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Math Help - Wave Equation In Spherical Coordinates

  1. #1
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    Wave Equation In Spherical Coordinates

    Let u be the solution of the Initial Value Problem:

    \Delta u-u_{tt}=0
    u(x,0)=\phi (r)
    u_t(x,0)=\psi (r)

    Where \phi and \psi are continuous. Show that the solution can be represented in the region {r>0, t>0} as:

    u(r,t)=\frac{1}{2r}[(r+t)\phi(r+t)+(r-t)\psi(r-t)+\int_{r-t}^{r+t}\tau\psi(\tau)d\tau].

    I honestly have no idea where to start. :-(
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  2. #2
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    Quote Originally Posted by mathematicalbagpiper View Post
    Let u be the solution of the Initial Value Problem:

    \Delta u-u_{tt}=0
    u(x,0)=\phi (r)
    u_t(x,0)=\psi (r)

    Where \phi and \psi are continuous. Show that the solution can be represented in the region {r>0, t>0} as:

    u(r,t)=\frac{1}{2r}[(r+t)\phi(r+t)+(r-t)\psi(r-t)+\int_{r-t}^{r+t}\tau\psi(\tau)d\tau].

    I honestly have no idea where to start. :-(
    Hint:

    Use the change of coordinates

    \alpha=r-t \quad \beta=r+t

    and rewrite the PDE using

    u(\alpha,\beta)
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