  We solve the Cauchy problem defined by the fractional partial differential
equation $[\partial_{tt}-\kappa\mathbb{D}]u=0$, with $\mathbb{D}$ the
pseudo-differential Riesz operator of first order, and the initial conditions
$u(x,0)=\mu(\sqrt{\pi}x_0)^{-1}e^{-(x/x_0)^2}$, $u_t(x,0)=0$. The solution of
the Cauchy problem resulting from the substitution of the Gaussian pulse
$u(x,0)$ by the Dirac delta distribution $\varphi(x)=\mu\delta(x)$ is obtained
as corollary.
