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Suppose $u(x,t)$ is a solution to the wave equation with initial data $u_t(x,0)=g(x)$ and $u(x,0)=f(x)$ where $f$ and $g$ are smooth with compact support in $\mathbb{R}^3$. Given $x\in\mathbb{R}^3$ and $\xi\in{S^2}$, how do I compute the limit $\lim_{t\rightarrow\infty}{tu(x+tc\xi,t)}$?

My attempt: It is clear from the Huygens principle that $\lim_{t\rightarrow\infty}{u(x,t)}=0$. Moreover, the Kirchoff formula for a solution is given by- $u(x,t)=\frac{t}{4\pi}\intop_{|\xi|=1}g(x+ct\xi)dS_{\xi}+\frac{1}{4\pi}\intop_{|\xi|=1}f(x+ct\xi)dS_{\xi}+\frac{t}{4\pi}\sum_{i=1}^{3}\intop_{|\xi|=1}\frac{\partial f}{\partial x_{i}}(x+ct\xi)dS_{\xi}$

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