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PDEs: reducing wave equation

  1. May 13, 2012 #1

    K29

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    1. The problem statement, all variables and given/known data

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


    3. The attempt at a solution

    [itex]\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}[/itex] (chain rule)
    [itex]\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})[/itex]
    After some use of chain rule and product rule I get:
    [itex]\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}}[/itex] (1)
    Similarly
    [itex]\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}}[/itex] (2)
    Making the substitutions:
    [itex]\frac{\partial \xi}{\partial t}=-\alpha[/itex], [itex]\frac{\partial^{2}\xi}{\partial t^{2}}=0[/itex]
    [itex]\frac{\partial \xi}{\partial x}=1[/itex], [itex]\frac{\partial^{2}\xi}{\partial x^{2}}=0[/itex]
    and
    [itex]\frac{\partial \eta}{\partial t}=\alpha[/itex], [itex]\frac{\partial^{2}\xi}{\partial t^{2}}=0[/itex]
    [itex]\frac{\partial \eta}{\partial x}=1[/itex], [itex]\frac{\partial^{2}\xi}{\partial x^{2}}=0[/itex]

    Substituting these into (1) and (2) and substituting (1) and (2) into the original wave equation we get:
    [itex]\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[/itex]

    [itex]0=0[/itex]

    Has something gone wrong here? Please help. Thanks
     
  2. jcsd
  3. May 13, 2012 #2

    tiny-tim

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    Homework Helper

    Hi K29! :wink:

    I don't follow this. :confused:

    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. :smile:
     
  4. May 13, 2012 #3
    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:

    [tex]\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}[/tex]

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

    [tex]\frac{\partial}{\partial t}\left(\frac{\partial u}{\partial \xi}\right)[/tex]

    What exactly is that?
     
  5. May 13, 2012 #4

    K29

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    Thanks for the help. Solved :)
     
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