## 1 Introduction  
Scientific studies of LSCs have been conducted in various research communities for many years, including the military, space, and commercial sectors. The main research interest has been the penetration performance of LSCs, since, except for specially designed large LSCs, they exhibit poor cutting performance of a matter of several inches (approximately 20-50 mm) at maximum. This tendency to low penetration becomes more apparent when the penetration depth is compared with CSCs, where the penetration performance is significantly better. There have been many engineering attempts to overcome the limited penetration performance of LSCs using various approaches including novel liner materials or liner configurations [1; 2; 3]. Unfortunately, while there have been considerable advances, there have been very few meaningful breakthroughs in this field of study, because the jet formation process of LSCs has not been fully studied or understood. This is due to the complex nature of the jet formation during the collapse of the LSC liner. Note that the fundamental LSC jet formation process is different from that of CSCs due to the different geometry and detonation front propagation [4]. There have been multiple attempts to observe the nature of jet projection of LSCs using the flash X-ray imaging technique. The initial attempts were made by [PERSON] and [PERSON] (unpublished), and these images elicited a great deal of interest in the scientific community because it was the first time the jet projection pattern of LSCs had been imaged (commercially manufactured 600 and 900 g/ft LSCs) [5; 6]. Note that the unit of g/ft has been used in this field of study as customary, and 1 g/ft = 212.6 mg/m. These images have revealed several unique features typical of LSCs. First, no solid jet was observed before particulation. The liner collapse of the LSC produces severely particulated jet segments from the start without any solid jet. Second, the particulated LSC jet segments always project in a line-grouping pattern, and each jet group has its own angle of projection, due to the axial directional detonation propagation in the LSC charge [5; 6; 7]. In order to explain the nature of LSC jet projection, [PERSON] proposed several engineering approaches, along with supplemental numerical simulation. [PERSON] identified several unique features of LSCs in numerical simulation, including the LCL (liner collapse line) [5]. The LCL is a collection of liner collapse points parallel to the detonation propagation direction. Note that in CSCs there is only a single liner collapse point because detonation is axially symmetrical. The LCL provides a clue to the LSC jet grouping trends [5; 6]. [PERSON] also identified the flight pattern of the LSC jet. Because each LSC jet group has different jet projection properties, including different angles, the jet does not tend to project at a right angle. Instead, an LSC jet projects slightly forward, making it hard to aim if the flight to target is long. This hinders proper design of special LSCs where a long flight before the hit-on-target is required; the problem has been investigated experimentally (unpublished) [5]. A three-dimensional configuration and analysis approach has been adopted for theoretical study of the jet formation process of LSCs, although a two-dimensional approach would be reasonable for the theoretical analysis of CSCs [8; 9]. This is because the jet formation and projection direction of the LSC is not in-line (or not parallel) with the detonation direction. In terms of the theoretical study LSCs, most research in LSCs has been inspired by the work of [PERSON], who first addressed the theory of shaped charges [10]. [PERSON]'s theory assumes a steady-state condition; however, a more advanced Pugh-Eichelberger-Rostoker (PER) theory was developed later [11].