  The main object of this paper is the planar wave equation
\[\bigg(\frac{\partial^2}{\partial t^2}-a^2\varDelta\bigg)U(x,t)=f(x,t),\quad
t\ge0, x\in \mathbb {R}^2,\] with random source $f$. The latter is, in certain
sense, a symmetric $\alpha$-stable spatial white noise multiplied by some
regular function $\sigma$. We define a candidate solution $U$ to the equation
via Poisson's formula and prove that the corresponding expression is well
defined at each point almost surely, although the exceptional set may depend on
the particular point $(x,t)$. We further show that $U$ is H\"{o}lder continuous
in time but with probability 1 is unbounded in any neighborhood of each point
where $\sigma$ does not vanish. Finally, we prove that $U$ is a generalized
solution to the equation.
