2-d laplace equation with corrugated isothermal boundary

Question

Consider a 2-d laplace equation $\Delta\Theta(x,z)=0$ with a corrugated boundary $ \Theta(x,f(x))=\Theta_0$. You can assume $f(x)$ to be a sinusoidal function.

1.My idea is to set $p=z-f(x)$. But then how to transform my laplace equation into the new coordinate $(x,p)$? Kind of get lost in applying the chain rules.

2.If there is a better way to solve this problem please do let me know.

Thanks a lot.

Answer 1

Using the transformation $(x,z) \to (\xi,\eta)$, you must re-write the Laplace's equation in terms of the new variables by using the chain rule:

$$\begin{align} \partial_x\Theta & = \frac{\partial \Theta }{\partial \xi} \frac{\partial \xi}{\partial x} + \frac{\partial \Theta}{\partial \eta} \frac{\partial \eta}{\partial x}, \\ \partial_x\Theta & = \frac{\partial \Theta }{\partial \xi} \frac{\partial \xi}{\partial z} + \frac{\partial \Theta}{\partial \eta} \frac{\partial \eta}{\partial z}, \\ \partial_{xx} \Theta = \partial_x (\partial_x \Theta) & = \partial_x \left( \frac{\partial \Theta }{\partial \xi} \frac{\partial \xi}{\partial x} + \frac{\partial \Theta}{\partial \eta} \frac{\partial \eta}{\partial x} \right) \\ & = \left( \frac{\partial \xi}{\partial x} \right)^2 \frac{\partial^2 \Theta}{\partial \xi^2} + 2 \frac{\partial \xi}{\partial x} \frac{\partial \eta}{\partial x} \frac{\partial \Theta }{\partial \xi \partial \eta} + \left(\frac{\partial \eta}{\partial x} \right)^2 \frac{\partial^2 \Theta}{\partial^2 \eta}, \\ \partial_{zz} \Theta = \partial_z (\partial_z \Theta) & = \partial_z \left( \frac{\partial \Theta }{\partial \xi} \frac{\partial \xi}{\partial z} + \frac{\partial \Theta}{\partial \eta} \frac{\partial \eta}{\partial z} \right) \\ & = \left( \frac{\partial \xi}{\partial z} \right)^2 \frac{\partial^2 \Theta}{\partial \xi^2} + 2 \frac{\partial \xi}{\partial z} \frac{\partial \eta}{\partial z} \frac{\partial \Theta }{\partial \xi \partial \eta} + \left(\frac{\partial \eta}{\partial z} \right)^2 \frac{\partial^2 \Theta}{\partial^2 \eta}. \end{align} $$ Substitute your data in these equations by making $\xi = x$, $\eta = z - f(x)$. I'm sure you can take it from here.

I hope this is useful.

Cheers!