We compute the near-field, high-frequency radiation from a circular crack expanding with constant rupture velocity and discuss the characteristics of the stopping phases. We then introduce rupture velocity jumps in the fracture process. The computed accelerations show the dominant role played by the rupture front kinematics. The high acceleration pulses are associated with sudden changes of the rupture velocity. For a sudden jump (or a sudden stop), there is no theoretical high-frequency limit to the spectral density of acceleration. In order to account for fmax, we introduce a smooth deceleration of the rupture front over a time t′ in place of a sudden stop. This results in a spectral fall-off for frequencies greater than 1/t′ and supports the interpretation of fmax as a source effect.