The 1d wave equation $$\frac{\partial^2 y}{\partial t^2} - \frac{\partial^2 y}{\partial x^2} = 0$$ has a retarded Green's function given by (according to this table) $$G_{ret} = \frac{1}{2} \Theta(t - |x|)$$ where $\Theta$ denotes the heaviside step function. Now suppose I want to check whether substituting $G_{ret}$ in place of $y$ in the PDE gives $\delta(t) \delta(x)$, which is what we would expect if it really is a Green's function of the PDE. But I am not sure how to properly do that. if I differentiate the Heaviside step function once, I get the Dirac delta. But I do not know how to proceed from there.
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