Water waves equations in sallow and deep waters (Master thesis)

Γκαρτζονίκα, Δανάη

In the present thesis we study the equations that govern the generation and propagation of waves in water. Water wave phenomena are described by a set of quasi-linear hyperbolic equations governing adiabatic and inviscid flow termed the Euler equations. These equations represent equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier-Stokes equations with zero viscosity and zero thermal conductivity. Solutions of these equations are difficult to obtain as they are coupled equations with a free boundary (meaning that the solution is also part of the boundary conditions). For this reason the usual way to study their solution is using numerical techniques. Our approach will reduce the Euler system, using perturbation techniques and multiple scales methods, to other equations capable of describing water wave phenomena which are mathematically significantly less complex. In doing so, two distinct limits will be considered: shallow and deep waters. The distinction between deep and shallow water waves is determined by the ratio of the water’s depth to the wavelength of the wave. In layman’s terms, in shallow water, waves, begin to be affected by the ocean bottom whereas in deep water the depth of the ocean is taken to be infinite. In the first case, the Korteweg-de Vries (KdV) equation is obtained whereas in the latter the nonlinear Schrodinger (NLS) equation. For each equation we provide examples from observable (real world) phenomena that can be modelled with one or the other system. We also briefly discuss a special type of solution, the soliton (a special solitary wave that maintains its shape and velocity during propagation even after it collides with other solitons), and provide a perturbative scheme to directly connect the two systems and their relative solutions.
Institution and School/Department of submitter: Πανεπιστήμιο Ιωάννινων. Σχολή Θετικών Επιστημών. Τμήμα Μαθηματικών
Subject classification: Εξισώσεις -- Άλγεβρα
Keywords: Υδάτινα κύματα,Βαθιά,Αβαθή,Εξισώσεις,Water waves equations,kdv equation,nls equation,Sallow and deep waters
URI: https://olympias.lib.uoi.gr/jspui/handle/123456789/29289
Appears in Collections:Διατριβές Μεταπτυχιακής Έρευνας (Masters)

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