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Wave dispersion relation and parametric representation of a streched circle

  1. Jul 1, 2009 #1
    Wave dispersion relation and parametric representation of a "streched" circle


    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 [tex]\vec{U}(U_x, U_y)[/tex]. Assuming "infinite" depth d of water [tex]\tanh{(\|k\|d)}[/tex] can be approximated by 1 and the dispersion relation becomes :

    [tex]\omega_0 = \sqrt{g\|k\|} + \vec{k}.\vec{U}[/tex]

    where :
    [tex]\omega_0[/tex] is the wave pulsation
    [tex]g[/tex] the gravitational constant
    [tex]\vec{k}(k_x, k_y)[/tex] the wave number vector and [tex]\|k\| = \sqrt{k_x^2 + k_y^2}[/tex]
    [tex]\vec{U}(U_x, U_y)[/tex] the surface current, with components assumed to be constant and known.

    In the 2D space [tex]k_x, k_y[/tex] (slice for a given [tex]\omega_0[/tex] in 3D space [tex](k_x, k_y, \omega_0)[/tex]) and without current U the dispersion relation is a circle with parametric representation :

    [tex]k_x(t) = r.cos(\theta)[/tex] and [tex]k_y(t) = r.sin(\theta)[/tex], with [tex]r = \frac{\omega_0^2}{g}[/tex] and [tex]\theta[/tex] 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 [tex]\vec{U}[/tex] 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 [tex]\sqrt{k_x^2 . U_x^2 + k_y^2 . U_y^2}[/tex]

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

  2. jcsd
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