  The electric field of the Cherenkov radio pulse produced by a single charged
particle track in a dielectric medium is derived from first principles. An
algorithm is developed to obtain the pulse in the time domain for numerical
calculations. The algorithm is implemented in a Monte Carlo simulation of
electromagnetic showers in dense media (specifically designed for coherent
radio emission applications) as might be induced by interactions of ultra-high
energy neutrinos. The coherent Cherenkov radio emission produced by such
showers is obtained simultaneously both in the time and frequency domains. A
consistency check performed by Fourier-transforming the pulse in time and
comparing it to the frequency spectrum obtained directly in the simulations
yields, as expected, fully consistent results. The reversal of the time
structure inside the Cherenkov cone and the signs of the corresponding pulses
are addressed in detail. The results, besides testing algorithms used for
reference calculations in the frequency domain, shed new light into the
properties of the radio pulse in the time domain. The shape of the pulse in the
time domain is directly related to the depth development of the excess charge
in the shower and its width to the observation angle with respect to the
Cherenkov direction. This information can be of great practical importance for
interpreting actual data.
