  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 $C^1$-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.
