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This paper considers the propagation of neurological shock-waves in the human head due to improvised explosive devices (IEDs). The models adopted here use various mathematical techniques, including adoption and application of the two most important partial differential equations (PDEs) in this area, such as the Burgers' and Transport equations—together with a discussion of the inherent mechanics witnessed during experiments. In particular, only a one-dimensional model of the propagation of an intense acoustic compression wave—known as a shock-wave—traveling from air into the human head, is analyzed, using experimental data taken from existing literature. Computer simulations of these models also reproduce published experimental measurements of these acoustic dynamic pressures within the human brain, with graphs describing shock-wave motion in both two- and three- dimensions. There follows analysis and explanations of this phenomena, developed more thoroughly, to explain in detail features of experimentally-observed dynamic-pressures resulting in the brain, after a transitory shock-wave rapidly passes through the human head. The final part of this paper leads to further mathematical exposition—intended to be discussed within future publications—of dynamic-pressures, the latter being explained more comprehensively, and in greater detail, for clarity, especially in terms of the inherent physics and mechanical properties of the ensuing dynamics.

This second paper adopts a more rigorous, in-depth approach to modeling the resulting dynamic-pressures in the human brain, following a transitory improvised explosive device (IED) shock-wave entering the head. Determining more complicated boundary conditions, a set of particular-solutions for both Burgers' and the Transport equations has been obtained to describe the highly damped neurological pressures, complete with respective graphical plots. Many of these two-dimensional solution-curves closely resemble the Friedlander curve [1–4], not only illustrating enormous over-pressures that result almost immediately after the initial impact, but under-pressures experimentally depicted in all cases, due to oscillatory motion. It appears, given experimental evidence, that most—if not all—of these models can be aptly described by damped sinusoidal functions, these facts being further corroborated by existing literature, referencing models expounded by Friedlander's seminal work [1–4]. Using other advanced mathematical techniques, such as the Hopf-Cole Transformation, application of the Dirac-delta function and the Heat-Diffusion equation, expressions have been determined to model and predict the associated energies and temperatures within this paper.