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I Normal vector on complex function

  1. Apr 5, 2016 #1
    Hi,

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

    My approach is that ([itex]\overline{z}[/itex] beeing the conjugate of [itex]z[/itex]):
    [tex]
    \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}
    [/tex]

    and
    [tex]
    \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}
    [/tex]

    So I have [itex] \frac{\partial z}{\partial x} = i\omega A e^{i(\omega t + \alpha x)} [/itex]. Do I now choose [itex]\partial x = 1 + i[/itex] 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. jcsd
  3. Apr 5, 2016 #2

    Ssnow

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    Gold Member

    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...
     
  4. Apr 5, 2016 #3

    BvU

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    Science Advisor
    Homework Helper
    Gold Member

    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 ?
     
  5. Apr 5, 2016 #4
    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 [itex]p[/itex] and surface displacement [itex]z[/itex]. Therefore, I first need the aerodynamic force [itex]\textbf{f} = \oint_A p \mathbf{n} dA[/itex]. My problem is now that I need a correct normal vector [itex]\mathbf{n}[/itex].

    Edit: Sorry, N and n are the same...
     
    Last edited: Apr 5, 2016
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