Self-organization of bubbles in acoustic fields, or self-action of the acoustic waves in bubbly liquids is a strongly nonlinear phenomenon due to two-way interaction of the bubbles and the acoustic field. Theoretical model and preliminary computations predict that waves of self-induced acoustic transparency may exist. Such effect is confirmed in the experiments presented in this paper. Formation of a wave of void fraction which rapidly propagates through the bubbly medium leaving a region almost free of bubbles behind its front is observed in the experiments. Measurements of the dynamics of such a wave at different acoustic frequencies and amplitudes are carried out. A three dimensional model of self-organization of a polydisperse bubble continuum in acoustic field is developed and the results of simulations are compared with experiments. A good agreement of the theory and experiment is found.