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 \vec{U}(U_x, U_y). Assuming "infinite" depth d of water \tanh{(\|k\|d)} can be approximated by 1 and the dispersion relation becomes :
\omega_0 = \sqrt{g\|k\|} + \vec{k}.\vec{U}
where :
\omega_0 is the wave pulsation
g the gravitational constant
\vec{k}(k_x, k_y) the wave number vector and \|k\| = \sqrt{k_x^2 + k_y^2}
\vec{U}(U_x, U_y) the surface current, with components assumed to be constant and known.
In the 2D space k_x, k_y (slice for a given \omega_0 in 3D space (k_x, k_y, \omega_0)) and without current U the dispersion relation is a circle with parametric representation :
k_x(t) = r.cos(\theta) and k_y(t) = r.sin(\theta), with r = \frac{\omega_0^2}{g} and \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 \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 \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
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 \vec{U}(U_x, U_y). Assuming "infinite" depth d of water \tanh{(\|k\|d)} can be approximated by 1 and the dispersion relation becomes :
\omega_0 = \sqrt{g\|k\|} + \vec{k}.\vec{U}
where :
\omega_0 is the wave pulsation
g the gravitational constant
\vec{k}(k_x, k_y) the wave number vector and \|k\| = \sqrt{k_x^2 + k_y^2}
\vec{U}(U_x, U_y) the surface current, with components assumed to be constant and known.
In the 2D space k_x, k_y (slice for a given \omega_0 in 3D space (k_x, k_y, \omega_0)) and without current U the dispersion relation is a circle with parametric representation :
k_x(t) = r.cos(\theta) and k_y(t) = r.sin(\theta), with r = \frac{\omega_0^2}{g} and \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 \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 \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