Viscosity Effects on the Propagation of Acoustic Transients

Abstract

The propagation of an impulsive excitation applied at the origin of a lossy viscous medium is studied by operational techniques as the excitation advances through the medium. The solution of the governing partial differential equation (PDE) for such transient propagation problems has been elusive. Such solution is found as an infinite sum of properly weighted successive integrals of the complementary error function, and it is quantitatively examined here using a one-dimensional model in space and time. As expected, as the transient advances through space, its amplitude decreases, and its width broadens. Such is the damping effect of viscosity that one would anticipate from elementary considerations in related disciplines such as electrodynamics. Such is also the smoothing-out effect of dispersion. We also obtain an approximate solution of the present boundary-initial value problem based on the method of steepest descents. This approximation agrees with the first term of the complete analytic solution given here. The pertinent dispersion relation associated with the governing parabolic PDE is shown to impose a restrictive condition on the allowable values of the propagation speed and the kinematic viscosity coefficient, thus assuring that propagation with attenuation does take place. Various numerical results illustrate and quantitatively describe the propagation of the transient pulse in several nondimensional graphs.