Streamlines - Continuum mechanics

[Matt atkinson]

Homework Statement

In Cartesian coordinates $x$, $y$, where $x$ is the horizontal and $y$ the vertical coordinate,
the velocity in a small-amplitude standing surface wave on water of depth $h$ is given
by;
$$v_x = v_0 sin(\omega t) cos(kx) cosh[k(y + h)]$$
$$v_y = v_0 sin(\omega t) sin(kx) sinh[k(y + h)]$$
where $v_0$, $\omega$ and $k$ are constants. Find the equation of streamlines written in the
form $F(x, y) = const$.

Homework Equations

$$\frac{dx_i}{d\lambda}=v_i (\lambda,t)$$

The Attempt at a Solution

Look being honest I have no idea what to do, I noticed that;
$$\frac{dx}{d\lambda}=v_0 sin(\omega t) cos(kx) cosh[k(y + h)]$$
$$\frac{dy}{d\lambda}=v_0 sin(\omega t) sin(kx) sinh[k(y + h)]$$
I tried doing;
$$\frac{dy}{d\lambda} \frac{d\lambda}{dx}=\frac{1}{tan(kx)tanh[k(y+h)]}$$
I don't believe that is the correct way to do it, I think I am supposed to try and write them as parametric equations but I am not sure how.

 

[pasmith]

If $F$ is to be constant on streamlines, then its gradient must be orthogonal to $v$. Thus you need to solve $$\frac{\partial F}{\partial x} = v_y, \\ \frac{\partial F}{\partial y} = -v_x.$$

 

[Matt atkinson]

Oh thankyou! it appears i wasnt thinking about streamlines at all, not sure what i was trying to do