Supervisors info:
Ιωάννης Στρατής Καθηγητής (επιβλέπων), Δημήτριος Φραντζεσκάκης Καθηγητής, Νικόλαος Καραχάλιος Καθηγητής
Summary:
This thesis has as main object to study water wave equations and their
connection with extreme phenomena of the sea, such as tsunamis, rogue waves and
super rogue waves.
Initially, we mention the fundamental equations of continuity, momentum and
pressure, and also the framework of Euler equation in order to introduce the
linear wave equations.
Afterwards, we focus on the theory of shallow water. Here we discuss solitary
waves, called solitons. Naval architect J. S. Russell, through experiments
found, inter alia, that the wave speed depends on the height and thus concluded
that there must be a nonlinear effect. J. Boussinesq, D. Korteweg and G. de
Vries produced approximate nonlinear equations for shallow water waves and also
individual and periodic solutions of nonlinear wave equations for them
mentioned above. They also found that the wave speed is proportional to
amplitude, i.e. longer waves travel faster. Kadomtsev and Petviashvili included
in their equations weak transverse variation and so dispersion was connected.
The inverse rate of the increase of the amplitude of the linear shallow water
wave approaching a shore and gentle slope is given by the Bessel equation and
quantified by Green's lαw.
Then we will discuss the production of Nonlinear Schrodinger (NLS) equation in
the theory of shallow and deep waters and the relationship with solitons, as
well. In addition, we will refer to multidimensional water waves and the
equations generated not only without the influence of surface tension but also
under it.
Keywords:
Wave, Tsunamis, Water wave equations, Rogue wave, Shallow water
