I Normal vector on complex function

1. Apr 5, 2016

MarkoA

Hi,

I'm not sure about the the normal vector N on a complex function
$$z(x,t) = A e^{i(\omega t + \alpha x)}$$

My approach is that ($\overline{z}$ beeing the conjugate of $z$):
$$\Re{(\mathbf{N})} = \frac{1}{\sqrt{\frac{1}{4}(\partial x + \overline{\partial x} )^2 + \frac{1}{4}(\partial z + \overline{\partial z} )^2 }} \begin{bmatrix} -\frac{1}{2}(\partial z + \overline{\partial z}) \\ \frac{1}{2}(\partial x + \overline{\partial x}) \end{bmatrix}$$

and
$$\Im{(\mathbf{N})} = \frac{1}{\sqrt{-\frac{1}{4}(\partial x - \overline{\partial x} )^2 -\frac{1}{4}(\partial z - \overline{\partial z} )^2 }} \begin{bmatrix} -\frac{1}{2i}(\partial z - \overline{\partial z}) \\ \frac{1}{2i}(\partial x - \overline{\partial x}) \end{bmatrix}$$

So I have $\frac{\partial z}{\partial x} = i\omega A e^{i(\omega t + \alpha x)}$. Do I now choose $\partial x = 1 + i$ for the complex x-component? Can I see the imaginary part of the normal vector as the normal vector on the imaginary part of z? Or is my approach wrong?

Thanks!

2. Apr 5, 2016

Ssnow

Your approach seems interesting, personally with the identification of $\mathbb{R}^{2}$ with $\mathbb{C}$ the function is $z(x,t)=A(\cos{(\omega t+\alpha x)},\sin{(\omega t+\alpha x)})$, from $\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ and the procedure is real ... but I didn't try ... , I don't know your procedure I must examine in details but if its works its ok...

3. Apr 5, 2016

BvU

Hello,

Good question ? I didn't know such a thing existed and can't find it (at least not here).
Could you tell us what you think constitutes a normal vector to a complex function ?

4. Apr 5, 2016

MarkoA

Actually the real part of N looks good (plotting it in Matlab). However, I'm not sure about my imaginary part. It's more like a trial and error approach...

To the background: I want to compute the the aerodynamic work on a traveling wave. I have a complex pressure $p$ and surface displacement $z$. Therefore, I first need the aerodynamic force $\textbf{f} = \oint_A p \mathbf{n} dA$. My problem is now that I need a correct normal vector $\mathbf{n}$.

Edit: Sorry, N and n are the same...

Last edited: Apr 5, 2016