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PDE: Nontrivial solution to the wave equation

  1. Feb 20, 2016 #1


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    1. The problem statement, all variables and given/known data
    Consider the wave equation:

    u_{tt} - c^2u_{xx} = f(x,t),
    0 < x < l \\
    u(0,t) = 0 = u(l,t) \\
    u(x,0) = g(x), u_t(x,0) = f(x) \\
    Find a nontrivial solution.
    2. Relevant equations

    3. The attempt at a solution

    Here's what I did, but I have little understanding of it other than I know that I am using boundary conditions and some previous material to get here:

    We form a series solution:

    u(x,t) = \sum_{n=1}^{\infty} u_n(x,t) = \sum_{n=1}^{\infty} sin{\frac{nπx} {l}}u_n(t) \\
    g(x) = \sum_{n=1}^{\infty}b_nsin{\frac{nπx}{l}} \\
    f(x) = \sum_{n=1}^{\infty} \frac{cnπb_n*}{l} sin{\frac{nπx}{l}} \\
    b_n = \frac{2}{l} \int_{0}^{l} g(x)sin{\frac{nπx}{l}}dx \\
    b_n* = \frac{2}{l} \int_{0}^{l} f(x)sin{\frac{nπx}{l}}dx \\

    All together:

    u(x,t) = \sum_{n=1}^{\infty} sin{\frac{nπx} {l}} [b_n cos(\frac{cnπt}{l}) + b_n*sin(\frac{cnπt}{l})]

    From my understanding, inputting b_n and b_n* would give us the nontrivial solution. Yes?
  2. jcsd
  3. Feb 20, 2016 #2


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    Gold Member

    Yes, although you should consider using the notation ##b_n^*## instead, it would be less confusing.

    If you would like a better understanding of what is going on, I suggest reading up on function spaces and orthogonal functions.
  4. Feb 20, 2016 #3


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    Gold Member

    Thank you kindly for the confirmation and suggestion.
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