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D'Alembert question - boundary conditions parts

  1. Aug 29, 2012 #1
    1. The problem statement, all variables and given/known data

    I have a general wave equation on the half line
    utt-c2uxx=0
    u(x,0)=α(x)
    ut(x,0)=β(x)
    and the boundary condition;
    ut(0,t)=cηux
    where α is α extended as an odd function to the real line (and same for β)
    I have to find the d'alembert solution for x>=0; and show that in general it doesn't exist for η=-1
    2. Relevant equations

    The d'alembert solution is
    u(x,t)=1/2(α(x+ct)+α(x-ct))+1/2c [itex]\int[/itex]x+ctx-ctβ(y) dy
    for x>ct
    3. The attempt at a solution

    I know that to restrict it to the whole of the x>=0, t>=0 region, I need to use the boundary condition; but I get that
    u(0,t)=0 because α and β are odd, which makes α(ct)+(-ct)
    and the integral from -ct to ct of β(y) zero; and so u_t(0,t) is zero which is supremely not useful...
     
  2. jcsd
  3. Aug 30, 2012 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    Are you sure you copied down the problem statement correctly? It would make a lot more sense if [itex]\beta[/itex] were an even function, and your boundary condition was [itex]u_t(0,t)=c\eta u_x(0,t)[/itex] for some constant [itex]\eta[/itex].
     
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