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What does "radial solution" for the wave equation mean?

E.g. if I am to find a radial solution for

$$ \left\{\begin{array}{rcl} u_{tt}-u_{xx}&=&0 \\ u(x,0)&=&0 \\ u_t(x,0)&=&e^{-|x|^2} \end{array}\right. $$

Teddy38
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mavavilj
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    It's not my area, but I believe that "radial solution" refers to a solution that is a function only of $|x|$ and $t$. That is, the solution's $x$ component will depend only on how far $x$ is from $0$, and not the direction $x$ is in. – Theo Bendit Nov 14 '17 at 03:30

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Theo is correct: a radial solution of an evolution equation posed in $\mathbb{R}^d$ is a solution $u(t,x)$ whose spatial dependence is only on the magnitude of the spatial variable, i.e. $u(t,x) = u(t,|x|)$. In the particular example provided, since the spatial dimension is $1$, you are looking for a solution that is an even function.

Gyu Eun Lee
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  • Okay and there's some formulation for radially symmetric wave equation? – mavavilj Nov 14 '17 at 18:55
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    @mavavilj I don't understand your question. Could you elaborate? – Gyu Eun Lee Nov 16 '17 at 05:49
  • How does one discover radial solution to the wave equation? – mavavilj Nov 16 '17 at 09:14
  • @mavavilj To find radial solutions in general, you can change to polar coordinates, which involves writing the Laplacian in polar coordinates, and turn the PDE into an ODE. In the specific example you provided, the question is moot, because you are also given the initial data, and you can solve the IVP for the wave equation on $\mathbb{R}^d$ using d'Alembert's formula. – Gyu Eun Lee Nov 17 '17 at 04:30