We derive the effective medium theory for the linearized time-domain acoustic waves propagating in a bubbly media. The analysis is done in the time-domain avoiding the need to use Fourier transformation. This allows considering general incident waves avoiding band limited ones as usually used in the literature. Most importantly, the outcome is as follows:
1. As the bubbles are resonating, with the unique subwavelength Minnaert resonance, the derived effective wave model is dispersive. Precisely, the effective acoustic model is an integro-differential one with a time-convolution term highlighting the resonance effect.
2. The periodicity in distributing the cluster of bubbles is not needed, contrary to the case of using traditional two-scale homogenization procedures. Precisely, given any C1-smooth function K, we can distribute the bubbles so that locally the number of such bubbles is dictated by K. In addition to its dispersive character, the effective model is affected by the function K. Such freedom and generality is appreciable in different applied sciences including materials sciences and mathematical imaging