An in-depth characterization of coherent radio Cherenkov pulses from particle showers in dense dielectric media, referred to as the Askaryan effect, is presented. The time-domain calculation developed in this article is based on a form factor to account for the lateral dimensions of the shower. It is computationally efficient and able to reproduce the results of detailed particle shower simulations with high fidelity in most regions of practical interest including Fresnel effects due to the longitudinal development of the shower. In addition, an intuitive interpretation of the characteristics of the Askaryan pulse is provided. We expect our approach to benefit the analysis of radio pulses in experiments exploiting the radio technique.