# 1-D wave-equation and change of variables

## Homework Statement

Hi all.

I have the 1-D wave-equation, and I wish to make a change of variables, where a = x+ct and b = x-ct. I get:

$$\begin{array}{l} c^2 \frac{{\partial ^2 u}}{{\partial x^2 }} = c^2 \left[ {\frac{{d^2 u}}{{da^2 }}\left( {\frac{{da}}{{dx}}} \right)^2 + \frac{{du}}{{da}}\frac{{d^2 a}}{{dx^2 }}} \right] + c^2 \left[ {\frac{{d^2 u}}{{db^2 }}\left( {\frac{{db}}{{dx}}} \right)^2 + \frac{{du}}{{db}}\frac{{d^2 b}}{{dx^2 }}} \right] = c^2 \frac{{d^2 u}}{{da^2 }} + c^2 \frac{{d^2 u}}{{db^2 }} \\ \frac{{\partial ^2 u}}{{\partial t^2 }} = \left[ {\frac{{d^2 u}}{{da^2 }}\left( {\frac{{da}}{{dt}}} \right)^2 + \frac{{du}}{{da}}\frac{{d^2 a}}{{dt^2 }}} \right] + \left[ {\frac{{d^2 u}}{{db^2 }}\left( {\frac{{db}}{{dt}}} \right)^2 + \frac{{du}}{{db}}\frac{{d^2 b}}{{dt^2 }}} \right] = c^2 \frac{{d^2 u}}{{da^2 }} + c^2 \frac{{d^2 u}}{{db^2 }} \\ \end{array}$$

For this I have used the chain rule for higher derivates (for second derivates, from Wikipedia: http://en.wikipedia.org/wiki/Chain_rule). The result I wish to get is:

$$\frac{{\partial ^2 u}}{{\partial a\partial b}} = 0$$

I can't quite see how I would get this. Am I on the right track here?

Cheers,
Niles.

EDIT: Ok, now this is the second thread in a row I am doing this: I preview the thread, and the title resets itself. The original title was: "1-D wave-equation and change of variables". If a moderator can insert the proper title, I would be grateful.

cristo
Staff Emeritus
I've changed the title. Thanks for letting us know that it was the preview that causes the problem.

Ok, my first attempt is quite bad. Instead of using the expression from Wikipedia, I can just derive du/dx and du/dt with respect to dx and dt, respectively. But still, this gives me:

$$c^2 \frac{{\partial ^2 u}}{{\partial x\partial a}} + c^2 \frac{{\partial ^2 u}}{{\partial x\partial b}} = c\frac{{\partial ^2 u}}{{\partial t\partial a}} - c\frac{{\partial ^2 u}}{{\partial t\partial b}}$$

Is it possible to go from this to

$$\frac{{\partial ^2 u}}{{\partial a\partial b}} = 0$$
?