One-dimensional retarded wave Green function ($G_R(x,t)=\frac{\Theta (t-|x|/c)}{2c}$)

For $({\partial }_t^2-c^2{\partial }_x^2)G_R=\delta (t)\delta (x)$, zero response before the impulse gives the displayed retarded fundamental solution. Its initial displacement is zero and its initial velocity is $\delta (x)$. The response fills the interior of the forward causal cone for the constant-speed wave equation, rather than living only on its boundary. Changing the overall sign of the differential operator changes the Green function's sign. Equivalently, the spatial Fourier transform is $\Theta (t)\sin (c|k|t)/(c|k|)$, whose inverse is a rectangular profile. Values assigned exactly on the wave fronts do not change the associated distribution.