# Modelling moving loads on floating ice plates

For several centuries now, scientists and engineers have been prompted to undertake mathematical modelling of deflexions in flexible beams, plates or shells. Almost two hundred years ago the Swiss mathematical school in Basel proceeded to introduce important new concepts such as virtual displacement, and Daniel Bernoulli and Leonhard Euler developed classical beam theory (Timoshenko, 1953). Outstanding investigators (including Rayleigh and Timoshenko) further pursued the mathematical modelling of flexible beams; and in 1887 Greenhill considered a floating elastic plate to model an ice sheet, which was initially followed in discussing the consequences of both stationary and moving loads.

In the early 1980's, Physics personnel at the University of Waikato (NZ) returned from McMurdo Sound in the Antarctic with data obtained from a device intended to measure the seasonal ice movement - and they recorded interesting oscillations when Hercules aircraft supplying the base were landing several kilometres away. Incidentally, pilots normally target a steady landing speed. This led to a seminal paper with John Davys and Alfred Sneyd on "Waves due to a steadily moving source on a ice plate" (J. Fluid Mech. 158, 269-287, 1985), which predicted the wave patterns produced in the ice sheet at various load speeds and explained the critical speed at which the response is most pronounced physically.

The mathematical model adopted was regarded as the simplest acceptable, where a uniform infinitesimally thin elastic plate rests on an incompressible inviscid fluid of finite depth - cf. also the monograph by Squire, Hosking. Kerr and Langhorne ("Moving Loads on Ice Plates", Springer, 1996 & 2012). Earlier theoreticians, most notably Kheisin in Russia and Nevel in the USA, had predicted an infinitely large response at one or two critical speeds - but 2D analysis for a distributed or point load published after the monograph was originally written demonstrated that only the load speed coincident with the minimum phase speed of the waves is critical, and the incorporation of anelasticity in the mathematical model was confirmed to render the critical response finite. Agreement between theory and field observations is excellent.

More recent work on the propagation of nonlinear waves where an incompressible fluid is bounded at its upper surface by a flexible elastic plate, has shown that solitons may occur at load speeds approaching critical. And much more recently, the response due to a non-uniformly moving load has been investigated, and in particular it has been shown from a time-dependent dispersive Whitham-type equation with loading, that the response may be magnified considerably when a moving load decelerates.