Theory of water waves

Programme theme

Water waves are a dramatic, potentially dangerous, yet beautiful phenomena that is omnipresent and impacts every aspect of life on the planet. At smaller length scales the ripples driven by surface tension affect remote sensing. At intermediate length scales waves in the mid-ocean affect shipping and near the shoreline they control the coastal morphology and the ability to navigate along shore. At larger length scales waves such as tsunamis and hurricane-generated waves can cause devastation on a global scale. Across all length scales an exchange of momentum and thermal energy between ocean and atmosphere occurs affecting the global weather system and the climate.

From a mathematical viewpoint water waves pose rich challenges.The governing equations for water waves are a widely accepted model and they have been the subject of a wide range of research. However, the equations are highly nonlinear and the level of difficulty is so great that theory has yet to scratch the surface of the subject. The solutions to the equations that describe fluid motion are elusive and whether they even exist in the most general case is one of the most difficult unanswered questions in mathematics.

On the other hand, there is good reason to be buoyant about the headway that mathematics can make in tackling the great open problems posed by water waves. In light of recent developments the questions are now clearer, new methodologies are emerging, computational approaches are becoming much more sophisticated and the number of researchers at the highest international level involved is growing. All these indicators point to an opportune time to have a focused conference on water waves. Particular themes will include:

The initial-value problem (IVP)

Well-posedness, theory in two versus three dimensions,Lagrangian versus Eulerian formulations
Vortex sheet formulations, singularities
Numerical methods for the IVP, simulations
Theoretical aspects of wave breaking

Existence and classification of waves

Traveling waves in two and three dimensions
Periodic, multi-periodic, and localized patterns
Surface waves versus interfacial waves
Numerical methods for steady waves and their bifurcations
Nonlinear time-periodic waves: Standing waves and beyond

Linear and nonlinear stability of waves

Stability and instability of the waves mentioned above (e.g., periodic and multi-periodic traveling waves, localized traveling waves, standing waves)
Beyond (symmetric) short-crested waves: Stability of genuinely three-dimensional traveling waves.
Wave interactions
Rogue waves and other extreme events
Energetic stability

Dynamical systems and geometric techniques

Hamiltonian and Lagrangian formulations, variational principles
Center-manifold reduction, spatial dynamics, and generalizations
Differential geometry of the sea surface
The effect of bottom topography

Beyond irrotational flow

Waves with vorticity
Weakly viscous potential flow
Waves with forcing and dissipation

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
Well-posedness of two-dimensional hydroelastic waves Authors: David Ambrose, M Siegel Download PDF	2014	TWW	21 October 2016

Title

Year

Programme

Well-posedness of two-dimensional hydroelastic waves

Authors: David Ambrose, M Siegel

Download PDF

2014

TWW

21 October 2016

Seminars

Monday 14th July 2014
10:00 to 11:00	Random time series analysis and extreme wave events	Room 2
11:00 to 12:00	Nonlinear dynamics of trapped waves on currents	Room 2
13:00 to 14:00	Tsunami	Room 2
14:00 to 15:00	Gravity - Capillary Solitary Waves and Lumps	Room 2
15:30 to 16:30	Birkhoff normal forms for the equations of water waves	Room 2
16:30 to 17:30	Standing and three-dimensional water waves	Room 2

Tuesday 15th July 2014
09:30 to 10:30	Short Course: HOPS Methods: Boundary Value Problems: I	Room 2
11:00 to 12:00	Short Course: The Unified Transform and Boundary Value Problems: I	Room 2
13:00 to 14:00	Nonlinear flexural and capillary-gravity waves	Room 2
14:00 to 15:00	Solitary waves with intrinsic wavelength	Room 2
15:30 to 16:30	New nonsymmetric gravity-capillary solitary waves and related problems	Room 2

Wednesday 16th July 2014
09:30 to 10:30	Short Course: High Order Perturbation of Surfaces (HOPS) Methods for Water Waves: II	Room 2
11:00 to 12:00	Short Course: The Unified Transform and Boundary Value Problems: II	Room 2
13:00 to 14:00	Nearly time-periodic water waves	Room 2
14:00 to 15:00	Stability of periodic travelling waves of the water wave problem in one dimension	Room 2
15:30 to 16:30	Stability of Traveling Waves with Vorticity	Room 2
16:30 to 17:30	Reconstructing the water surface from pressure measurements	Room 2

Thursday 17th July 2014
09:30 to 10:30	Short Course: High Order Perturbation of Surfaces (HOPS) Methods for Water Waves: III	Room 2
11:00 to 12:00	Short Course: The Unified Transform and Boundary Value Problem: III	Room 2
15:00 to 16:00	Dynamics of critical layer in turbulent shear flows above unsteady water waves	Room 1
16:00 to 17:00	New results on wind driven waves and negative tsunami waves at coastlines	Room 1
17:00 to 18:00	Wine Reception following Sir Julian Hunt's lecture	No Room Required

Friday 18th July 2014
09:30 to 10:30	Short Course: High Order Perturbation of Surfaces (HOPS) Methods for Water Waves: IV	Room 2
13:00 to 14:00	Boundary element and spectral methods for water waves	Room 2
14:00 to 15:00	Variational principles for water waves beyond perturbations	Room 2
15:30 to 16:30	Finite element method for nonlinear free surfaces water waves	Room 2

Tuesday 22nd July 2014
09:30 to 14:00	S Page - informal discussion	Discussion Room
14:00 to 15:00	Trajectories of fluid particles in a water-wave	Room 2
15:00 to 16:00	On the Benjamin-Feir instability	Room 2
16:30 to 17:30	Waves over periodic topographies	Room 2

Wednesday 23rd July 2014
10:30 to 11:10	Registration and Tea & Coffee	No Room Required
11:10 to 11:15	Welcome from John Toland (INI Director)	No Room Required
11:15 to 12:00	Validity and non-validity of the NLS approximation for the water wave problem - Recent developments and open problems	Room 1
12:00 to 14:00	Lunch - Sandwich lunch at INI	No Room Required
14:00 to 14:45	Interface singularities for the Euler equations	Room 1
14:45 to 15:30	Singularities of steady free surface water flows	Room 1
15:30 to 16:15	Afternoon Tea	No Room Required
16:15 to 17:00	Three-dimensional water waves	Room 1
17:00 to 18:00	Welcome Wine Reception	No Room Required

Thursday 24th July 2014
11:00 to 12:00	The Froude number and solitary waves with vorticity	Room 1
12:00 to 12:30	Violent wave motion due to impact	Room 1
14:00 to 15:00	Justification of the nonlinear Schrödinger equation as the modulation equation to the full water-wave problem	Room 2
15:00 to 16:00	On water waves with angled crests	Room 2

Monday 28th July 2014
14:00 to 15:00	Dynamic energy cascades in the theory of water waves	Room 1

Tuesday 29th July 2014
14:00 to 15:00	Resonance curves of finite-amplitude gravity waves in shallow-water limit	Room 2
15:00 to 16:00	Maximum finite depth waves: breaking, kinematics and particle drift	Room 2
16:30 to 17:30	Impacts from highly nonlinear waves on walls and substructures of wind turbines	Room 2

Wednesday 30th July 2014
14:10 to 14:40	Ocean Wave Measurements: The Challenges and Consequences for the Wave Energy Industry	Room 1
15:00 to 15:30	Teas & coffee	No Room Required
16:00 to 16:30	100-year and 10,000-year Extreme Significant Wave Heights - How Sure Can We Be of These Figures?	Room 1

Thursday 31st July 2014
14:00 to 15:00	Ill-posedness of truncated series models for water waves	Room 2
15:00 to 16:00	Global bifurcation for steady gravity water waves with constant vorticity and critical layers	Room 2
16:30 to 17:30	Hydrodynamic Surface Wave Analogues for Quantum Mechanics and Nonlinear Optics	Room 1

Tuesday 5th August 2014
14:00 to 15:00	Changing forms and sudden smooth transitions of tsunami waves	Room 2
15:00 to 16:00	Why bouncing droplets are a pretty good model of quantum mechanics	Room 2
16:30 to 17:30	Transverse instability of generalised solitary waves	Room 2

Wednesday 6th August 2014
09:15 to 09:45	Registration	No Room Required
09:45 to 10:00	Welcome from John Toland (INI Director) and Opening	No Room Required
10:00 to 12:30	Initial Value Problem and Vortex Sheets: Analysis and Computation	No Room Required
14:00 to 15:00	Forward looking / open problems session	No Room Required
15:00 to 15:30	Afternoon Tea - Served at INI	No Room Required
15:30 to 16:30	Forward looking / open problems session continued	No Room Required
16:45 to 17:45	Wine Reception - Served at INI	No Room Required

Thursday 7th August 2014
10:00 to 12:30	Well Posedness and Singularities of Water Waves	No Room Required
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Conformal Mapping and Complex Topography	No Room Required
15:00 to 15:30	Afternoon Tea - served at INI	No Room Required
15:30 to 16:30	Conformal Mapping and Complex Topography	No Room Required
19:30 to 22:00	BBQ at the Møller Centre	No Room Required

Friday 8th August 2014
10:00 to 12:30	Variational Water Waves: on Continuum and Discrete Modelling, and Experimental Validation	No Room Required
14:00 to 15:00	Modulation of Water Waves	No Room Required
15:00 to 15:30	Afternoon Tea - Served at INI	No Room Required
15:30 to 16:30	Modulation of Water Waves	No Room Required

Organisers

Tom Bridges University of Surrey
Mark Groves Loughborough University
Paul Milewski Aalborg Universitet
David Nicholls University of Notre Dame

Programme theme

Preprints

Well-posedness of two-dimensional hydroelastic waves

Seminars

Organisers

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