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Math Help - Solving 1-d wave equation

  1. #1
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    Solving 1-d wave equation

    Hey guys,
    i have a small question. 1D-Wave equation is given as:

    <br />
\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}

    i need to find u_s(x,t)=X(x)sinwt

    From the equation form above , U_xx= X''(x)sinwt
    U_tt= -w^2X(x)sinwt are obtained as given as a solution.

    My question is how -w^2 is obtained from u_s(x,t)=X(x)sinwt

    thank you,
    Last edited by Ali Sura; March 29th 2009 at 04:20 PM.
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  2. #2
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    What are your boundary conditions? That is what determines k.
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  3. #3
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    Two Boundary conditions are:

    1)
    u(0,t)=0<br />


    2)
    u(1,t)=sinwt<br />

    thanks,
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  4. #4
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    If u_s(x,t)= X(x)sin(\omega t)

    Then u_s(0,t)= X(0)sin(\omega t)= 0 so X(0)= 0.

    Also u_s(1,t)= X(1)sin(\omega t)= sin(\omega t) so X(1)= 1.

    Now, u_xx= X_xx sin(\omega t) and u_tt= -\omega^2 X sin(\omega t) so the partial differential equation becomes X_xx sin(\omega t)=  -\omega^2 X sin(\omega t) for all x and t and so we must have X_xx= -\omega^2 X for all x.\

    Can you solve the ordinary differential equation X_xx+ \omega^2 X with the boundary conditions X(0)= 0, X(1)= 1?
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