The decay of transverse current correlation functions in a supercooled
liquid is studied by taking into account the mode coupling effects du
e to dominant density fluctuations. At higher densities the large mode
coupling contributions due to cooperative dynamics can sustain propag
ating shear waves in the liquid. The crossover from a diffusive to osc
illatory behavior of the shear mode is analyzed. We compute the speed
of the shear wave in the liquid phase as well as the ideal glassy phas
e described in the self consistent mode coupling theory. The qualitati
ve behavior is in agreement with the computer simulation results and e
xperiments. (C) 1998 Elsevier Science B.V. All rights reserved.