4Let us say for the sake of simplicity, we have the $1$-d wave equation on the real line, $u_{tt} - u_{xx} = 0$ on $(0, \infty ) \times \mathbb{R}$ and $u(0,x) = f(x)$, $u_t(0,x) =g(x)$. My question, is, why exactly is the domain of dependence of the equation a cone? I understand the energy methods argument that $u =0$, $u_t = 0$ on some ball means that $u \equiv 0$ on the space-time cone whose height is the same as its radius, but why exactly is it the cone, and not some other shape i.e a cylinder?


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