The Westervelt equation with viscous attenuation versus a causal propagation operator: A numerical comparison

The numerical solution of acoustic pulse propagation through dispersive media requires the inclusion of attenuation and its causal companion, phase velocity. For the case of propagation in a linear medium, Szabo [Time domain wave equations for lossy media obeying a frequency power law, Journal of the Acoustic Society of America 96 (1994) 491–500] introduced the concept of a causal convolutional propagation operator that plays the role of a causal propagation factor in the time domain. The convolutional operator has been successfully employed in the linear wave equation for both isotropic and non-isotropic dispersive media. This operator was originally proposed by Szabo to replace the loss term responsible for attenuation due to thermal conduction and viscosity of the fluid in the Westervelt equation. The Westervelt equation with a traditional loss term fails to incorporate the full dispersive characteristics of the medium, a deficiency which can be removed, at least in principle by replacing the traditional loss term with the causal convolutional propagation factor. Previously no comparison has been made between the Westervelt equation with the traditional loss term and with Szabo's causal convolutional propagation operator. In this paper we show numerically that by employing the convolutional propagation operator the full dispersive characteristics of the media are properly incorporated into the solution, and that the results can differ significantly from the Westervelt equation with the traditional loss term. The equations will be solved via the method of finite differences.