Phase-field fracture and peridynamics are two non-local methods to compute fracturing solids efficiently. While in reality, cracks in a solid are sharp two-dimensional hypersurfaces, the phase-field approach regularizes the material discontinuities with smooth transitions between broken and unbroken states. Otherwise, it relies on the classical Boltzmann continuum. In contrast, peridynamics describes the material in a non-local form with a similar regularizing effect on cracks. One significant issue for the simulation of dynamic fracture and spallation of a structure is the correct handling of elastic waves, like the pressure and tension waves inside a body induced by dynamic boundary conditions from an impact or impulse. This contribution investigates finite-element computations of phase-field fracture and different peridynamic formulations. By studying longitudinal pressure waves inside an elastic body, we compare the propagation and reflection of linear and non-linear waves at phase-field cracks to bond-based and state-based peridynamics. We show that both the phase-field fracture approach and the peridynamic formulations can reproduce the classical solutions to a different extent. By means of some numerical examples, it will be illustrated that both approaches offer a very efficient calculation method for dynamic crack propagation and spallation.