## 4 Discussion  
The calculated incident peak overpressures from the free-field hemispherical blast simulations show good agreement with the KB predictions in Figure 10. Interestingly, the largest deviation from the KB curve appears to at the points closest to the explosions yielding the largest overpressure; these first three points lie closer to the von-Neumann-Taylor-Sedov prediction, which rapidly deviates from the Swisdak (KB) curve. The analytical prediction is only valid at an intermediate distance from large explosions; it breaks down near the explosion, as the point source assumption washes out details of the actual detonation event, but also in the very far-field, where the assumption that \(p_{0}=0\) in the ambient gas begins to corrupt the results as the incident overpressure approaches the ambient atmospheric pressure. Since the deviation of the data points from the empirical curve is likely within the experimental errors of the original fits, the fact that the data seem to jump from the analytical to the empirical curves may be coincidental. In the very far-field, the KB predictions seem to be doing a reasonable job at predicting the calculated overpressures, despite the fact that it is fit to data from explosions that were orders of magnitude larger. As expected, the KB predictions do not do well at predicting the overpressures near the detonation cord. The asymmetrical blast wave from a long, thin cylindrical cord lit at one end reaches a nearby point at different times, making the peak pressure smaller than that predicted from a localized (hemispherical) source. However, with greater distance this time delay becomes smaller and the data appears to converge onto the KB curve. There is a larger discrepancy between the KB-predicted and calculated incident impulses in Figure 11. There is very good agreement in the slope of the data versus the curve, but the free-field simulations appear to uniformly under-predict the impulse relative to the KB curve by a relatively small amount. Given the better agreement in the peak overpressures, there may be some discrepancy in the shape or duration of the whole incident pressure wave. The source of this error could be numerical or physical. There may be low pressure effects to the waves which we not captured in the very large Kingery tests. Note the oscillations that appear in the smaller pressure time histories in Figure 3. These appear after the sharp shock-like pressure spikes decay into more smooth waves traveling at sound speed. While these oscillations could be numerical effects, subsequent calculations were  
Figure 15: Results from a plane strain analysis of a ‘deep-roof’ type mitigation barrier loaded by a wave generated from 0.45359 kg of TNT at approximately 36.576 m. (**a**) A snapshot of the wave reflecting over the barrier. The color gradient is set so that max (red) values are above the 0.4516 kPa overpressure threshold. (**b**) The “bubble” behind the barrier for which max overpressure was beneath 0.4516 kPa. conducted to investigate this by changing mesh size and the position of the tracer nodes, which appeared to have no effects on the oscillations at distance. Thus it is possible that the oscillations in the far-field small pressure waves is in fact a physical phenomenon. This could in part explain the discrepancy in impulse when there is good agreement in the peak pressure magnitude. It is noted that similar but less pronounced oscillations also seem to appear in the farthest-field pressure histories in the works of [PERSON] et al., [PERSON] et al. (2020, 2021). There do not appear to be other curves available from similarly small charges at distance to compare with Figure 4. Taking experimental measurements of incident pressures and impulses much smaller that atmospheric pressure very far away from small explosives is quite difficult. It is further noted that the overall predictive accuracy in the subsequent blast-mitigation dynamic simulations is in part dependent on the accuracy of the predicted incident waves.