<p>We investigate a wave equation augmented with a fractional damping term that involves a Caputo fractional derivative in time. The regularity of the solution is analyzed for both the homogeneous problem with smooth and nonsmooth initial data, as well as a nonhomogeneous problem with vanishing initial data. The spatial derivatives are discretized using the Galerkin finite element method. By employing a semigroup approach, we derive error estimates that are optimal with respect to the regularity of the initial data. The semidiscrete problem is then discretized in the temporal direction using a convolution quadrature generated by the second-order backward difference method, and error estimates for the fully discrete scheme are established. Finally, several numerical tests are conducted to validate the theoretical results.</p>