A rapidly moving crack in a brittle material is often idealized as a one-dimensional object with a singular tip, moving through a two-dimensional material. However, in real three-dimensional materials, tensile cracks form a planar surface whose edge is a rapidly moving one-dimensional singular front. The dynamics of these fronts under repetitive interaction with material inhomogeneities (asperities) and the morphology of the fracture surface that they create are not yet understood. Here we show that perturbations to a crack front in a brittle material result in long-lived and highly localized waves, which we call 'front waves' These waves exhibit a unique characteristic shape and propagate along the crack front at approximately the Rayleigh wave speed (the speed of sound along a free surface). Following interaction, counter-propagating front waves retain both their shape and amplitude. They create characteristic traces along the fracture surface, providing cracks with both inertia and a new mode of dissipation. Front waves are intrinsically three-dimensional, and cannot exist in conventional two-dimensional theories of fracture. Because front waves can transport and distribute asperity-induced energy fluctuations throughout the crack front, they may help to explain how cracks remain a single coherent entity, despite repeated interactions with randomly dispersed asperities.