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Math Help - wave equation

  1. #1
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    wave equation

    Let f and g be differentiable functions of one variable

    set \phi = f(x-t)+g(x+t)

    a) prove that \phi satisfies the wave equation : \frac{\partial^2 \phi}{\partial t^2} = \frac{\partial^2 \phi}{\partial x^2}

    b) sketch the graph of \phi against t and x if f(x)=x^2 and g(x)=0

    is part a) as simple as ( I can't see it being) f''(x-t)+g''(x+t)=f''(x-t)+g''(x+t)
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  2. #2
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    Re: wave equation

    Sort of. You need to evaluate $\displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial t^2} \end{align*}$ and $\displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial x^2} \end{align*}$ and show they're equal, which is what you have tried to do. But you just need to set it out better. Also you can't use Newton's notation " ' " for a derivative here, because these are partial derivatives...
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    Re: wave equation

    Quote Originally Posted by Prove It View Post
    Sort of. You need to evaluate $\displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial t^2} \end{align*}$ and $\displaystyle \begin{align*} \frac{\partial ^2 \phi}{\partial x^2} \end{align*}$ and show they're equal, which is what you have tried to do. But you just need to set it out better. Also you can't use Newton's notation " ' " for a derivative here, because these are partial derivatives...
    \frac{ \partial^2 f}{\partial t^2} + \frac{\partial^2 g}{\partial t^2} = \frac{ \partial^2 f}{\partial x^2} + \frac{\partial^2 g}{\partial x^2}

    like that??
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  4. #4
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    Re: wave equation

    And now you actually have to show that these things are equal...
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  5. #5
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    Re: wave equation

    Quote Originally Posted by Prove It View Post
    And now you actually have to show that these things are equal...
    is that

    f_{xx} = 0; f_{tt} = 0; g_{xx} = 0; g_{tt} = 0;

    so 0+0=0+0 ??
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