# 2d SWE - Linear Gravity Wave

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• Keccogrin
In summary, the 2D SWE (Shallow Water Equations) model the behavior of shallow water waves and have an exact analytical solution for a linear surface gravity wave known as the Gerstner wave. This solution can be found in various textbooks and papers, such as "Geophysical Fluid Dynamics" by Joseph Pedlosky and "Shallow Water Waves on the Rotating Earth" by C. J. K. Knight. The paper "Exact Solutions of the Shallow-Water Equations and Their Application to Nonlinear Wave Interactions" by T. J. Bridges and P. K. Newton provides a comprehensive review of analytical solutions for the 2D SWE.
Keccogrin
Hi,
I have some questions about the 2D SWE (Shallow Water Equations).
I would like to know if an exact (analytical) solution is available in the case of the (linear) surface "gravity" wave.
Indeed, my example of interest is given by a square [0,L]2 with initial velocity field u = v = 0 and an initial Gaussian perturbation of the free surface elevation $$\displaystyle h$$ placed in the center of the domain:
h(x,y,t=0) = H + η' exp [- (x - L/2)2 / (2 σx2 )] exp [- (y - L/2)^2 / (2 σy2) ]

I know that, in 1D, a simple exact solution is available, which derives from a linearization of the Sain-Venant equations.

Can you give me some references? (I am studying on Leveque, "Finite Volume methods for hyperbolic Problems" 2004)

Thank you,Francesco

Hi Francesco,

Thank you for your question about the 2D SWE (Shallow Water Equations). The 2D SWE is a set of equations that model the behavior of shallow water waves, such as ocean waves, in two dimensions. The equations are derived from the Navier-Stokes equations and are commonly used in numerical simulations to study the behavior of these waves.

To answer your question, yes, there is an exact analytical solution for a linear surface gravity wave in the 2D SWE. This solution is known as the Gerstner wave and was first derived by the German mathematician Johann Gerstner in 1802. It describes the behavior of a single, periodic wave in shallow water, and has the form of a cosine function.

The exact solution for the Gerstner wave in the 2D SWE can be found in various textbooks and papers, such as "Geophysical Fluid Dynamics" by Joseph Pedlosky and "Shallow Water Waves on the Rotating Earth" by C. J. K. Knight. In addition, the paper "Exact Solutions of the Shallow-Water Equations and Their Application to Nonlinear Wave Interactions" by T. J. Bridges and P. K. Newton provides a comprehensive review of analytical solutions for the 2D SWE, including the Gerstner wave.

I hope this helps answer your question and provides you with some useful references for your studies. Best of luck in your research!
Scientist in the field of Shallow Water Equations

## 1. What is a 2D SWE?

A 2D SWE, or two-dimensional shallow water equation, is a mathematical model used to study the behavior of surface waves in a body of water. It takes into account the effects of gravity, surface tension, and bathymetry (changes in depth) on the motion of the water's surface.

## 2. What is a linear gravity wave?

A linear gravity wave is a type of wave that occurs on the surface of a body of water due to the force of gravity. It is characterized by its sinusoidal shape and constant wavelength, and it travels at a constant speed. This type of wave is often used in the study of 2D SWEs.

## 3. How are 2D SWEs used in scientific research?

2D SWEs are used in scientific research to better understand the behavior of water waves and how they are affected by various factors such as gravity, wind, and obstacles. This information can then be applied to real-world scenarios such as predicting storm surges and designing wave energy converters.

## 4. What are some limitations of the 2D SWE model?

One limitation of the 2D SWE model is that it assumes the water is shallow, meaning the depth is much smaller than the wavelength of the wave. This assumption may not hold true in certain scenarios, such as when studying tsunamis or deep ocean waves. Additionally, the model does not take into account the effects of turbulence, which can significantly impact the behavior of water waves.

## 5. How can 2D SWEs be solved?

2D SWEs can be solved using numerical methods, such as finite difference or finite element methods. These methods involve dividing the domain into a grid and solving the equations at each grid point. This allows for the simulation of complex scenarios and the visualization of the wave's behavior over time.

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