Solitary gravity wave computation

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These Matlab scripts compute the steady irrotational surface solitary gravity wave solution of the Euler equations (homogeneous, incompressible and perfect fluids). The wave is defined by its Froude number Fr and the result is about fifteen digits accurate. The method works for all but the highest waves, i.e. for all amplitude/depth ratio less than 0.796.

Another source:

Please, note that this script can be also downloaded from the Matlab Central server.

Synopsis:

• SolitaryGravityWave(Fr, [], 1); % plot results only
• [zs, ws, fs, SWP] = SolitaryGravityWave(Fr); % output results at the surface and parameters
• [zs, ws, fs, SWP, W, F, P, A] = SolitaryGravityWave(Fr,Z); % surface and bulk output
• [zs, ws, fs, SWP, W, F, P, A] = SolitaryGravityWave(Fr,Z,1);

Input:

• Fr : Froude number (must be a scalar).
• Z : Complex abscissa where fields are desired inside the fluid (default Z = []).
  • Z should be strictly below the surface, i.e., -1 <= imag(Z) < eta(real(Z))
  • y = eta(x) being the equation of the free surface.
• PF : Plot Flag. If PF = 1 the final results are plotted, if PF ~= 1 nothing is plotted (default).

Output (dimensionless quantities):

• zs : Complex abscissa at the surface, i.e., x + i*eta.
• ws : Complex velocity at the surface, i.e., u - i*v.
• fs : Complex potential at the surface, i.e., phi + i*psi.
• SWP : Solitary Wave Parameters, i.e.
  • SWP(1) = wave amplitude, max(eta)
  • SWP(2) = wave mass
  • SWP(3) = circulation
  • SWP(4) = impulse
  • SWP(5) = kinetic energy
  • SWP(6) = potential energy
• W : Complex velocity in the bulk at abscissas Z.
• F : Complex potential in the bulk at abscissas Z.
• P : Pressure in the bulk at abscissas Z.
• A : Complex acceleration in the bulk at abscissas Z (A = dW/dt).

Example:

zs = SolitaryGravityWave(1.25); plot(real(zs), imag(zs));

Parametrization by the amplitude:

The script SolitaryGravityAmplitude.m can be used in a similar way with the only difference is that the first parameter is the wave amplitude/depth ratio which has to be less than 0.796, as before.

References:

More details on the methods used in these scripts can be found in the following references:

• D. Clamond & D. Dutykh. Fast accurate computation of the fully nonlinear solitary surface gravity waves. Computers & Fluids, 84, 35-38, 2013

• D. Dutykh & D. Clamond. Efficient computation of steady solitary gravity waves. Wave Motion, 51, 86-99, 2014

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Authors

D. Dutykh & D. Clamond
www.denys-dutykh.com
math.unice.fr/~didierc/

Licence

Open source under the MIT licence.