  We study a wave equation with a nonlocal time fractional damping term that
models the effects of acoustic attenuation characterized by a frequency
dependence power law. First we prove existence of a unique solution to this
equation with particular attention paid to the handling of the fractional
derivative. Then we derive an explicit time stepping scheme based on the finite
element method in space and a combination of convolution quadrature and second
order central differences in time. We conduct a full error analysis of the
mixed time discretization and in turn the fully space time discretized scheme.
Error estimates are given for both smooth solutions and solutions with a
singularity at $t = 0$ of a type that is typical for equations involving
fractional time-derivatives. A number of numerical results are presented to
support the error analysis.
