1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: PDEs: reducing wave equation

  1. May 13, 2012 #1


    User Avatar

    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)
    [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]
    [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]


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


    User Avatar
    Science Advisor
    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


    User Avatar

    Thanks for the help. Solved :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook