  We theoretically study the formation of Brillouin precursors in Debye media.
We point out that the precursors are visible only 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 a simple examination of the long-time behavior of the overall signal.
These expressions are in good agreement with the signals obtained in numerical
or real experiments performed on water in the radio-frequency domain and
explain in particular some observed shapes of the precursor. Results are
obtained for other remarkable incident waves. In addition, we show quite
generally that the shape of the Brillouin precursor appearing alone at
sufficiently large 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
again given in the limit of weak and strong conductivities.
