  In an ultrasonic experiment, we have investigated the temperature profile of
the velocity of longitudinal elastic waves propagating along a direction
perpendicular to the layers in the organic superconductors kappa-(BEDT-TTF)_2X,
X = Cu(SCN)_2 and Cu[N(CN)_2]Br. Although a small decrease of the velocity is
observed at the superconducting transition, the most anomalous behavior is
obtained in the normal metallic state where an important softening is
identified around 40-50 K. In order to characterize the origin of this anomaly,
we have studied its behavior under the application of hydrostatic pressure. The
observed behavior is found to mimic those of the transport and magnetic
properties of these materials which have been attributed to the magnetic
fluctuations. Following the example of one-dimensional insulating systems where
coupling between longitudinal acoustic waves and magnetic fluctuations is known
to occur, our results suggest that the pseudo-gap regime of these
two-dimensional organic superconductors is dominated by a similar mechanism.
