speed of gravity waves in shallow water. The asymptotic estimates also show that the decay or the growth rate of the transient component of the ice deflection does not depend on either the relative orientation of the observation point and the load or on the distance between the load and the observation point.

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Plates (Engineering), Mathematical models, Fluid dynamics, Finite element method, Mathematical analysis