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Math Help - 1d wave eqution (help urgent)

  1. #1
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    Exclamation 1d wave eqution (help urgent)

    1d wave eqution (help urgent)-untitled.png

    Basically, I have to use u = Real (f(x)e^-iwt) and sub it into the wave equation.

    Simplify it etc,

    then use say u= e^at to solve the characteristics equation.

    However, I get stuck.

    Can someone show me how to do it.

    Thank you
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  2. #2
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    Re: 1d wave eqution (help urgent)

    Quote Originally Posted by princessmath View Post
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    Basically, I have to use u = Real (f(x)e^-iwt) and sub it into the wave equation.

    Simplify it etc,

    then use say u= e^at to solve the characteristics equation.

    However, I get stuck.

    Can someone show me how to do it.

    Thank you
    If you assume

    u(x,t)=f(x)e^{-i \omega t} \implies u_{tt}=(i\omega)^2f(x)e^{-i\omega t}

    and

    u_{x}= f'(x)e^{-i \omega t} \text{ and } u_{xx}=f''(x)e^{-i \omega t}


    Now just expand out the right hand side of the eqation

    \frac{\partial }{\partial x} \left( A(x) \frac{\partial u}{\partial x}\right) = \frac{\partial A}{\partial x}\frac{\partial u}{\partial x}+A(x)\frac{\partial^2 u}{\partial x^2}=A'u'+Au''

    Now just put all of this into the equation

    (i \omega)^2f(x)e^{-i \omega t} = \frac{c^2}{A}\left(A'f'e^{- i \omega t}+f''e^{-i \omega t} \right)

    Can you finish from here?
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  3. #3
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    Re: 1d wave eqution (help urgent)

    Yeah, that what I did. Btw you forgot the A(x) next to f''e^-iwt

    You can cancel e^-iwt etc.

    But I'm stuck on how do I find f(x).
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  4. #4
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    Re: 1d wave eqution (help urgent)

    In short, I'm stuck on how do I solve the second order differential equation.
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  5. #5
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    Re: 1d wave eqution (help urgent)

    Quote Originally Posted by princessmath View Post
    In short, I'm stuck on how do I solve the second order differential equation.
    The equation can be written in the form

    -\frac{d}{dx}\left( A(x)\frac{df(x)}{dx}\right)=\frac{\omega^2}{c^2}f(  x)

    The is a Sturm-Liouville ODE (it is in its self-adjoint form). Does the equation have boundary conditions?
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  6. #6
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    Re: 1d wave eqution (help urgent)

    No, the question doesn't have any boundary conditions.

    Just said to find the ode f(x).
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  7. #7
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    Re: 1d wave eqution (help urgent)

    How would you go about solving that pde?

    See what I get is

    A(x)F''(x) + A'(x)F'(x) + w^2/c^2 F(x) = 0

    and then I try letting m = e^bx

    then solving for the charateristic, but then I get confused
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