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acoustic theory Aeronautics and Space alti analytical procedure based angle APPLICATION OF ACOUSTIC ballistic wave theory beneath the aircraft bow shock caustic coordinates cutoff Mach number D-1860 National Aeronautics decreasing sound speed distance disturbance travels dive Donald effects typical flight equa feet Figure flight direction flight path given which show graphical procedure involving ground locus ground shock pattern hit the ground intersects the ground inverse function involving ray tracing Langley Research Center Lansing level flight linearly decreasing sound minimum altitude obtained from equation parameter PATTERNS FROM MANEUVERING point of origin problem of determining procedure involving ray propagation of weak ray directly beneath ray theory reach the ground shock front shock originating shock to travel shock travels shock wave produced shock waves originating sonic booms Space Administration speed is applied speed of sound spherical coordinates standard atmosphere statute miles supersonic aircraft turbances typical flight maneuvers vertical cross section wave are discussed xq(t