Transform a pde into rotating frame

[climbon]

Hi,

I have an equation of the form;

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

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;

$$\frac{\partial}{\partial x} = -cos(\omega t) \frac{\partial}{\partial x'} - sin(\omega t) \frac{\partial}{\partial y'} \\ \frac{\partial}{\partial y} = -cos(\omega t) \frac{\partial}{\partial y'} + sin(\omega t) \frac{\partial}{\partial x'} \\ \frac{\partial^2}{\partial x^2} = -cos^(2)(\omega t) \frac{\partial^2}{\partial x'^2} - sin^(2)(\omega t) \frac{\partial^2}{\partial y'^2}$$

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.

 

[gtoguy97]

Yes, you need to take into account the time derivative in the transformation. The time derivative should be transformed using the chain rule as follows: \frac{d}{dt}=\frac{\partial x'}{\partial t}\frac{\partial}{\partial x'}+\frac{\partial y'}{\partial t}\frac{\partial}{\partial y'}\\= -\omega(x'\frac{\partial}{\partial x'}+y'\frac{\partial}{\partial y'})Substituting the above derivatives and the transforms into the original equation will give the equation in the rotating frame.