I should derive
$\int_{R^{3}}e^{-2\pi ix \cdot \xi}\left(\frac{1}{4\pi}\int_{S^2}f(x-\gamma t)d\sigma(\gamma)\right)dx=\hat{f}(\xi)\frac{\sin(2\pi |\xi| t)}{2\pi |\xi|t}$
I already calculate the useful lemma : $\frac{1}{4\pi}\int_{S^2}e^{-2\pi i \xi \cdot \gamma}d\sigma (\gamma)=\frac{\sin(2\pi |\xi|)}{2\pi |\xi|}$ .