2d SWE - Linear Gravity Wave

[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 σ_x² )] exp [- (y - L/2)^2 / (2 σ_y²) ]

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

 

[jvicens]

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