Analytical study of sonic boom from supersonic projectiles

Whitham's first-order theory for steady flow at moderate supersonic Mach numbers around slender axisymmetric bodies is reviewed and applied to determine sonic boom overpressure signatures from bodies of various shapes, particularly those of projectiles in steady and rectilinear flight. Based on his theory, certain closed-form solutions are derived, including extended first-order decay laws for signature evolution with increasing distance from the flight path. Also, for the case of arbitrary axisymmetric bodies of simple and smooth shape, whose axial profile can be defined piecewise by segments of polynomial curves, a new analytical solution is presented for Whitham's ‘smooth-body’ F-function integral. An alternate formulation for the F-function that was derived by Lighthill for non-smooth bodies is shown to be better for actual smooth and non-smooth projectile shapes. A computer code was developed to implement the analysis with various F-functions and compute sonic boom signatures for specified body shapes. These numerical studies were conducted to investigate and illustrate important features of these signatures and their evolution with increasing miss distance. The effects of body shape such as surface protuberances and afterbody flow are two examples.