Shallow Water Waves: Stability and Tsunamis

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Description

This lecture covers topics related to shallow water waves, including numerical stability, Fourier transforms, and the derivation of wave equations in shallow waters. The instructor demonstrates simulations of tsunamis and discusses the stability conditions for finite difference schemes. The lecture also explores the impact of disturbances on ocean surfaces, linearization in shallow waters, and the analytical solutions using the WKB method.

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https://mediaspace.epfl.ch/media/0_kmjlwl1b

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In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

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In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations. The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity.