We theoretically study the formation of Brillouin precursors in Debye media. We point out that the precursors are only visible at propagation distances such that the impulse response of the medium is essentially determined by the frequency-dependence of its absorption and is practically Gaussian. By simple convolution, we then obtain explicit analytical expressions of the transmitted waves generated by reference incident waves, distinguishing precursor and main signal by physical arguments. These expressions are in good agreement with the signals obtained in numerical or real experiments performed on water and explain some features of these signals that remained mysterious or unnoticed. In addition, we show quite generally that the shape of the Brillouin precursor appearing alone at large enough propagation distance and the law giving its amplitude as a function of this distance do not depend on the precise form of the incident wave but only on its integral properties. The incidence of a static conductivity of the medium is also examined and explicit analytical results are given in the limit of weak and strong conductivity.