2.3.3 Doppler Radar Method

Christian Johann Doppler was an Austrian physicist and mathematician who first described the Doppler effect in 1842. He found that when a radio wave, light wave or sound wave is transmitted between objects moving with respect to each other, the frequency of the wave is shifted in proportion to the speed of one object relative to the other. In a Doppler radar system, a transmitting antenna transmits a radar beam toward a moving object. The moving object reflects the beam back to a receiving antenna, which is co-located with the transmitting antenna. Because the object is moving, the reflected beam arriving at the receiving antenna has a frequency that is shifted a small but measurable amount from the frequency of the transmitted beam. This frequency shift is proportional to the speed of the moving object relative to the antennas. In our case, the moving object is the bullet, and the radar antennas are located at the firing position. Doppler radar tracks the bullet as it flies and provides measurements of the radial velocity of the bullet with respect to the antennas; that is, with respect to the firing point. The data from the radar are processed mathematically in a computer using very sophisticated software. At any point in the bullet trajectory, the results of these computations are bullet position coordinates (downrange, crossrange, and vertical directions), bullet velocity components in these directions, and even drag deceleration, all versus time of flight from the firing point. These data are available almost continuously as the bullet flies from the firing point until it impacts the ground. A firing elevation angle of several degrees can be used so that each bullet is tracked continuously as its velocity decreases from the muzzle through the supersonic, transonic and subsonic velocity regions before impact. Knowing the position and velocity of the bullet at any two points along the trajectory makes possible the calculation of a BC value for bullet performance between those two points. Infinity can be used for the BC calculation.

The Doppler radar method is far and away the best method of measuring ballistic coefficients, mainly because it provides measurements of bullet performance throughout bullet flight from supersonic velocity levels through subsonic velocity levels. However, Doppler radars are just not readily available. The radar system is very expensive, and a large computer complex is necessary to process the radar data to produce position and velocity data. A crew of several experts is required to operate the instrumentation and process the data. The cost of these capabilities exceeds the affordability limits of all sporting bullet manufacturers, and Doppler radar facilities are available only at some military sites.

For the past several years, these authors and other Sierra representatives have been privileged to participate annually for two days in a series of tests conducted at the U.S. Army Yuma Proving Ground near Yuma, Arizona. The Gun Position (shooting site) used for these tests is equipped with a high performance Doppler radar. The facilities are provided by the U.S. Army for tests planned and conducted by the Association of Firearm and Toolmark Examiners (AFTE), which is an association of forensic criminalists from U.S. and international law enforcement crime laboratories. The authors are technical advisory members of AFTE and have suggested tests to be conducted at the Yuma Proving Ground. Measurements of ballistic coefficients versus velocity for a number of bullets of different shapes have been performed over the past three years, and examples will be described in a later subsection.