Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 72 / 1 / b

The sea ice acts as a frequency-selective filter. Over a long path, wave attenuation in sea ice is generally much stronger for shorter surface gravity waves, through repeated scattering, internal ice losses and water-side dissipation. Their amplitude can fall below the tiltmeter's detection level even if the source initially generated them. Thus the long-period swell survives while the shorter-period tail does not.

The disappearance near 14 s is an attenuation and detectability limit, not a forbidden-frequency interval of the ideal elastic plate. The flexural-gravity wave dispersion relation admits a real positive wavenumber for every positive angular frequency. In an impulsive-source picture, shorter-period deep-water gravity waves also arrive later because their group velocity is smaller, but delayed arrival alone does not explain a persistent observed cutoff.

Use a common emission time $t_0$ and a path length $L$, and treat each observed period as a narrow wave packet. For deep-water gravity waves, its group velocity is $gT/(4\pi )$, not the phase velocity $gT/(2\pi )$. Therefore the dispersive swell source inversion givesThus period decreases hyperbolically, while frequency $1/T$ increases linearly. The elapsed time between the two observations is $40\mathrm{h}40\min =146400\mathrm{s}$, soThe frequency slope is $2.6021\times 10^{-7}\mathrm{Hz}{\mathrm{s}}^{-1}$. Initially $dT/dt=-0.843\mathrm{s}{\mathrm{h}}^{-1}$; at 14 s it has slowed to $-0.184\mathrm{s}{\mathrm{h}}^{-1}$. The inferred emission was $C/30=35\mathrm{h}35\min$ before the first arrival, namely 19 March at 1710 UT. Both observed endpoints give this same time.

A northward-propagating swell with this distance scale points to a remote energetic storm south of the high Arctic rather than local wind acting on continuous sea ice. Along a meridional route, 3,000 km is about $27^{\circ }$ of latitude, putting the source on the scale of the northern North Atlantic and adjacent open seas. A route through Fram Strait is plausible, but longitude and refraction are not supplied, so no particular storm or unique source position follows. Long swell arriving first and steadily increasing frequency are the expected signatures of remote wave dispersion. The distance and time are conditional on an approximately impulsive source and open-water propagation speeds; passage through ice, currents and finite storm duration produce corrections.