Variational & asymptotic methods in the study of nonlinear, free-surface waves

Mavroeidis, Konstantinos; Μαυροειδής, Κωνσταντίνος

dc.contributor.author	Mavroeidis, Konstantinos	en
dc.contributor.author	Μαυροειδής, Κωνσταντίνος	el
dc.date.accessioned	2019-06-18T10:24:39Z
dc.date.available	2019-06-18T10:24:39Z
dc.date.issued	2019-06-18
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/48842
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.16647
dc.rights	Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα	*
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/gr/	*
dc.subject	Nonlinear gravity waves	en
dc.subject	Multiple scales	en
dc.subject	Averaged Lagrangian	en
dc.subject	Nonlinear Schrödinger	en
dc.subject	Symbolic programming	en
dc.subject	Μη γραμμικοί κυματισμοί βαρύτητας	el
dc.subject	Πολλαπλές κλίμακες	el
dc.subject	Μέση Λαγκραντζιανή	el
dc.subject	Μη γραμμική εξίσωση Schrödinger	el
dc.subject	Συμβολικός προγραμματισμός	el
dc.title	Variational & asymptotic methods in the study of nonlinear, free-surface waves	en
heal.type	bachelorThesis
heal.classification	Fluid dynamics	en
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2019-03-04
heal.abstract	In the present thesis, we investigate the application of certain asymptotic and variational methods to the classical water-wave problem, when the assumptions of weak nonlinearity and periodicity or narrowbandedness are made. Particularly, the methods of interest are the Multiple-Scales Method (MSM) [(Nayfeh 2008; Holmes 2012)] and Whitham’s Averaged Variational Principle (AVP) [(Whitham 1965a, 1965b, 1974; Jeffrey and Kawahara 1982)]. Our focus lies primarily in the derivation, via those methods, of simpler, but nonlinear nonetheless, model equations that govern the propagation of weakly nonlinear, narrow-banded (i.e. slowly modulated) wavetrains, such as the nonlinear Schrödinger (NLS) equation. Although there is a standard, and well understood, procedure to achieve that by implementing the MSM [(Mei, Stiassnie, and Yue 2005)], in the case of the AVP there occur some issues that render its applicability and its connection with other, established methods, such as the MSM, unclear. The matter of that connection has been answered to a great degree by (Yuen and Lake 1975) and (Sedletsky 2012, 2013, 2015, 2016), who showed that, given suitable ansatzes, the AVP leads to the NLS and other evolutionary equations, for waves in water of either infinite or arbitrary depth. However, in both cases, the vertical dependence of the velocity potential is a priori incorporated into the respective ansatz, and is inspired by the results of other formal perturbation methods. Namely, up to now, it seems that the AVP is not self-contained and, in order to ensure its consistency with other acknowledged results, a significant part of the solution (i.e. the vertical structure of the potential) has to be supplemented by “external” means. The main result of our work is that, in fact, the AVP is self-contained, as it can yield the appropriate vertical dependence by considering the admissible variations of arbitrary vertical functions. First, we apply our modification of the method to the case of weakly nonlinear, uniform wavetrains (Stokes waves), where we rederive the results of (Fenton 1985) variationally. Interestingly, in the context of the AVP, the two definitions of (Stokes 1847), regarding the wave celerity, arise naturally. Next, we do the same for slowly modulated wavetrains, where, relying again solely on the AVP, we conclude to results that are in complete agreement with those of (Sedletsky 2012, 2013). Therefore, our approach may be considered as a generalization of the works of Yuen, Lake and Sedletsky that renders Whitham’s AVP an autonomous and consistent method for the study of periodic or nearly periodic waves.	en
heal.advisorName	Αθανασούλης, Γεράσιμος	el
heal.committeeMemberName	Αθανασούλης, Γεράσιμος	el
heal.committeeMemberName	Μπελιμπασάκης, Κωνσταντίνος	el
heal.committeeMemberName	Σπύρου, Κωνσταντίνος	el
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Ναυπηγών Μηχανολόγων Μηχανικών. Τομέας Ναυτικής και Θαλάσσιας Υδροδυναμικής	el
heal.academicPublisherID	ntua
heal.numberOfPages	155 σ.
heal.fullTextAvailability	true