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I know that the characteristics for the 1-d space wave equation $u_{tt}=u_{xx}$ is $x=\pm t+c$. But what is the situation for 2-d space wave equation $u_{tt}=u_{xx}+u_{yy}$? Are the characteristics now hyperplanes?

henryforever14
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1 Answers1

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In higher spatial dimensions there are no characteristics. The information travels along to the light cone: $$ c^2t^2=x^2+y^2. $$

Yiorgos S. Smyrlis
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  • What do you mean by there are no characteristics in higher spatial dimensions? I thought we need to just find a function $\phi(t,x,y)$, s.t. $\phi_t^2=\phi_x^2+\phi_y^2$. And the set${\phi=const}$ would be the family of characteristic surfaces. When I tried to solve the equation, I got planes. But my intuition tells me that it should be something like the cone you mentioned. I don't know where I went wrong. – henryforever14 Feb 10 '14 at 17:51