Transform a pde into rotating frame

  1. Hi,

    I have an equation of the form;

    [tex]
    \frac{d}{dt}(W) = \omega \left(x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right) W + g \frac{\partial}{\partial y} W + k x \frac{\partial^2}{\partial y^2} W
    [/tex]

    I want to change it into the rotating frame using the transform;

    x = x' cos(wt) - y' sin(wt)
    y = x' sin(wt) + y' cos(wt)

    I have calculated the derivatives of these transforms to be;

    [tex]
    \frac{\partial}{\partial x} = -cos(\omega t) \frac{\partial}{\partial x'} - sin(\omega t) \frac{\partial}{\partial y'}
    \\
    \frac{\partial}{\partial y} = -cos(\omega t) \frac{\partial}{\partial y'} + sin(\omega t) \frac{\partial}{\partial x'}
    \\
    \frac{\partial^2}{\partial x^2} = -cos^(2)(\omega t) \frac{\partial^2}{\partial x'^2} - sin^(2)(\omega t) \frac{\partial^2}{\partial y'^2}
    [/tex]

    I am assuming I can just substitute these transforms for x, y and their derivatives into the original equation and this will give me the original equation in the rotating frame...but do I have to do something with the time derivative on the L.H.S of the original equation??

    Thank you.
     
  2. jcsd
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