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I know that for the canonical one dimensional wave equations $u_{\mu\eta}=0$, the fundamental solution $F$ satisfies $\frac{\partial}{\partial\mu\partial\eta}F=\delta(\mu,\eta)=\delta(\mu)\delta(\eta)$. Thus one solution would be $F=H(\mu)H(\eta)$. However, my question is if I introduce change of variables by $\mu=x-ct, \eta=x+ct$, what would the fundamental solution be? The one dimensional wave operator corresponds to $\partial^{2}_t-c^2\partial^{2}_x=-\frac{c^2}{4}\partial^2_{\mu\eta}$, what would the fundamental solution $F$ correspond to?

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Unless I'm mistaken (entirely possible) this is just D'Alembert's solution to the wave equation http://mathworld.wolfram.com/dAlembertsSolution.html. So to answer your question, the solution would be f(x-ct) + g(x+ct) with f and g chosen according to initial conditions.

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  • $\begingroup$ I know d'Alembert solution, but what I am trying to do is to find fundamental solution which is a distribution such that $(\partial^2_t-c^2\partial^2_x)F=\delta$, where $\delta$ is the Dirac function. $\endgroup$ – Xuxu Oct 10 '13 at 0:05

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