## 4 Discussion  
[PERSON] discussed the \(\alpha\)-\(\beta\) condition and subsequent changes of the original equation in the literature, and it opens another possibility of the application of such theory into LSCs. The comparison between the \(\alpha\)-\(\beta\) condition in the literature and the modified Birkhoff theory described herein exhibits a definite difference. The jet velocity is almost identical in the range of over 25\({}^{\circ}\) of original apex angle representing the jetting plane is quite similar angle with the detonation front plane in the modified Birkhoff. However this tendency is interrupted below 25\({}^{\circ}\) of the original apex angle in the \(\alpha\) = \(\beta\) condition (Figure 9).  
The \(\alpha\)-\(\beta\) condition creates a significant increase of the jet velocity near 0\({}^{\circ}\) original apex angle. This occurrence does not follow the theoretical results from the original [PERSON] theory, limiting its application, and the jet velocity in the range of large original apex angle decreases in general. In terms of the geometrical perspective, a LSC with a large original apex angle (close to 90\({}^{\circ}\)) cannot be a type of shaped charge anymore because the liner angle then creates no definite jetting. It then to EFP (Explosively Formed Projectiles) formation, with no clear jetting behavior, and the projectile (or whatever comes out of the liner surface) velocity should then be identical to the pure liner velocity calculated from Gurney analysis. Figure 10 exhibits a general jet velocity tendency of the modified  
\begin{table}
\begin{tabular}{c c c|c|c c|c c c|c} \hline \multicolumn{2}{c|}{**Original Liner**} & \multicolumn{2}{c|}{**Thickness (mm)**} & \multicolumn{2}{c|}{**M/C**} & \multicolumn{3}{c|}{**Modified Birkhoff**} & \multicolumn{3}{c|}{**Jet Tip vel. in Simulation**} & \multicolumn{1}{c|}{**Jet Tip vel.**} \\ \cline{3-10}
**Apex Angle, 2a (\({}^{\circ}\))** & **Liner** & **Explosives** & & **Jet vel.** & **Sluq vel.** & **25\%** & **50\%** & **75\%** & **Avg.** & **Difference** \\ \hline
60 & 1 & 4 & 1.39 & 4.32 & 0.31 & 5.64 & 5.25 & 5.01 & 5.30 & 0.98 \\
80 & 1 & 4 & 1.39 & 3.23 & 0.43 & 4.1 & 3.8 & 3.7 & 3.87 & 0.64 \\
100 & 1 & 4 & 1.39 & 2.54 & 0.55 & 2.87 & 2.69 & N/A * & 2.69** & 0.15 \\
120 & 1 & 4 & 1.39 & 2.06 & 0.69 & 2.11 & 2.13 & N/A * & 2.13 ** & 0.07 \\ \hline \end{tabular} all velocities are in km/s. \({}^{\circ}\) simulation stopped due to too much deformation.** 50\% liner collapse completion value was used.  
\end{table}
Table 3: Modified Birkhoff and numerical simulation comparison.  
Birkhoff theory depending on the original apex angle, and it follows the liner projection tendency described above.  
This point can be another favorable aspect of the modified Birkhoff theory because it covers both shaped charges and EFPs regime, providing all possible scenarios from the liner configuration in terms of the projectile velocity. When the original apex angles reaches to 90\({}^{\circ}\) the jetting process cannot occur, but the center of the flat liner bulged out due to the incoming rarefaction from the edge of the explosives surface creating a large chunk of EFP slug. In this case, the use of jetting criteria in understanding of the penetration performance wouldn't be reasonable, but a different penetration concept with an aerodynamically stable huge EFP slug capable of long distance travel should be applied. This approach is quite similar with the jet _vs._ no-jet criteria in the explosives welding application [10] in a rather different perspective.