Wave dispersion relation and parametric representation of a "streched" circle

**ScribeFI** · Jul1-09, 07:35 AM

Hello,

Not sure if that question goes better into mathematics or physics section.

It's related to the dispersion relation of a wave in ocean with current $LaTeX Code: \\vec{U}(U_x, U_y)$ . Assuming "infinite" depth d of water $LaTeX Code: \\tanh{(\\|k\\|d)}$ can be approximated by 1 and the dispersion relation becomes :

$LaTeX Code: \\omega_0 = \\sqrt{g\\|k\\|} + \\vec{k}.\\vec{U}$

where :
$LaTeX Code: \\omega_0$ is the wave pulsation

the gravitational constant
$LaTeX Code: \\vec{k}(k_x, k_y)$ the wave number vector and $LaTeX Code: \\|k\\| = \\sqrt{k_x^2 + k_y^2}$
$LaTeX Code: \\vec{U}(U_x, U_y)$ the surface current, with components assumed to be constant and known.

In the 2D space

(slice for a given $LaTeX Code: \\omega_0$ in 3D space $LaTeX Code: (k_x, k_y, \\omega_0)$ ) and without current U the dispersion relation is a circle with parametric representation :

$LaTeX Code: k_x(t) = r.cos(\\theta)$ and $LaTeX Code: k_y(t) = r.sin(\\theta)$ , with $LaTeX Code: r = \\frac{\\omega_0^2}{g}$ and $LaTeX Code: \\theta$ the angle between x axis and the given point on the circle (counter-clockwise).

What does the parametric equations become when current U is not null ? The figure must ressemble a sort of circle "stretched" along the $LaTeX Code: \\vec{U}$ direction, as seen in a scientific publication. I would need the equations to be able to plot this curve in Matlab, but cannot find a way to parametrize the dispersion relation with the additionnal $LaTeX Code: \\sqrt{k_x^2 . U_x^2 + k_y^2 . U_y^2}$

Thanks in advance for your help or any hint. Sorry for misalignment of formulas and text, don't know what is going on.

Matthieu