Determining the Position and Trajectory of Supersonic Projectiles from Acoustic Measurements

Locating the trajectory of an incoming projectile has important military and civilian applications. The origin of the trajectory can be used to locate sniper fire, and the end-point can be used to determine target hit coordinates on appropriately instrumented target ranges used by armed forces as well as civilian agencies (police and sports). Such non-contact determination of hit coordinates can be integrated into a completely computerized instrumentation scoring system. All supersonic projectiles are accompanied by a bow wave, often referred to as the “Mach Cone” which is virtually attached to the ogive of the projectile and which can be detected acoustically. When a distributed array of microphones detect the arrival of this cone, it is possible to calculate the position and angle of the trajectory from the microphone data. This paper discusses the analysis and models developed by the author, which enable computation of the trajectory from distributed arrays of acoustic microphones. The mathematical models described here can serve as the basis for an algorithmic procedure, which can be used to obtain the trajectory data in near real time. We also discuss the principal sources of error and the inherent limitations in accuracy of locating trajectories by this method. Introduction All supersonic projectiles are accompanied by a bow wave, an approximately hyperbolic shaped shock front, which travels attached to the projectile. The shock wave gradually degrades into a cone shaped acoustic wave-front with a sharp leading edge. This wave is often called the “mach cone” and can be considered as formed by the envelope of the distributed acoustic disturbances from the tip of the moving source (Figure 1). The wave front is well structured and well described as a cone with a half angle, which is only a dependent on the Mach number of the projectile. If the projectile passes near an acoustic transducer, the pressure jump at the wave front can be easily sensed. This leading edge must be the first acoustic wave to arrive because of the supersonic speed of the projectile. The nearly discontinuous leading edge marks a sharply defined and unique acoustic event. At any point in the wave front, there also exists a well-defined direction The Mach Cone is the envelope of the expanding acoustic waves ...