  To model wave propagation in inhomogeneous media with frequency-dependent
power-law attenuation, it is needed to use the fractional powers of symmetric
coercive elliptic operators in space and the Caputo tempered fractional
derivative in time. The model studied in this paper is semilinear stochastic
space-time fractional wave equations driven by infinite dimensional
multiplicative white noise and fractional Gaussian noise, because of the
potential fluctuations of the external sources. The purpose of this work is to
discuss the Galerkin finite element approximation for the semilinear stochastic
fractional wave equation. We first provide a complete solution theory, e.g.,
existence, uniqueness, and regularity. Then the space-time multiplicative white
noise and fractional Gaussian noise are discretized, which results in a
regularized stochastic fractional wave equation while introducing a modeling
error in the mean-square sense. We further present a complete regularity theory
for the regularized equation. A standard finite element approximation is used
for the spatial operator, and the mean-square priori estimates for the modeling
error and for the approximation error to the solution of the regularized
problem are established.
