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Water waves with moving boundaries
- Athanasios S. Fokas, Konstantinos Kalimeris
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- Journal:
- Journal of Fluid Mechanics / Volume 832 / 10 December 2017
- Published online by Cambridge University Press:
- 26 October 2017, pp. 641-665
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The unified transform, also known as the Fokas method, provides a powerful methodology for studying boundary value problems. Employing this methodology, we analyse inviscid, irrotational, two-dimensional water waves in a bounded domain, and in particular we study the generation of waves by a moving piecewise horizontal bottom, as it occurs in tsunamis. We show that this problem is characterised by two equations which involve only first-order derivatives. It is argued that under the assumptions of ‘small amplitude waves’ but not of ‘long waves’, the above two equations can be treated numerically via a recently introduced numerical technique for elliptic partial differential equations in a polygonal domain. In the particular case that the moving bottom is horizontal and under the assumption of ‘small amplitude waves’, but not of ‘long waves’, these equations yield a non-local generalisation of the Boussinesq system. Furthermore, under the additional assumption of ‘long waves’ the above system yields a Boussinesq-type system, which however includes the effect of the moving boundary.
5 - A Novel Non-Local Formulation of Water Waves
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- By Athanassios S. Fokas, University of Cambridge, Cambridge, Konstantinos Kalimeris, Greece & Johann Radon Institute for Computational
- Edited by Thomas J. Bridges, University of Surrey, Mark D. Groves, Universität des Saarlandes, Saarbrücken, Germany, David P. Nicholls, University of Illinois, Chicago
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- Book:
- Lectures on the Theory of Water Waves
- Published online:
- 05 February 2016
- Print publication:
- 04 February 2016, pp 63-77
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Summary
Abstract
An introduction to the new formulation of the water wave problem on the basis of the unified transform is presented. The main presentation is on the three-dimensional irrotational water wave problem with surface tension forces included. Examples considered are the doubly-periodic case, the linear case, the case of a variable bottom, and the case of non-zero vorticity.
Introduction
There have been numerous important developments in the study of surface water waves that date back to the classical works of Stokes and his contemporaries in the nineteenth century. A new reformulation of this problem was presented in [1]. This reformulation is based on the so-called unified transform or the Fokas method, which provides a novel approach for the analysis of linear and integrable nonlinear boundary value problems [2, 3].
This chapter is organised as follows: section 5.2 presents the novel formulation of the 3D water waves in the case of a flat bottom. This formulation is used in section 5.3 for the derivation of the associated linearized equations as well as a 2D Boussinesq equation and the KP equation. The case of the 2D periodic water waves is discussed in section 5.4. 3D water waves in a variable bottom are discussed in section 5.5. Finally, the case of 2D water waves with constant vorticity are considered in section 5.6.
A non-local formulation governing two fluids bounded above either by a rigid lid or a free surface is presented in [4]. The case of three fluids bounded above by a rigid lid is considered in [5]. A hybrid of the novel formulation and an approach based on conformal mappings is presented in [6].
3D Water Waves with Flat Bottom
Let the domain Ωf (where the subscript f denotes flat bottom) be defined by
where η denotes the free surface of the water. One of the major difficulties of the problem of water waves is the fact that η is unknown.
Let ϕ denote the velocity potential. The two unknown functions η(x1, x2, t) and ϕ(x1, x2, y, t) satisfy the following equations:
where is the gravitational acceleration, σ and ρ denote the constant surface tension and density respectively, and h is the constant unperturbed fluid depth.
Development and validation of the Greek Severe Impairment Battery (SIB)
- Anastasia Konsta, Eleni Bonti, Eleni Parlapani, Loukas Athanasiadis, Petros Kechayas, Maria Karagiannidou, Konstantinos Fokas
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- Journal:
- International Psychogeriatrics / Volume 26 / Issue 4 / April 2014
- Published online by Cambridge University Press:
- 20 January 2014, pp. 591-596
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Background:
Most neuropsychological batteries, especially those most often used, are unsuitable for the assessment of patients with severe dementia. The Severe Impairment Battery (SIB) was developed for the evaluation of preserved cognitive functions in these patients. The aim of this study was to formulate a Greek version of the SIB and to conduct a first assessment of its use of patients with mild, moderate, or severe Alzheimer's disease (AD), compared to the Mini-Mental State Examination (MMSE).
Methods:A convenience sample of 42 dementia patients according to DSM-IV-TR criteria and 23 healthy participants was selected. Patients were assessed twice using a Greek translation of the SIB and the Greek version of MMSE. Patients were divided into three severity groups based on grouped by Clinical Dementia Rating (CDR) score and the SIB and MMSE scores were compared.
Results:The validity of the SIB was confirmed by evaluating the correlation coefficients between the SIB and Greek-MMSE, grouped by CDR, which were found to be significant. Cronbach's α for the total SIB score and each subscale score showed high significance, and the item-total correlation for each subscale was also acceptable. The test-retest correlation for the total SIB score and subscale scores were significant. The total SIB score and subscale scores were examined according to CDR.
Conclusion:The Greek SIB is reliable and valid in differentiating patients with moderate or severe dementia, whereas MMSE loses sensitivity due to a floor and ceiling effect.