Brillouin precursors in Debye media

We theoretically study the formation of Brillouin precursors in Debye media. We point out that the precursors are visible only at propagationdistances such that the impulse response of the medium is essentially determined by the frequency dependence of its absorption and is practicallyGaussian. By simple convolution, we then obtain explicit analytical expressions of the transmitted waves generated by reference incidentwaves, distinguishing precursor and main signal by a simple examination of the long-time behavior of the overall signal. These expressionsare in good agreement with the signals obtained in numerical or real experiments performed on water in the radio-frequency domain and explainin particular some observed shapes of the precursor. Results are obtained for other remarkable incident waves. In addition, we show quite generallythat 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.