## 2 Steady State Equation of Motion of LSCs  
In this section, the liner motion of LSCs is identified first, and a detailed steady state equation of motion will be developed based on the geometrical uniqueness of LSCs. Upon detonation of the core charges in LSCs, the flat liner interacts with the detonation gas in a side direction creating a simultaneous movement of the entire liner at a given station along the length, maintaining the \(\alpha=\beta\) condition. This is different than that of progressive liner collapse starting from top to bottom of the liner occurred in CSCs (Figure 1, Top). This occurs at the detonation front plane at \(\Delta\)OGF. However, due to the apex angle and the liner distance, the liner gradually closes the gap between liners generating a diagonal collapse line \(\overline{\text{OB}}\) along the liner while the liner collapse angle is maintained in a constant value. Assume that the diagonal collapse line \(\overline{\mathrm{OB}}\) is perpendicular to the jetting plane, and the jetting plane is under different angle with the detonation front plane \(\Delta\mathrm{0 GF}\). Because the jetting plane and the detonation plane is in different angle, the \(\alpha=\beta\) condition cannot be maintained requiring a different set of apex and collapse angles (Figure 1). The \(\alpha=\beta\) condition described above is only applicable in the detonation front plane \(\Delta\mathrm{0 GF}\). In order to develop a new set of apex and collapse angles, the specific liner collapse condition should be identified. The original apex angle \(\alpha\) of the LSCs should be in the same plane as the detonation front plane \(\Delta\mathrm{0 GF}\). The jetting should be perpendicular to the line \(\overline{\mathrm{OB}}\) creating two jetting planes of \(\Delta\mathrm{0 ED}\) (before collapse starts) and A'HE'D(liner collapse in progress). The two planes of \(\Delta\mathrm{0 ED}\) and A'HE'D'I are in parallel, and these two planes bears the two important liner angles of modified apex angle before liner collapse, \(\alpha\) and collapse angle \(\beta\) (Figure 2). Once these two angles are identified, they will be applied to the Birkhoff theory later. In the detonation front plane of \(\Delta\mathrm{0 GF}\) where the liner collapse starts, the liner collapse completion time should be the same as time to take the liner reaches the center (1/2 \(\overline{\mathrm{GF}}\)). Because we assume that the liner projects perpendicular to the liner surface with velocity \(V_{p}\), the following approximated relation about \(\tau_{m}\) can be made from the Figure 3.  
Figure 1: Schematic diagram of LSC liner movement during detonation. Figure 3: Detonation front plane and the liner projection path. Figure 2: Original apex angle \(\alpha\), modified apex angle before collapse \(\alpha^{\prime}\), and collapse angles \(\beta\) (from left to right). \[t_{m}=\frac{l\;sin\alpha}{V_{p}} \tag{3}\]  
where, \(l\) is the liner length, and \(t_{m}\) is the liner collapse completion time. The detonation front travel distance while the liner collapse completes can be calculated from the liner collapse completion time because the detonation front moves along the lateral length of the liner with the detonation velocity, \(D\). The detonation front travel distance is \(\overline{\text{BD}^{\prime}}\) and \(Dt_{m}\) (Figure 4). In order to find the collapse point angle, \(h\) should be identified first from the geometrical descriptions in Figure 4, the following relations can be made  
\[\overline{OB}\times h=\overline{OB^{\prime}}\times\overline{BD^{\prime}} \tag{4}\]  
\[h=\frac{lDt_{m}}{\sqrt{L^{2}+(Dt_{m})^{2}}}=A^{\prime}D^{\prime} \tag{5}\]  
From the jetting plane \(\Delta\)A\({}^{\prime}\)E\({}^{\prime}\)D\({}^{\prime}\)in Figure 4, the collapse point angle \(\beta\) can be calculated from the following,  
\[ \begin{array}{c}\includegraphics[width=142.26378 pt]{fig-1.eps}\end{array}  \tag{6}\]  
The next step is to find the modified apex angle \(\alpha^{\prime}\) which is different than that of the original apex angle \(\alpha\).