Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    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 [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.

    Matthieu
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Wave dispersion relation and parametric representation of a streched circle
  1. Parametrizing a curve (Replies: 4)

  2. Parametric equations (Replies: 4)

Loading...