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## Solitary Water Wave

version 1.3.0.0 (10.7 KB) by Didier Clamond

### Didier Clamond (view profile)

Solitary gravity wave solution of the free surface Euler equations

Updated 05 Mar 2014

Computes 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.
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))

Edit the m-file for more details and
look at the reprint:
http://hal.archives-ouvertes.fr/hal-00786077/