  In this paper, a multi-dimensional fractional wave equation that describes
propagation of the damped waves is introduced and analyzed. In contrast to the
fractional diffusion-wave equation, the fractional wave equation contains
fractional derivatives of the same order $\alpha,\ 1\le \alpha \le 2$ both in
space and in time. This feature is a decisive factor for inheriting some
crucial characteristics of the wave equation like e.g. a constant phase
velocity of the damped waves that are described by the fractional wave
equation. Some new integral representations of the fundamental solution of the
multi-dimensional wave equation are presented. In the one- and
three-dimensional cases, the fundamental solution is obtained in explicit form
in terms of elementary functions. In the one-dimensional case, the fundamental
solution is shown to be a spatial probability density function evolving in
time. However, for the dimensions grater than one, the fundamental solution can
be negative and therefore does not allow a probabilistic interpretation
anymore. To illustrate analytical findings, results of numerical calculations
and numerous plots are presented.
