# Wave dispersion relation and parametric representation of a streched circle

1. Jul 1, 2009

### ScribeFI

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 $$\vec{U}(U_x, U_y)$$. 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