Causal transient propagation in media with classical or power-law loss [US transmission in tissue]

The manner in which transient waves propagate in a medium with classical viscous losses is generally addressed is by determining solutions to the wave equation originally derived by Stokes. Exact solutions are difficult to obtain even for plane waves and simple transient forms and consequently, approximations are generally made. A related problem, of particular interest in relation to ultrasound transmission in soft tissue, is propagation in a medium, whose absorption coefficient obeys power-law frequency dependence, i.e., α ∝ ω". By using a recently obtained solution to a causal convolution wave equation, expressions are obtained for one-dimensional transient propagation for n=2 and n=1. Analytical expressions are obtained for a sinusoidal step function source when the effects of dispersion are ignored. When the effects of dispersion are accounted for, it is shown that the propagation can be expressed in terms of Fourier transforms. Examples are used to illustrate the results for both dispersive and non-dispersive media.