  Surveys on wave propagation in dispersive media have been limited since the
pioneering work of Sommerfeld [Ann. Phys. 349, 177 (1914)] by the presence of
branches in the integral expression of the wave function. In this article, a
method is proposed to eliminate these critical branches and hence to establish
a modal expansion of the time-dependent wave function. The different components
of the transient waves are physically interpreted as the contributions of
distinct sets of modes and characterized accordingly. Then, the modal expansion
is used to derive a modified analytical expression of the Sommerfeld precursor
improving significantly the description of the amplitude and the oscillating
period up to the arrival of the Brillouin precursor. The proposed method and
results apply to all waves governed by the Helmholtz equations.
