  The frequency-dependent attenuation typically obeys an empirical power law
with an exponent ranging from 0 to 2. The standard time-domain partial
differential equation models can describe merely two extreme cases of frequency
independent and frequency-squared dependent attenuations. The otherwise
non-zeor and non-square frequency dependency occuring in many cases of
practical interest is thus often called the anomalous attenuation. In this
study, we developed a linear integro-differential equation wave model for the
anomalous attenuation by using the space fractional Laplacian operation, and
the strategy is then extended to the nonlinear Burgers, KZK, and Westervelt
equations. A new definition of the fractional Laplacian is also introduced
which naturally includes the boundary conditions and has inherent
regularization to ease the hyper-singularity in the conventional fractional
Laplacian. Under the Szabo's smallness approximation where attenuation is
assumed to be much smaller than the wave number, our linear model is found
consistent with arbitrary frequency dependencies. According to the fact that
the physical attenuation can be understood a statistic process, the empirical
range [0,2] of the power law exponent is explained via the Levy stable
distribution theory. It is noted that the power law attentuation underlies
fractal microstructures of anomalously attenuative media.
