The dispersive waves (DWs) emission has emerged as an efficient way to extend the spectral range of a light source. However, the applications of the DWs are limited because of its narrowband and low power. It is crucial to find novel ways to manipulate the DWs radiation. We investigate a kind of catastrophe pulses, called swallowtail pulses, which can radiate DWs with broadband and high power. By exploiting the dependence of the temporal and spectral structure of swallowtail pulses on the initial control parameter within the catastrophe integral, we can effectively manipulate the emission of DWs in terms of resonant frequency and conversion efficiency. Furthermore, we observe that the swallowtail pulses can shed two solitons, and that both of them radiate DWs with different frequencies. In comparison to fundamental solitons, the minimum value of third-order dispersion required for the onset of DWs emission becomes significantly smaller for that of swallowtail pulses. The results of this study clearly reveal the impact of the inherent behaviors of swallowtail pulses on the DWs emission, which should be relevant for applications requiring broadband light sources and optical frequency combs based on the mechanism of DWs emission.