-Military and Civil Particle Accelerator Energy-
In the Republic of Aurelia, located on the southern tip of the Osean Continent, before the year 2020 Aurelian Researchers began experiments to explore the capabilities of particles. Muon and Meson Technology was originally meant to provide the energy demanded by research projects in new medicine or materials, research focusing on the development of atomic-nucleus high-energy physics and the capabilities of Muon and Meson Technology powering metropolitan areas using Griswall as a sort of testing facility.

-the Atmos Ring around Griswall-
Griswall had a large ring like structure constructed around it under the orders of the Aurelian Government. The structure is officially known as the Aurelia Superconducting Synchrotron Laboratory (Abbreviation: ASSL). A synchrotron is a particular type of cyclic particle accelerator in which the magnetic field and the electric field are synchronized with a travelling particle beam. The ASSL produces 6.4 TeV (tera-electron volt / bolt) making it the world's most powerful circular (cyclic) accelerator. The Aurelian Government used the Atmos Ring to run their Muon Catalyst Fusion Reactor experiment.
This experiment would lead to the utilization of Muon and Meson particles for various purposes. During their experiments Aurelian Researchers developed a meson stabilizing system because of how considerably difficult it was to continuously irradiate a stable beam. The Meson Beam Stabilizing Regulator (MSBR) was produced in the Monte Breeze Industrial Area and delivered to the ASSL. When large energy protons are formed in the accelerator the MSBR regulates these protons and gradually generates a proton beam. Without the MSBR, this proton beam would grow larger and larger until it was too large to control, but with the MSBR, this beam can be sustained for extended periods of time. The Aurelian Military discovered that the Meson Beam Stabilizing Regulator could be installed on certain points of the Atmos Ring that could create a positive adolescent beam. The MSBRs could then discharge or 'fire' a continual beam of directed proton energy. The Aurelian Military dubbed this weapon as a cannon because of its capabilities. During the Leasath-Aurelia War, the MSBR was stolen by the Leasath Military, installed onto the Atmos Ring. The Meson Beam Stabilizing Regulator was then used as a defensive weapon by the Leasath Military, by having it fire Proton Beams at any Aurelian Forces that neared Griswall.
The Democratic Republic of Leasath developed similar technology, most likely during the same time Aurelia was doing its own research. Unlike Aurelia, the technology developed in Leasath was meant to be deployed as Military weaponry. Research with meson particles and a development of a Meson beam ignition system, Leasath developed its Shock Wave weaponry, most notably the Shock Wave Ballistic Missile and the Shock Cannon. Each of these weapons were deployed on the Glepnir.
Of the two weapons, the Shock Cannon was installed on the underside of the oversized aircraft in a disk like structure, which would be considered the cannon barrel of the weapon, which held a special sort of fuel for Meson discharges. The Glepinir itself would need a small-scale cyclotron accelerator that would essentially fire a Meson beam into the circular structure under the Glepnir and give the Shock Cannon its destructive energy. Because of the large amount of energy need to achieve the Meson Reaction in the weapon, sustained beam or firing is not possible, meaning that the weapon can only be fire in short bursts and must recharge.
The exact firing process used in the Shock Wave Ballistic Missile is unconfirmed, but theoretically it would essentially operate the same way the Shock Cannon System would. The weapon's discharged state would be similar to that of a Fuel Air Explosive. The Ballistic Missile most likely held a small, expednable cyclotron accelerator or a device that would operate in a similar state, its needed support systems and Meson Particles to form the destructive power of the missile.
Sources
Ace Combat X: Skies of Deception
Official Ace Combat X Website
NamcoCH. ACX Website

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