Wave on a string and the chain rule Argh

[physmurf]

Wave on a string and the chain rule...Argh

So, I am working through the wave equation for a review before my friend and I go off to grad school. It has been a couple of years since we both graduated with our BS in Physics.

So, here is the question:

Suppose I want to solve the wave equation using a change of variables. Let's use $$\alpha = x+ct$$, and $$\beta = x-ct, and\: u = \alpha + \beta$$

The wave equation is
$$\frac{\partial^{2} u}{\partial t^{2}}=c^{2}\frac{\partial^{2} u}{\partial x^{2}}$$


Now, if we take the partial derivative of u with respect to x and applying the chain rule one gets:

$$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial \alpha}\frac{\partial \alpha}{\partial x}+\frac{\partial u}{\partial \beta}\frac{\partial \beta}{\partial x}$$

Now if we evaluate $$\frac{\partial \alpha}{\partial x}, and \frac{\partial \beta}{\partial x}$$ we get
$$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \alpha} + \frac {\partial u}{\partial \beta}$$

So, what and how do I evaluate the second partial differential with respect to x? I get

$$\frac{\partial^{2}u}{\partial x^{2}} = \frac{\partial^{2}u}{\partial \alpha^{2}}\ \frac{\partial \alpha}{\partial x}+\frac{\partial^{2}u}{\partial \beta^{2}}\ \frac{\partial \beta}{\partial x}$$

Now, I know this isn't quite right. I am supposed to get:
$$\frac{\partial^{2}u}{\partial x^{2}} = \frac{\partial^{2}u}{\partial \alpha^{2}}\ \frac{\partial \alpha}{\partial x}+ \frac{\partial^{2}u}{\partial \alpha \partial \beta}}\ \frac{\partial \beta}{\partial x} + \frac{\partial^{2}u}{\partial \beta \partial \alpha}}\ \frac{\partial \alpha}{\partial x} +\frac{\partial^{2}u}{\partial \beta^{2}}\ \frac{\partial \beta}{\partial x}$$

Can anyone help me? Thanks.

 

[Ben Niehoff]

Hmm...I think there is an error at the beginning. It seems to be that you ought to have

$$u = f(\alpha) + g(\beta)$$

as this would correspond to the general solution of the one-dimensional wave equation,

$$u = f(x - ct) + g(x + ct)$$

Then, taking derivatives, you get

$$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \alpha} \frac{\partial \alpha}{\partial x} + \frac{\partial u}{\partial \beta} \frac{\partial \beta}{\partial x}$$

and then

$$\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial \alpha} \left(\frac{\partial u}{\partial x}\right) \frac{\partial \alpha}{\partial x} + \frac{\partial}{\partial \beta} \left(\frac{\partial u}{\partial x}\right) \frac{\partial \beta}{\partial x}$$

I think this is where you made your mistake. You have to apply the chain rule twice in order to take the second derivative!