Normal vector on complex function

[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!

 

[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...

 

[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 ?

 

[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...