Wave simulation in dissipative media described by distributed-order fractional time derivatives

We develop and solve a dissipative model for the propagation and attenuation of two-dimensional dilatational waves, using a new modeling algorithm based on distributed-order fractional time derivatives. We consider two distributions. The first has n powers of the order of differentiation as the weight function, and the second is based on a generalized Dirac's comb function. The wave equation is solved with the fractional derivative by means of a generalization of the Grunwald-Letnikov approximation. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general material-property variability. We verify the results by comparison with the two-dimensional analytical solution obtained for wave propagation in homogeneous media. Moreover, we illustrate the use of the modeling algorithm by simulating waves in the presence of an interface separating two dissimilar media.