An analysis of a new approximate model for two-dimensional water waves with constant vorticity
(2026) In Master's Theses in Mathematical Sciences MATM03 20251Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- In this thesis, we investigate the water-wave problem for two-dimensional steady waves with constant vorticity. We introduce a new approximate model that, in its simplest form, reduces to a planar dynamical system. We then analyze this system for various energy levels and vorticity constants, studying its phase portraits to determine the locations and types of solutions that arise.
In the unidirectional regime, where the flow preserves its direction, we recover classical periodic and solitary wave solutions, including both smooth profiles and extreme waves with sharp crests. Additionally, we identify new classes of waves propagating over flows that contain a critical layer. These solutions exhibit a discontinuity in the derivative of... (More) - In this thesis, we investigate the water-wave problem for two-dimensional steady waves with constant vorticity. We introduce a new approximate model that, in its simplest form, reduces to a planar dynamical system. We then analyze this system for various energy levels and vorticity constants, studying its phase portraits to determine the locations and types of solutions that arise.
In the unidirectional regime, where the flow preserves its direction, we recover classical periodic and solitary wave solutions, including both smooth profiles and extreme waves with sharp crests. Additionally, we identify new classes of waves propagating over flows that contain a critical layer. These solutions exhibit a discontinuity in the derivative of the surface profile at a point along the side of the wave. We therefore regard them as potential prototypes of overhanging waves for the full Euler equations. (Less) - Popular Abstract
- In this thesis, we investigate how surface waves behave when the underlying water flow has a constant rotational motion, known as vorticity. To study this, we develop a new approximate mathematical model that captures the essential dynamics of such waves. In its simplest form, this model reduces the problem to a planar dynamical system, allowing us to explore the possible wave behaviors by analyzing its trajectories.
By varying the energy and the strength of the vorticity, we map out different regimes of wave motion. In situations where the flow moves consistently in one direction, the model predicts familiar wave patterns, such as periodic waves and solitary waves. These include both smooth profiles and extreme waves with sharply... (More) - In this thesis, we investigate how surface waves behave when the underlying water flow has a constant rotational motion, known as vorticity. To study this, we develop a new approximate mathematical model that captures the essential dynamics of such waves. In its simplest form, this model reduces the problem to a planar dynamical system, allowing us to explore the possible wave behaviors by analyzing its trajectories.
By varying the energy and the strength of the vorticity, we map out different regimes of wave motion. In situations where the flow moves consistently in one direction, the model predicts familiar wave patterns, such as periodic waves and solitary waves. These include both smooth profiles and extreme waves with sharply pointed crests, which are known to occur in strongly nonlinear water-wave systems.
The model also reveals a new class of waves that arise when the flow contains a critical layer—a region where the horizontal velocity within the fluid matches the wave’s speed. In this case, the wave surface develops a discontinuity in its slope at a specific point on the profile. These features suggest that such solutions may serve as simplified prototypes of overhanging waves, complex structures predicted by the full Euler equations but difficult to analyze directly.
Overall, the work demonstrates that our new approximate model provides a reasonable approximation to the water wave problem with vorticity, capturing the most important features at the nonlinear level. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/student-papers/record/9221994
- author
- Cederholm, Christian LU
- supervisor
- organization
- course
- MATM03 20251
- year
- 2026
- type
- H2 - Master's Degree (Two Years)
- subject
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUNFMA-3162-2026
- ISSN
- 1404-6342
- other publication id
- 2026:E1
- language
- English
- id
- 9221994
- date added to LUP
- 2026-02-27 14:17:11
- date last changed
- 2026-02-27 14:17:11
@misc{9221994,
abstract = {{In this thesis, we investigate the water-wave problem for two-dimensional steady waves with constant vorticity. We introduce a new approximate model that, in its simplest form, reduces to a planar dynamical system. We then analyze this system for various energy levels and vorticity constants, studying its phase portraits to determine the locations and types of solutions that arise.
In the unidirectional regime, where the flow preserves its direction, we recover classical periodic and solitary wave solutions, including both smooth profiles and extreme waves with sharp crests. Additionally, we identify new classes of waves propagating over flows that contain a critical layer. These solutions exhibit a discontinuity in the derivative of the surface profile at a point along the side of the wave. We therefore regard them as potential prototypes of overhanging waves for the full Euler equations.}},
author = {{Cederholm, Christian}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{An analysis of a new approximate model for two-dimensional water waves with constant vorticity}},
year = {{2026}},
}