Related Experiment Video
Updated: Jun 8, 2026
08:54
Measurements of Waves in a Wind-wave Tank Under Steady and Time-varying Wind Forcing
Published on: February 13, 2018
Canonical variables for steep planar water waves over nonuniform beds
1Landau Institute for Theoretical Physics, 2 Kosygin Street, 119334 Moscow, Russia. ruban@itp.ac.ru
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 28, 2010
Summary
Researchers derived a Hamiltonian functional for nonlinear fluid dynamics, enabling precise analysis of water waves over uneven depths. This work advances the understanding of fluid motion and wave dynamics.
Area of Science:
- Fluid Dynamics
- Nonlinear Dynamics
- Wave Theory
Background:
- Understanding nonlinear fluid dynamics is crucial for modeling complex wave phenomena.
- Previous work established a noncanonical conformal description for water waves over undulating bottoms.
Purpose of the Study:
- To derive an explicit Hamiltonian functional for two-dimensional potential flows with free surfaces over arbitrary nonuniform depths.
- To establish canonical variables for describing fully nonlinear water wave dynamics.
- To explore alternative approaches for weakly nonlinear Hamiltonian models.
Main Methods:
- Derivation of canonical variables from a noncanonical conformal description.
- Application of Hamiltonian formalism to fluid dynamics.
- Development of alternative methods for weakly nonlinear models.
Main Results:
- An explicit expression for the Hamiltonian functional in terms of canonical variables was obtained.
- The derived variables describe the fully nonlinear dynamics of potential flows.
- Weakly nonlinear Hamiltonian models for both free-surface and interfacial waves were discussed.
Conclusions:
- The study provides a rigorous Hamiltonian framework for analyzing nonlinear water waves.
- The findings facilitate a deeper understanding of fluid behavior over complex geometries.
- The developed methods offer versatile tools for modeling various wave phenomena.
Related Concept Videos
01:21Partial Differential Equations
A stone dropped into a still pond generates waves that propagate outward in circular patterns, creating a dynamic surface whose elevation depends on both position and time. At any given location, the water level oscillates as the wave passes, while at any fixed moment, the surface exhibits smooth, curved structures extending across space. This dual dependence requires a mathematical description that accounts for variation in multiple variables simultaneously.At a fixed point on the water...
01:27Uniform Depth Channel Flow
Uniform depth channel flow keeps fluid depth consistent along channels such as irrigation canals. In natural channels, such as rivers, approximate uniform flow is often assumed. This condition occurs when the channel’s bottom slope matches the energy slope, balancing potential energy lost from gravity with head loss due to shear stress. This balance prevents depth changes along the channel length, resulting in a steady, uniform flow.Uniform flow in open channels with a constant cross-section...
01:17Steady, Laminar Flow Between Parallel Plates
Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
01:30Typical Model Studies
Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.
01:10Wave Parameters
The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be modeled using a combination of sine and cosine functions. Consider a simplified surface water wave that moves across the water's surface. Unlike complex ocean waves, in surface water waves, water moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. If a seagull is floating on the...
01:18Uniform Depth Channel Flow: Problem Solving
To calculate the flow rate for a trapezoidal channel, first, identify the bottom width, side slope, and flow depth of the channel. The cross-sectional area (A) corresponding to the depth of flow (y), channel bottom width (B), and side slope (θ) is determined by:Next, calculate the wetted perimeter, which includes the bottom width and the sloped side lengths in contact with the water. Using the values of the cross-sectional area and the wetted perimeter, determine the hydraulic radius by...