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.
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