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Semi-infinite wave equation

  1. Aug 26, 2010 #1
    1. The problem statement, all variables and given/known data

    solve the wave equation (dx^2 -dt^2)[tex]\Phi[/tex] = 0 on the semi-infinite line x<=0 with boundary conditions [tex]\Phi[/tex] at x=0 = 0 and initial conditions [tex]\Phi[/tex] at t=0 = tanh(x)

    2. Relevant equations

    solution of the wave equation is of the form [tex]\Phi[/tex] = f(x-t) + g(x+t).

    [tex]\Phi[/tex] at t=0 = tanh(x)
    tanh(x) is an odd function

    3. The attempt at a solution

    i know that [tex]\Phi[/tex] at t=0 = tanh(x) which i will call A
    i know that [tex]\Phi[/tex] at x=0 is 0, which i will call B

    am i correct in thinking that i can simply add the two equations together to get 2[tex]\Phi[/tex] = f(x-t) + g(x+t) = A + B
    therefore giving [tex]\Phi[/tex] = [tanh(x+t) tanh(x-t)]/2
    also what happens if tanh(x) is not an odd function? how can i turn it into an odd function?
  2. jcsd
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