  A thorough analysis of the transverse current autocorrelation function
obtained by molecular dynamics simulations of a dense Lennard-Jones fluid
reveals that even such a simple system is characterized by a varied dynamical
behavior with changing length scale. By using the exponential expansion theory,
we provide a full account of the time correlation at wavevectors $Q$ between
the upper boundary of the hydrodynamic region and $Q_p/2$, with $Q_p$ the
position of the main peak of the static structure factor. In the $Q$ range
studied we identify and accurately locate the wavevector at which shear wave
propagation starts to take place, and show clearly how this phenomenon may be
represented by a damped harmonic oscillator changing, in a continuous way, from
an overdamped to an underdamped condition. The decomposition into exponential
modes allows one to convincingly establish not only the crossover related to
the onset of transverse waves but, surprisingly, also the existence of a second
pair of modes equivalent to another oscillator that undergoes, at higher $Q$
values, a similarly smooth over- to underdamped transition.
