A correction to sonic boom theory

Abstract

NOMENCLATURE aspeed of sound (ms -1 ) C p specific heat of cool air at constant pressure, 1004 J.kg -1 .K -1 f source function (m) F Whitham's function (m ½ ) g local gravity (ms -2) ) h geopotential altitude (m) H potential function, (J.kg -1 ) k function of M, Equation (22) L reference length (m) M free stream Mach number P pressure (Pa) p Riemann invarient (ms -1 ) q radial function, Equation (32) (m -½ ) r radius (m) R gas constant of air, (287 J.kg -1 .K -1 ) R b radius of body (m) S entropy (J.kg -1 .K -1 ) t time, or running variable, (s or m) T temperature (K) u xcomponent of velocity (ms -1 ) u perturbation of u (ms -1 ) v ycomponent of velocity (ms -1 ) V total velocity (ms -1 ) ABSTRACTCurrent sonic boom theory is based on linear midfield solutions coupled with acoustic propagation models. Approximate corrections are made within the theory to account for non-linearities, in particular for the coalescence of compression waves and the formation of weak shocks. A very large adjustment is made to account for the increasing acoustic impedance that the waves encounter as they propagate from the low density air at cruise altitude to the high density air at sea level. Typically this correction reduces the calculated over pressure levels by a factor of three. Here the method of characteristics (MOC) is used to prove that the density gradient within a hydrostatic atmosphere has no direct effect on the propagation or intensity of the wave. However gravity and ambient temperature both affect the wave propagation and the combined pressure level attenuation is not dissimilar to that previously attributed to acoustic impedance. Although the flawed acoustic theory has given reasonable predictions of measured sonic booms, the omission of gravity from the equation of motion and the inclusion of a false impedance modification, makes the model unreliable for prediction of future designs, particularly those focused on boom minimisation. As an aid to quiet supersonic aircraft design, Whitham's theory is extended to include gravity and ambient temperature variation and shown to be in good agreement with a MOC solution for the real atmosphere.propagation. It is this supposed non linear effect that results in delayed development of the N wave and increases the possibility of achieving an overpressure below that calculated by Jones. Once an N wave has formed the George-Seebass-Jones' minimisation theory reduces to that of Jones, and hence the extension of the theory that underpinned the boom minimisation work in the US High Speed Research programme (9) , and the DARPA/NASA Shaped Sonic Boom Demonstrator (SSBD) (10) is removed. This is not to say that N waves are inevitable, on the contrary, even a cursory exploration of the problem using the MOC reveals the possibility of: signatures with character; and bow shock overpressures much less than predicted by Jones. At this stage it is necessary to mention that this is not the first application of MOC to the sonic boom problem. Ferri and colleagues (11) at New York University wrote ...

Ballistic Wave from Projectiles and Vehicles of Simple Geometry

“…[13] indicates the "shape factor" F W  0.104 without other detail about the calculation, while the value F W  0.095 can be deduced from a numerical result given in Ref. [28]. In fact, the system made up of Eq.…”

“…Shape N° 1 is quoted as a bullet shape in Ref. [9,13,28], shape N° 5 is given as a typical shell shape in Ref. [8].…”

“…The theory developed for projectiles of simple shape (symmetry of revolution) has then been extended to supersonic aircraft, by taking into account the lift effects and the theoretical problems relating to the propagation of the "sonic boom" through the atmosphere as far as the observer location [20][21][22][23][24][25][26][27][28][29]. Thus, the sonic boom is the detonation resulting from a ballistic wave of high amplitude.…”

The effect of gravity on a shaped sonic boom

“…. (1) and shown in green is used for both wakes, where χ is greater than the measured signature limit.…”

“…Proof that gravity is directly responsible for the attenuation of sonic booms and that the density gradient within a hydrostatic atmosphere has no direct effect, was presented in a recent paper (1) . That paper challenged the assumptions on which most boom prediction codes are based; these typically omit gravity from the acoustic propagation equations but include an artificial acoustic impedance effect.…”