On Steady Water Waves and Their Properties
(2008) In Doctoral Theses in Mathematical Sciences 2008:7. Abstract
 Abstract: This thesis consists of four papers related to various aspects of steady water waves.
Paper I: Deepwater waves with vorticity: symmetry and rotational behaviour.
We show that for steady, periodic, and rotational gravity deepwater waves, a monotone surface profile between troughs and crests implies symmetry. It is observed that if the vorticity function has a bounded derivative, then it vanishes as one approaches great depths.
Paper II: Linear water waves with vorticity: rotational features and particle paths.
Steady linear gravity waves of small amplitude travelling on a current of constant vorticity are found. For negative... (More)  Abstract: This thesis consists of four papers related to various aspects of steady water waves.
Paper I: Deepwater waves with vorticity: symmetry and rotational behaviour.
We show that for steady, periodic, and rotational gravity deepwater waves, a monotone surface profile between troughs and crests implies symmetry. It is observed that if the vorticity function has a bounded derivative, then it vanishes as one approaches great depths.
Paper II: Linear water waves with vorticity: rotational features and particle paths.
Steady linear gravity waves of small amplitude travelling on a current of constant vorticity are found. For negative vorticity we show the appearance of internal waves and vortices, wherein the particle trajectories are not any more closed ellipses. For positive vorticity the situation resembles that of Stokes waves, but for large vorticity the trajectories are affected.
Paper III: On the streamlines and particle paths of gravitational water waves.
We investigate steady symmetric gravity water waves on finite depth. For nonpositive vorticity it is shown that the particles display a mean forward drift, and for a class of waves we prove that the size of this drift is strictly increasing from bottom to surface. This includes the case of particles within irrotational waves. We also provide detailed information concerning the streamlines and the particle trajectories.
Paper IV: Travelling waves for the Whitham equation.
The existence of travelling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves on finite depth. It is found that there exist smallamplitude periodic travelling waves with subcritical speeds. As the period of these travelling waves tends to infinity, their velocities approach the limiting longwave speed c0, and the waves approach a solitary wave. It is also shown that there can be no solitary waves with velocities much greater than c0. Finally, numerical approximations of some periodic travelling waves are presented. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1215331
 author
 Ehrnström, Mats ^{LU}
 supervisor

 Adrian Constantin ^{LU}
 opponent

 Professor Groves, Mark, Universität des Saarlandes, Saarbrücken
 organization
 publishing date
 2008
 type
 Thesis
 publication status
 published
 subject
 keywords
 Particle trajectories, Water waves, Vorticity, Maximum principles, Elliptic equations
 in
 Doctoral Theses in Mathematical Sciences
 volume
 2008:7
 pages
 96 pages
 publisher
 KFS AB
 defense location
 Matematikcentrum, Sölvegatan 18, sal MH:C
 defense date
 20080912 13:00
 ISSN
 14040034
 ISBN
 9789162875534
 language
 English
 LU publication?
 yes
 id
 02176b6bb2c64105b512424e3cd398f8 (old id 1215331)
 date added to LUP
 20080818 12:02:17
 date last changed
 20160919 08:44:47
@phdthesis{02176b6bb2c64105b512424e3cd398f8, abstract = {Abstract: This thesis consists of four papers related to various aspects of steady water waves.<br/><br> <br/><br> <br/><br> Paper I: Deepwater waves with vorticity: symmetry and rotational behaviour. <br/><br> <br/><br> We show that for steady, periodic, and rotational gravity deepwater waves, a monotone surface profile between troughs and crests implies symmetry. It is observed that if the vorticity function has a bounded derivative, then it vanishes as one approaches great depths.<br/><br> <br/><br> <br/><br> Paper II: Linear water waves with vorticity: rotational features and particle paths.<br/><br> <br/><br> Steady linear gravity waves of small amplitude travelling on a current of constant vorticity are found. For negative vorticity we show the appearance of internal waves and vortices, wherein the particle trajectories are not any more closed ellipses. For positive vorticity the situation resembles that of Stokes waves, but for large vorticity the trajectories are affected.<br/><br> <br/><br> <br/><br> Paper III: On the streamlines and particle paths of gravitational water waves. <br/><br> <br/><br> We investigate steady symmetric gravity water waves on finite depth. For nonpositive vorticity it is shown that the particles display a mean forward drift, and for a class of waves we prove that the size of this drift is strictly increasing from bottom to surface. This includes the case of particles within irrotational waves. We also provide detailed information concerning the streamlines and the particle trajectories.<br/><br> <br/><br> <br/><br> Paper IV: Travelling waves for the Whitham equation.<br/><br> <br/><br> The existence of travelling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves on finite depth. It is found that there exist smallamplitude periodic travelling waves with subcritical speeds. As the period of these travelling waves tends to infinity, their velocities approach the limiting longwave speed c0, and the waves approach a solitary wave. It is also shown that there can be no solitary waves with velocities much greater than c0. Finally, numerical approximations of some periodic travelling waves are presented.}, author = {Ehrnström, Mats}, isbn = {9789162875534}, issn = {14040034}, keyword = {Particle trajectories,Water waves,Vorticity,Maximum principles,Elliptic equations}, language = {eng}, pages = {96}, publisher = {KFS AB}, school = {Lund University}, series = {Doctoral Theses in Mathematical Sciences}, title = {On Steady Water Waves and Their Properties}, volume = {2008:7}, year = {2008}, }