Finite-difference time-domain simulation of acoustic propagation in dispersive medium: An application to bubble clouds in the ocean

Accurate modeling of pulse propagation and scattering is a problem in many disciplines (i.e. electromagnetics and acoustics). For the case of an acoustic wave propagating in a two-dimensional non-dispersive medium, a routine 2nd order in time and space Finite-Difference Time-Domain (FDTD) scheme representation of the linear wave equation can be used to solve for the acoustic pressure. However when the medium is dispersive, one is required to take into account the frequency dependent attenuation and phase speed. Until recently to include the dispersive effects one typically solved the problem in the frequency domain and not in the time domain. The frequency domain solutions were Fourier transformed into the time domain. However by using a theory first proposed by Blackstock [D.T. Blackstock, J. Acoust. Soc. Am. 77 (1985) 2050. [1]], the linear wave equation has been modified by adding an additional term (the derivative of the convolution between the causal time-domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium. In the case of acoustic propagation through water, the water environment becomes strongly dispersive due to the presence of air bubbles that are present below the air-water interface.