Reducing the Wave Equation: Change of Variables

[K29]

Homework Statement

Show that the wave equation $u_{tt}-\alpha^{2}u_{xx}=0$ can be reduced to the form $\phi_{\xi \eta}=0$ by the change of variables
$\xi=x-\alpha t$
$\eta=x+\alpha t$

The Attempt at a Solution

$\frac{\partial u}{\partial t}=\frac{\partial \xi}{\partial t}\frac{\partial \phi}{\partial \xi}+\frac{\partial \eta}{\partial t}\frac{\partial \phi}{\partial \eta}$ (chain rule)
$\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial}{\partial t}(\frac{\partial \xi}{\partial t}\frac{\partial \phi}{\partial \xi} + \frac{\partial \eta}{\partial t} \frac{\partial \phi}{\partial \eta})$
After some use of chain rule and product rule I get:
$\frac{\partial^{2} u}{\partial t^{2}}= \frac{\partial^{2}\xi}{\partial t^{2}}\frac{\partial\phi}{\partial\xi} + \left(\frac{\partial \xi}{\partial t}\right)^{2} \frac{\partial^{2}\phi}{\partial \xi^{2}} + \frac{\partial^{2}\eta}{\partial t^{2}}\frac{\partial\phi}{\partial\eta} + \left(\frac{\partial \eta}{\partial t}\right)^{2} \frac{\partial^{2}\phi}{\partial \eta^{2}}$ (1)
Similarly
$\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial\phi}{\partial\xi} + \left(\frac{\partial \xi}{\partial x}\right)^{2}\frac{\partial^{2}\phi}{\partial \xi^{2}}+\frac{\partial^{2}\eta}{\partial x^{2}}\frac{\partial\phi}{\partial\eta} + \left(\frac{\partial \eta}{\partial x}\right)^{2}\frac{\partial^{2}\phi}{\partial \eta^{2}}$ (2)
Making the substitutions:
$\frac{\partial \xi}{\partial t}=-\alpha$, $\frac{\partial^{2}\xi}{\partial t^{2}}=0$
$\frac{\partial \xi}{\partial x}=1$, $\frac{\partial^{2}\xi}{\partial x^{2}}=0$
and
$\frac{\partial \eta}{\partial t}=\alpha$, $\frac{\partial^{2}\xi}{\partial t^{2}}=0$
$\frac{\partial \eta}{\partial x}=1$, $\frac{\partial^{2}\xi}{\partial x^{2}}=0$

Substituting these into (1) and (2) and substituting (1) and (2) into the original wave equation we get:
$\alpha^{2}\frac{\partial^{2}\phi}{\partial \xi^{2}}+\alpha^{2}\frac{\partial^{2} \phi}{\partial \eta^{2}}-\alpha^{2}\left(\frac{\partial^{2}\phi}{\partial \xi^{2}}+\frac{\partial^{2}\phi}{\partial\eta^{2}}\right)=0$

$0=0$

Has something gone wrong here? Please help. Thanks

 

[tiny-tim]

Hi K29!

I don't follow this.

First, u is not necessarily equal to φ … keep with u until the end!

Second, you seem to have used the chain rule for ∂/∂t (and ∂/∂x) the first time, but not for the second time (which you haven't copied)

Start again. ​

 

[jackmell]

Homework Statement

Show that the wave equation $u_{tt}-\alpha^{2}u_{xx}=0$ can be reduced to the form $\phi_{\xi \eta}=0$ by the change of variables
$\xi=x-\alpha t$
$\eta=x+\alpha t$

The Attempt at a Solution

$\frac{\partial u}{\partial t}=\frac{\partial \xi}{\partial t}\frac{\partial \phi}{\partial \xi}+\frac{\partial \eta}{\partial t}\frac{\partial \phi}{\partial \eta}$ (chain rule)
$\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial}{\partial t}(\frac{\partial \xi}{\partial t}\frac{\partial \phi}{\partial \xi} + \frac{\partial \eta}{\partial t} \frac{\partial \phi}{\partial \eta})$
After some use of chain rule and product rule I get:
$\frac{\partial^{2} u}{\partial t^{2}}= \frac{\partial^{2}\xi}{\partial t^{2}}\frac{\partial\phi}{\partial\xi} + \left(\frac{\partial \xi}{\partial t}\right)^{2} \frac{\partial^{2}\phi}{\partial \xi^{2}} + \frac{\partial^{2}\eta}{\partial t^{2}}\frac{\partial\phi}{\partial\eta} + \left(\frac{\partial \eta}{\partial t}\right)^{2} \frac{\partial^{2}\phi}{\partial \eta^{2}}$ (1)
Similarly
$\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial\phi}{\partial\xi} + \left(\frac{\partial \xi}{\partial x}\right)^{2}\frac{\partial^{2}\phi}{\partial \xi^{2}}+\frac{\partial^{2}\eta}{\partial x^{2}}\frac{\partial\phi}{\partial\eta} + \left(\frac{\partial \eta}{\partial x}\right)^{2}\frac{\partial^{2}\phi}{\partial \eta^{2}}$ (2)
Making the substitutions:
$\frac{\partial \xi}{\partial t}=-\alpha$, $\frac{\partial^{2}\xi}{\partial t^{2}}=0$
$\frac{\partial \xi}{\partial x}=1$, $\frac{\partial^{2}\xi}{\partial x^{2}}=0$
and
$\frac{\partial \eta}{\partial t}=\alpha$, $\frac{\partial^{2}\xi}{\partial t^{2}}=0$
$\frac{\partial \eta}{\partial x}=1$, $\frac{\partial^{2}\xi}{\partial x^{2}}=0$

Substituting these into (1) and (2) and substituting (1) and (2) into the original wave equation we get:
$\alpha^{2}\frac{\partial^{2}\phi}{\partial \xi^{2}}+\alpha^{2}\frac{\partial^{2} \phi}{\partial \eta^{2}}-\alpha^{2}\left(\frac{\partial^{2}\phi}{\partial \xi^{2}}+\frac{\partial^{2}\phi}{\partial\eta^{2}}\right)=0$

$0=0$

Has something gone wrong here? Please help. Thanks

How come you not getting any mixed-partials in there. I don't see a single one. Gonna' need some right? I think I know what the problem is but not sure but it's one that gets lots of students and also, I don't know about you but that phi thing just gets in the way for me. Isn't it really just:

$$\frac{\partial u}{\partial t}=\frac{\partial u}{\partial \xi} \frac{\partial \xi}{\partial t}+\frac{\partial u}{\partial \eta} \frac{\partial \eta}{\partial t}$$

Now when you do the second partial you get terms like:

$$\frac{\partial}{\partial t}\left(\frac{\partial u}{\partial \xi}\right)$$

What exactly is that?

 

[K29]

Thanks for the help. Solved :)