Ship generated mini-tsunamis

Abstract

Very long waves are generated when a ship moves across an appreciable depth change ∆h comparable to the average and relatively shallow water depth h at the location, with ∆h/h ≃ 1. The phenomenon is new and the waves are recently observed in the Oslofjord in Norway. The 0.5-1 km long waves, extending across the 2-3 km wide fjord, are observed as run-ups and run-downs along the shore, of periods of 30-60 seconds, where a wave height up to 1.4 m has been measured. The waves travelling with the shallow water speed, found ahead of the ships moving at subcritical depth Froude number, behave like a mini-tsunami. A qualitative explanation of the linear generation mechanism is provided by an asymptotic analysis, valid for ∆h/h << 1 and long waves, expressing the generation in terms of a pressure impulse at the depth change. Complementary fully dispersive calculations for ∆h/h ≃ 1 document symmetries of the waves at positive or negative ∆h. The wave height grows with the ship speed U according to U n with n in the range 3 − 4, for ∆h/h ≃ 1, while the growth in U is only very weak for ∆h/h << 1 (the asymptotics). Calculations show good agreement with observations.