Commercial Space Operations Noise and Sonic Boom Modeling and Analysis (2018)

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2-1 CHAPTER 2. Technical Reference 2.1. Rocket Noise Modeling Methodology The RUMBLE noise modeling methodology was developed to produce accurate acoustic estimates relevant to environmental analysis of commercial space operations. The model is applicable to inflight and static operations of vertical and horizontal launch vehicles. Launch vehicle propulsion systems, such as liquid-propellant rocket engines and solid rocket motors, generate high amplitude, broadband noise. The majority of the noise is created by the rocket plume, or jet exhaust, interacting with the atmosphere, and combustion noise of the propellants. This results in noise that radiates in all directions. However, it is highly directive, meaning that a significant portion of the source’s acoustic power is concentrated in specific directions. The emitted sound is modified in several ways as it propagates outward. These effects include the source directivity, forward flight effects, Doppler effect, geometric spreading, atmospheric absorption, and ground interference to a receiver location. The received one-third octave (OTO) band sound levels from a source can be expressed as the sum of source components and propagation effects: Source Propagation where: Source sound power level; Forward flight effects; Source directivity, azimuthal symmetry is assumed; Doppler effect; Geometrical spherical spreading loss (point source); Atmospheric absorption; and Ground interference (interaction between direct and reflected acoustic rays). Rocket propulsion noise is calculated based on a specific source (vehicle trajectory point) to a receiver geometry (grid point). The position of the rocket and the receiver grid are provided in latitude and longitude, defined relative to a reference coordinate system (the World Geodetic System, WGS84). Implementation of this geo-referenced coordinate system ensures that large-distance geometric calculations are completed with greater accuracy than traditional flat Earth models. The core components of the proposed model are described in the following subsections. A conceptual overview of the rocket noise prediction model methodology is presented in Figure 2-1.

2.1.1. Sou The the rocket 2.1.1 for the sou its total th Ridge Re between f 2014]. Th The that is con using Gue modeled Therefore modeled a boosters o contributi FIGURE FIGU rce definition of , forward flig .1. Acoustic rce characte rust, exhaus search and our separate e results of th acoustic effic verted into a st’s variable as a single c , launch veh s a single en r cores (tha on from each 2-2 Valida level fo RE 2-1 Con a rocket nois ht effects, dir Power. Eldr rization. The t-velocity, an Consulting ( full scale roc is validation iency of the coustic powe acoustic eff ompact sour icle propulsio gine with an t are not con booster/core tion of NAS r various ro ceptual over e source’s st ectivity, and ed’s Distribu DSM-1 mod d the engine BRRC) of t ket noise me are presented rocket engine r. The acous iciency [Gue ce located at n systems w effective exi sidered to b . A SP-8072’s ckets - (righ view of RUM rength and ch Doppler effe ted Source M el determines /motor’s aco he DSM-1 m asurements an in Figure 2- /motor speci tic efficiency st, 1964]. In the nozzle ith multiple t diameter an e tightly clu DSM-1 emp t) normalize BLE model aracteristics ct. ethod 1 (DS the launch v ustic efficien odel demon d the empiri 2. fies the perce of the rocke the far-field exit with an tightly cluste d total thrust stered) are h irical curve d relative po methodology involves the M-1) [Eldred ehicle’s soun cy. Recent v strated very cal source cu ntage of the t engine/moto , distributed equivalent t red equivale [Sutherland, andled by s s - (left) over wer spectru . acoustic pow , 1971] is ut d power bas alidation by good agree rves [James mechanical p r will be mo sound source otal sound p nt engines c 1993]. Addi umming the all sound po m. 2-2 er of ilized ed on Blue ment et al., ower deled s are ower. an be tional noise wer

2.1.1 a static en of the rela 2012; Buc rocket exh airflow. C descendin velocity d the differe which red typically g a forward 2.1.1 specific d receiver. T made sign motor (RS frequencie the formu noise sou the Overa closely m these mod updated D RUMBLE h FIGUR .2. Forward vironment. A tive velocity kley et al., 1 aust. At the onversely, fo g rocket bod ifferential fr ntial betwee uces the corr enerated wh flight velocit .3. Directivi irections and he National ificant impr RM) [Hayn s and angles lation of the rce [James et ll Sound Pre atches measu ified DI is sh I are includ as the capab E 2-3. Pred Flight Effec standard me between the 984; Buckley onset of a lau r a vertical y and the je om the static n the forward esponding no ile the vehicl y of Mach 1. ty. Rocket no the sound Aeronautics ovements in es et al., 200 than any pr RSRM DI by al., 2014]. T ssure Level ( rements mad own in FIGU ed in the m ility to imple icted OASPL and dista t. A rocket in thod to quan jet and the et al., 1983] nch, the roc landing of a t exhaust ar condition, a flight veloc ise emission e is at subson ise is highly pressure obse and Space A determining 9]. The RSR eviously ava accounting f his improve OASPL) rad e in the far-f RE 2-3, wh odel. As fu ment updated with modif nce is indica forward flig tify this effe outside airflo . This outsid ket exhaust tr reusable laun e in opposin nd creating i ity and exhau . Notably, the ic speeds. Th directive, me rved will de dministration launch vehi M directivity ilable data. R or the spatia d formulation iation by app ield during la ere the nozzl ture measure DI sets spec ied DI - the ted in nozzl ht radiates le ct reduces ov w [Viswanat e airflow trav avels at far g ch vehicle, t g directions, ncreased jet st velocity d maximum o us, the mode aning the ac pend on the (NASA) Pro cle directivit indices (DI ecently, BR l extent and d substantiall roximately unches. An e e exhaust flo ments make ific to a spac downstream e diameters ss noise than erall sound l han et al., 20 els in the sam reater speed he ambient a yielding an mixing and r ecreases, jet verall sound led noise red oustic power angle from ject Constell y of the reu ) incorporate RC and NAS ownstream o y changes th 14°, from 51 xample soun ws in the dire additional ecraft’s engin direction of (De). the same roc evels as a fun 11; Saxena e direction s than the am irflow aroun increased re esultant nois mixing is red pressure leve uction is capp is concentra the source t ation Program sable solid r a larger ran A have imp rigin of the r e directional ° to 65° and d level map ction of 0°. T DI sets avai e(s). the rocket i 2-3 ket in ction et al., as the bient d the lative e. As uced, ls are ed at ted in o the has ocket ge of roved ocket ity of more using hese lable, s 0°

2.1.1 observer m by the m frequency identical During a sound as explained crest is em slightly le successive source of from the reducing t where a) moving a frequency sound lev frequencie FIGURE for highe velocity < 2.1.2. Pro The mode absorption 2.1.2 expanding every dou reduced b 2.1.2 modes of humidity Standards per unit d altitudes, the param layers. Th The ANS is perform .4. Doppler oving relati oving sound is higher (co at the instant rocket launc the distance as follows: w itted from a ss time to r wave crests waves is mo observer tha he frequency the source is t the speed is shifted lo el. For unwe s are accoun 2-4 Effect o r relative sp speed of so pagation led sound pr , and ground .1. Geometr outward as bling of the y 6 decibels ( .2. Atmosph air molecul of the air. At Institute (AN istance that is it will experi eters of the e total sound I sound atten ed on an O Effect. The D ve to its sour source and mpared to th of passing h, an observe from the sou hen the sour position clos each the obs at the observ ving away f n the previo . FIGURE 2- stationary, b of sound, an wer, the A-w ighted overal ted for equall f expanding eeds of the r und, (c) sour opagation fro interference ic Spreadin it travels. Th slant range d dB). eric Absorp es. Atmosph mospheric ab SI) standard a function o ence a wide r atmosphere attenuation i uation algorit TO-band ba oppler effec ce. The frequ by the spee e emitted fre by, and it is r on the gro rce to receiv ce of the wav er to the obse erver than th er is reduced rom the obse us wave; th 4 illustrates t ) the source d d) the sou eighting filte l sound level y. wavefronts ocket relativ ce velocity = m the source . g. When so is expansion istance betwe tion. Atmosp eric absorpti sorption is c S1.26-1995 f frequency a ange of atmo in each laye s the sum of hms are calc sis, the SAE t is defined a ency at the r d of the sou quency) if th lower if the und will hea er increases es is moving rver than the e previous w , causing an rver, then ea e arrival tim his spreading is moving le rce is movin ring on the s, the Dopple (decrease in e to the obse speed of so to a receive und leaves reduces the s en the sourc heric absorp on is a func alculated usin (R2004), wh nd atmosphe spheric cond r and the dis the absorptio ulated for pu Method [Ri s the change eceiver is rel rce relative e source is m source is m r a downwar . The relativ toward the previous wa ave, and th increase in th ch wave is e e between s effect for an ss than the s g faster tha spectrum res r effect does frequency) t rver for: (a) und, (d) sou r includes ge a source, it ound level th e and a recei tion arises f tion of temp g formulas f ich provides ric condition itions. The am tance that th n experience re-tone sound ckley et al., in frequenc ated to the fr to the recei oving towar oving away d shift in the e changes in observer, eac ve. Therefor e time betwe e frequency. mitted from uccessive w observer in peed of soun n the speed ults in a decr not change hat an obser stationary s rce velocity > ometric sprea travels out e farther the ver, the recei rom the exci erature, pres ound in the A a sound atten s. Since a roc ount of abso e sound trav d from each s. As the ro 2012] is use y of a wave f equency gene ver. The rec d the receiver from the rec frequency o frequency c h successive e, each wave en the arriv Conversely, a position f aves is incre a series of im d, c) the sou of sound. A eased A-wei the levels sin ver would n ource, (b) so speed of so ding, atmosp in all direc sound travel ved sound le tation of vib sure, and re merican Na uation coeff ket travels to rption depen els through atmospheric cket noise an d as a simp 2-4 or an rated eived , it is eiver. f the an be wave takes als of if the arther ased, ages, rce is s the ghted ce all otice urce und. heric tions, s. For vel is ration lative tional icient high ds on those layer. alysis lified

procedure standard. 2.1.2 provide a accurately to ground receiver a ground m energy to Embleton wind and included i ground wh FIGURE 2.1.2.4. R effects. R environm Exposure Level (CN 2.1.3. Val A input data noise mod rocket lau various di the launch measured FIGURE during the to calculate .3. Ground free-field so modeled as to receiver) nd will inter ay attenuate the receiver et al., 1983] temperature n the ground en calculatin 2-5 Sound (blu eceiver. The UMBLE calcu ental noise a Level (SEL EL). idation committed e , computation el, which u nches. Exam stances from event over d OASPL tim 2-7. The mo three measu the OTO-ban Interferenc und level at the combinat as shown in fere both con the sound e . The model when estim on the dire interference g ground int propagation e) and a refl received noi lates and pr nalysis: A-w ), Day-Night ffort was tak al resources ses the mode ples of com the launch p istances rang e histories fo deled time h red launches. d attenuation e. The calcu the receiver ion of a dire FIGURE 2-5 structively a nergy causin accounts for ating the rec ct and reflec [Chessel, 19 erference. near the gr ected wave ( se is estimate epares the m eighted and Average S en to ensure t , and run-tim ling method parative resu ad. The mode ing from les r distances o istories matc s utilizing th lated results . However, s ct wave (sour . The ground nd destructiv g the reflect the attenuati eived noise. ted wave, th 77; Daigle, 1 ound is mod red) from th d by combin odeled recei unweighted ound Level he best avail e. BRRC has ology describ lts are presen l-predicted S s than 1 km t f 1, 2, 4, and h the level, e pure-tone of the sou ound propag ce to receive will reflect ely with the ed wave to p on of sound To account e effect of a 979]. RUMBL eled as the c e source to t ing the sourc ved noise fo Sound Lev (DNL), and able science performed co ed above, to ted in FIGU EL values ag o 8.5 km. A 6.6 km from shape, and d sound attenu nd propagat ation near th r) and a refle sound energ direct wave ropagate a by the groun for the rando tmospheric E assumes a ombination he receiver. e componen r six noise m el (LAMAX Community is employed g mparisons o measured d RE 2-6 and ree well with comparison o the launch uration of th ation of the ion using D e ground is cted wave (s y back towar . Additionall smaller porti d [Chessel, m fluctuatio turbulence is homogeneou of a direct w ts and propag etrics releva , LMAX), S Noise Equiv iven the ava f its current r ata from mu FIGURE 2- measureme f the modele site are sho e levels rec 2-5 ANSI SM-1 most ource d the y, the on of 1977; ns of also s soft ave ation nt to ound alent ilable ocket ltiple 7, for nts of d and wn in orded

FIGURE set distan 2-6 Measu ces from th FIGURE red versus p e launch pad 2-7 Measur redicted laun - (right) Ma ed versus pr ch vehicle n ximum OAS edicted laun oise exposur PL at set di ch vehicle no e levels - (le stances from ise time hist ft) SEL valu the launch ories. 2-6 es at pad.

2.2. Soni The [Plotkin e model inc operations  C  A  H  B F An e tourism th ship at sl descends down” mo powered a flight dur SpaceShip characteri FIGURE The includes b presented and signa metrics ca 2.2.1. Son 2.2.1.1. S A so boom from bow wav extends in cone to on usually no c Boom Mod sonic boom t al., 2002] d luding focu : onventional v ir-launched s orizontal-tak allistic entry alcon 9 first s xample of th at will carry ightly under unpowered to de, where it scent and fin ing the final Two are e stics and sho 2-8 SpaceS sonic boom ackground o in Section 2. ture evolutio lculated by th ic Boom Bac onic Boom R nic boom is t an aircraft e of a boat. definitely, te e that refrac ted that boo eling Metho modeling me eveloped by W s analysis th ertical launc pacecraft, lik eoff, horizon vehicles, sim tage. ese operation six passenge 50,000 feet, a conventio resembles a al landing. part of ascen xamples of uld be accoun hipTwo, “fe modeling m n sonic boom 2.2 covering n. Section e model. kground ay Patterns he wave fiel in level flig While Figu mperature g ts upward, so m impinges dology thodology is yle. PCBo at can handl h vehicles; e Virgin Gala tal-landing sp ilar to Mer s, Virgin Gal rs and two c ascends und nal runway l conventional Figure 2-9 sh t after engin different ve ted for in the ather down” ethodology propagation the major ele 2.2.3 covers d about a sup ht. It shows re 2-10 show radients in th the ground i on a “carpet” based on th om4 is a sing e the four g ctic’s Space acecraft; and cury/Gemini actic’s Space rew. It is ai er rocket pow anding. Figu aircraft and ows its “feat e cutoff and hicle config modeling. . FIGU is presented and vehicle ments of the focal zones ersonic vehi a conical wa s the wave e atmospher ntercept goe correspondi e PCBoom4 le event, arbi eneral types ShipTwo; /Apollo mod ShipTwo is a r launched fr er to an alt re 2-8 show flies at low her up” mod most of des urations tha RE 2-9 Spac in the follo source charac model: sonic and Section cle. Figure 2 ve moving w as a simple e generally d s out to a fin ng to the wid sonic boom trary maneuv of commer ules, or Spa suborbital sp om a WhiteK itude above s SpaceShipT angle of attac e, used for h cent. The tw t effect so eShipTwo, wing section teristics. So boom gener 2.2.4 cover -10 is a clas ith the aircr cone, whose istort the wa ite distance o th of the gro prediction m er, full ray tr cial space ve ceX’s recove acecraft for nightTwo m 100 km, and wo in its “fe k, and is use igh angle of a o flight mod nic boom s “feather up” s. Section nic boom the ation, propag s the sonic sic sketch of aft, much lik ground inte ve from a p n either side und intercep 2-7 odel acing hicle rable space other then ather d for ttack es of ource . 2.2.1 ory is ation, boom sonic e the rcept erfect . It is t and

the length sketched recompres between. Figu shown at perspectiv perspectiv later time always as associated azimuths. perspectiv Figure 2- sketched generated further ma 10 that bo never fly FIGUR of the fligh in the figur sion at the r re 2-10 is dra a particular t e of rays es. The con s. The interc sociated with with rays g It is typical e is particul 12, or the ef in Figure 2- at a given tim neuver of th om follows a over its boom E 2-10 Son t track. Th e, where co ear generally wn from the ime, but was propagating e represents r ept of a given a particular enerated at p to denote th arly useful w fects of man 13. From th e will interc e vehicle wil long an aircr footprint. ic boom wav FIGURE e waveform mpression in coalesce int perspective o generated o relative to ays that are ray cone wit time along articular azim e azimuth as hen consider euvers, wher e ray perspe ept the groun l have no eff aft’s track. I e field. 2-12 Ray cu at the groun the forwar o a bow shoc f aircraft coo ver a time p ground-fixed generated at h the ground the vehicle’ uths at the φ, with φ = ing refractio e rays can co ctive it is a d ahead of th ect. This is c t is possible FIGUR rvatures in a d is genera d part of th k and a tail rdinates. No eriod. Boom coordinate a given time, is called an “ s trajectory. aircraft. Fig 0° downwar n by atmosp alesce into lso importan e vehicle at a ounter to the for an aircraf E 2-11 Wav real atmosp lly an “N-w e vehicle an shock, with te that the w s can also b s. Figure and which r isopemp.” A Points alon ure 2-11 sho d and φ = ±1 heric gradien high amplitu t to understa later time, a impression t to generate e versus ray here. ave” signatu d expansion a linear expa ave cone exi e viewed fro 2-11 shows each the grou given isope g the isopem ws rays at se 80° upward. ts, as sketch de focal zon nd that the nd once gene given by Figu a boom, turn viewpoints. 2-8 re, as and nsion sts as m the both nd at mp is p are veral Ray ed in es, as boom rated re 2- , and

Follo crossing a the associ the focus Focal zon properly r booms, bu FIGURE Figu associated moderate steep dive elliptical where des ray cone h vertical la ground un FIGU wing rays is nd tracing o ated isopemp region. Ther es are genera epresent. B t focal zone 2-13 Ray an accelera res 2-10 and with rays g climb and di , such as sh or even circu cent at times alf angle, ra unches, wher less refractio RE 2-15 R particularly ut the caustic pattern on th e is also no b lly associated oom amplitu dimensions te crossing and tion-induced 2-11 give enerated from ve angles, or own in Figur lar isopemp is near verti ys from that e ascent is at n by atmosph ay cone in le important wh line of a fo e ground for oom on the g with the edg des in focal z nd to be very convergenc focus. the impressi the bottom level flight a e 2-16, the e . This is of cal. Conver part of its tra a steep angl eric gradient vel flight. en an aircra cused superb the same m round before e of a boom ones can be small. e in FIG on that the of an aircra s shown in F ntire ray con particular im sely, if a veh jectory will e. In these ca s causes prop FIG ft is maneuv oom from ac aneuver. No the onset of footprint, an three to five URE 2-14 accelera ground foot ft. This is t igure 2-15. e may inters portance fo icle is climbi not reach the ses, sonic bo agation to th URE 2-16 R ering. Figur celeration. F te how the is the focal zo d often requi times that o Isopemp con tion-induce print of a b he case for For an aircra ect the groun r space fligh ng at an ang ground. Th om is not exp e ground. ay cone in e 2-13 shows igure 2-14 s opemps over ne at the left re refined de f nearby “ca vergence in d focus. oom is gen aircraft that ft in a suffic d, resulting t reentry ana le steeper tha is is importa ected to reac diving flight 2-9 rays hows lap in edge. tail to rpet” an erally are at iently in an lysis, n the nt for h the .

2-10 2.2.1.2. Vehicle Source Characteristics Sonic booms are the wave field of a supersonic vehicle, as heard on the ground. The boom begins as the near field pressure field around the vehicle, then propagates to the ground according to the ray patterns described in Section 2.2.1.1. The initial signature is generally defined by the F-function [Whitham, 1956], which is a normalized representation of the pressure field. This is related to the aerodynamic loads and flow around the vehicle. It can be computed from detailed knowledge of the geometry using several different methods as described in Section 2.2.2.1. 2.2.2. Sonic Boom Theory Sonic boom calculations can be divided into specific elements: Prediction of the pressure disturbance in the vicinity of the vehicle. This is required at a radius large enough that the disturbance is a locally axisymmetric acoustic wave, and small compared to atmospheric gradients. For supersonic slender bodies, this source function can be calculated directly from vehicle geometry via area rule methods. The source function can also be obtained empirically from wind tunnel or flight test measurements or analytically from CFD (Computational Fluid Dynamic) flow-field calculations. Section 2.2.2.1. describes the four currently available methods for using vehicle and signature data in PCBoom4. Calculation of linear acoustic propagation to large distances, accounting for atmospheric gradients. This is accomplished by the method of geometrical acoustics. The linear ray field computed depends only on the vehicle kinematics and atmospheric properties. The amplitude of the propagating acoustic disturbance is governed by the change in area of ray tubes (bundles of differentially separated rays) and local acoustic impedance. Section 2.2.2.2 characterizes the application of linear acoustic principles to sonic boom. Calculation of the non-linear steepening of the boom signature as it propagates. One technique employs a cumulative “advance” or “age” parameter which is computed as part of the geometrical acoustics solution. This age parameter defines the differences in arrival time, at various distances along a ray, between an infinitesimal strength acoustic wave and a unit-strength wave. The non-linear distortion of each part of the boom signature consists of an advance proportional to its original strength times the age parameter. Signature Aging and steepening techniques are described in Section 2.2.2.3. The three elements above are the standard components of sonic boom calculation programs. Figure 2-17 shows the typical computational flow. Note that elements (1) and (2) are carried out independently, and their results merge only in the non-linear signature calculation (3).

2-11 FIGURE 2-17 Logical flow of sonic boom calculations. There are additional elements to sonic boom computation which must be considered under certain circumstances: Prediction and Computation of Focal Zone Signatures. Focal zones exist where ray tube area vanishes and the linear acoustic solution is singular. Diffraction effects cause such focused superbooms to be finite, and validated local solutions are available. Focus solutions must be locally applied to focal zones, using the adjacent geometrical acoustics calculation as a boundary condition. PCBoom is the only sonic boom code to implement proper focal zone theory. Section 2.1.3 contains a brief description of theory and PCBoom computational technique for focal zones. Computing finite shock thickness. Shock waves have a finite thickness, dependent on shock amplitude and atmospheric conditions. This thickness is very small compared to overall signature length, and is usually neglected, with sonic boom shocks treated as zero-thickness jumps. Shock structure affects the upper frequency content of booms, and is important in loudness calculations. Shock structure is controlled by a combination of non-linear steepening and molecular relaxation processes. Sufficient data are available to empirically define nominal shock rise times, which can be applied in a simple manner to nominal thin-shock ground boom waveforms. The simplified finite shock thickness technique is included Aircraft Trajectory Atmosphere Ray Tracing Aircraft Geometry Flight Parameters Area Rule F-Function Ray Tube Area Age Parameter Acoustic Amplitude Aging/Steepening Shock Formation Ground Reflection Sonic Boom At Ground

2-12 in PCBoom. The main ray tracing and signature aging module, FOBoom, calculates signatures with thin shocks. The signature and footprint post-processing module PCBfoot applies a Taylor structure to each shock in the signature. Propagation of Sonic Booms through turbulence. While most sonic boom propagation physics is deterministic, turbulence effects are stochastic processes that affect waveforms passing through turbulent regions, most notably the turbulent boundary layer consisting of the last few thousand feet of propagation. In the context of sonic boom modeling, turbulence effects are applied as a statistical variation about the deterministic solution. Turbulence modeling is not a feature of PCBoom4, though it is employed as a post-processing function in later versions of the program. To improve turbulence modeling capabilities, NASA is currently conducting a research program, led by Wyle, to study the effects of turbulence on sonic boom propagation. 2.2.2.1. Sonic Boom Source F-Functions PCBoom4 has four input modes currently available for vehicle and signature data which are described in the following sections. These input modes include 1) Original Thomas form, 2) Simple F- function, 3) Carlson N-wave F-function, and 4) Hypersonic Blunt Body Reentry Model. Original Thomas form THOMAS is the original Thomas code [Thomas, 1972] input format, where NX pairs of the F- function DPP (∆P/P) and axial distance X are provided by the user. NX is the number of points in the signature. THOMAS is applicable only at one azimuth and flight condition. This input may come in two forms, an inline definition, and an external file definition. Axial coordinate X can be full scale for an aircraft of length AL, or subscale for a model length ML. The original Thomas model was intended for direct input of wind tunnel model data. Unless actual wind tunnel data is being used, it is common to use the full scale length for both. Supporting input data are NX, DPP, X, AL and ML. DPP is dimensionless, and the lengths are in feet. This mode is appropriate for a single azimuth, corresponding to that for which DPP applies. Simple F-function Simple F-Function Mode is useful if DPP is known only at zero (downward) azimuth and booms are expected to be N-waves. NX, DPP and X are read, but DPP is considered to be the F-function under the aircraft at flight condition EMREF (Mach number) and PVFFN (reference pressure) for steady level flight. The input F-function is scaled to other loads, flight parameters, and azimuths by use of Carlson's formulae. Supporting input data are AL, ML, WEIGHT, PVFFN and EMREF. WEIGHT is the aircraft weight in kilo pounds and the others are as described. Carlson N-wave F-function It was noted in the discussion of Figure 2-10 that the near field wave pattern generally coalesces into a simple N-wave shape at the ground. Carlson [Carlson, 1978] developed a simple semi-empirical method to compute N-wave booms from aircraft in steady flight. It was a combination of ray tracing analysis and fitting of flight test data. The aircraft source was represented by a shape factor that depended on general configuration, length, and a lift factor that was related to weight. He pointed out that if an N- wave was expected at the ground, an effective F-function could be constructed from those factors, as shown in Figure 2-18. Figure 2-19 shows shape factors for various modern aircraft types. SpaceShipTwo in its feather mode is essentially a lifting body, similar to the space shuttle. Its F-function source could thus be represented by the shuttle’s shape factor curve, using the length shown in Figure 2-9 and the weight provided.

FIGURE N-wav Laun As underexpa F-function parameter 1975] the universal parameter Hyp The Tiegerman It is best u weight an assumes a did not ge 2.2.2.2 P Soni signature propagate is based orthogona tracing yi 2-18 F-fun e, based on s ch Vehicle M part of the nded rocket for the veh s are based ory with dra plume mod s. ersonic Blun hypersonic ’s thesis ent sed for non- d the kinema ballistic ree nerate lift wh ropagation U c boom pred along linear s along a ray on the aircr l to wavefro elds the posit ction approx hape factor ode – Exte “Carlson N exhaust plum icle itself, im on a shape g correspond el [Jarvinen t Body Reen blunt body itled ‘Sonic B lifting hyper tics of the traj ntry. Most h ile on a balli sing Geome iction consi acoustic rays tube. The s aft's configu nts, are trace ion of the boo imated as a and length. nsion to Car -wave F-fu es on the bo mediately f factor corres ing to the e et al., 1970 try Model reentry mod ooms of Dr sonic vehicle ectory. This ypersonic ree stic reentry. trical Acous sts of the n . The proce hape of the a ration and d by the met m and also i n FIG lson nction” mod om. A startin ollowed by ponding to T ngine’s plum ]. Engine el or drag-d ag-Dominate s. In the blun is a volume ntry vehicle tics on-linear pro ss is illustra coustic signa aerodynamic hod of geom ts amplitude. URE 2-19 C variou e, PCBoom g signature i an F-function iegerman’s e drag, and thrust and p ominated b d Hypersoni t body mode only model th s have been b pagation of ted in Figure ture is given loads (Sec etrical acous arlson’s sha s aircraft ty accounts f s defined by for the plu hypersonic b length from lume drag a lunt body m c Vehicles’ [ , drag is dete at does not a lunt, capsule a near field 2-20. A ne by the F-fun tion 2.1.2.1) tics [Blokhin pe factors fo pes. or the effe a Carlson N- me. The Ca oom [Tiege the Jarvinen re required ode is base Tiegerman, 1 rmined by v ccount for li like vehicle aircraft pre ar field sign ction  F w . Acoustic tzev, 1946]. 2-13 r ct of wave rlson rman, -Hill input d on 975]. ehicle ft and s that ssure ature, hich rays, Ray

Figu cone that independe interest on plane con 1972] calc Acou Cons where P sound spe assumptio The amplitude including substitute phase fun FIGUR re 2-21 illustr is orthogona nt variables: that ray con taining the fl ulated boom stics and th ider a sound  trp , r is a spati ed. Wavefr ns or restrict acoustic equ acoustic pe flow. If E d, one can at ctions. In g E 2-20 Sch ates the coor l to the Mach the time t e. The sense ight path and for a single FIGUR e Eiconal field in whic          ti erP   al amplitude onts are loc ions; it is sim ations may rturbations a quation (1) tempt to solv eneral, the r ematic of son dinates of ray cone. For a at the aircraf of  is sho positive  ,t . E 2-21 Initi h the acoustic      a rW  function, W i of r wher ply a conven be written b ssumed), but and compar e for  WrP , esulting syste ic boom pro s from their given flight t, and the az wn in the fig out the left w al ray cone a pressure at f  r is a phas e  rW  = c ient form. y linearizin allowing ar able ones f  r and com m is not ea pagation al origin at the path and atm imuthal ang ure, with 0 d ing. The o nd coordina requency  e function, a onstant. Eq g the fluid bitrary gradi or velocity parable velo sily solved a ong a ray tub aircraft. Ray ospheric pro le  that ide egrees being riginal Thom tes. is written as nd a is a uation (1) d conservation ents of unpe and density city and den nd is freque e. s emanate in file, there ar ntifies the r down in a ve as code [Th : constant refe oes not carry equations ( rturbed quan perturbation sity amplitud ncy depende 2-14 a ray e two ay of rtical omas, ( rence any small tities, s are e and nt. If, 1)

2-15 however, the limit  is taken, such that the acoustic wavelength is small compared to the gradient scales of unperturbed flow quantities, the following two equations are obtained [Blokhintzev, 1946]: 2 0 22 c qW   (2) 02 00 2 2 00 2                sVqa aP qa aP t   (3) where WVaq s    0ˆ VnaV os   W Wn      ˆ is the unit vector normal to wavefronts and 00 ,, Va   are the local fluid density, sound speed, and velocity, respectively. Equation (2) is the eiconal equation, and shows that sound propagates along rays with velocity sV  and that successive wavefronts (surfaces of   rW  constant) are constructed by Huygen’s principle. It is somewhat more complex than the usual optical form, since a mean flow can convect sound, but there is no ether to convect light. Equation (3) has the form of a conservation equation along rays. It may be integrated to give the following acoustic amplitude relation: const qa AVp s 2 00 2  (4) where A is the normal cross-sectional area of a ray tube (bundle of rays, analogous to a fluid dynamic stream tube being a bundle of streamlines). This quantity is commonly referred to as the Blokhintzev invariant. Note that, in the presence of wind, the ray trajectories which transport energy are not necessarily normal to the constant phase wavefront surfaces. Ray Paths In PCBoom, ray paths are computed by direct numerical integration of the eiconal. The difference form for a horizontally stratified atmosphere over a flat Earth is given in [Thomas, 1972]. A simple linear predictor/corrector technique is used. An initial ray unit vector is taken as normal to the Mach cone at the aircraft, which is moving relative to the local wind. A ray tube is constructed from four corner rays, as sketched in Figure 2-22. The rays are separated by small but finite angle  (azimuth  about the flight path is relative to a vertical plane) and time t along the trajectory. The initial ray angles at the different time steps account for differences in flight path angle, heading, and Mach number, so that the ray tube implicitly accounts for maneuver effects. The integration time steps along all four rays are coordinated such that a given number of steps correspond to a constant phase on all four rays.

The diagonally a quantity local unit sectional a Ellip Trad flat Earth where pro gradients. boom car distances examples capability section an scheme th al., 2010] Ther traditiona where B i can comp implemen A fu 23 has be normal to FIGURE cross-section across the t proportional vector of th rea. soidal Earth itionally, son . This is ge pagation dis There has pet edges fr involved, the of long ran to PCboom4 d the one fo at have alrea . e is nothing l flat stratified pp  s an amplitud ute ray paths ting an appro ll 3-D ray tra en implemen the local sur 2-22 Ray t al area of a ube ( ,t to to the area o e ,t ray is ic boom pro nerally an a tances are s been recent om high alti Earth's curv ge boom pro to accommo llowing descr dy been impl implicit abou geometry.   B F  0 e parameter and B, subse priate ray tra cing in an Ea ted in later v face of the e ube outlined ray tube is   ,tt f the constan then formed pagation is c cceptable app hort relative interest in lo tude flight a ature is not n pagation. F date a full t ibe the Earth emented in l t a near fiel A near field s combining sc quent propag cing scheme. rth fixed geo ersions of PC llipsoid, and by four cor obtained by and  ,t t phase surfa , to yield a omputed for roximation to Earth cu ng range son nd for analy ecessarily n or this reas hree dimensi fixed geoce ater versions d signature o ignature may ale factors an ation and ste centric (EFG Boom. Latit longitude  i ner rays  first taking  to tt  ce. The dot quantity pro a horizontal for primary rvature and ic boom pro sis of over- egligible. Sp on, it is bei onal atmosph ntric coordin of PCBoom r Whitham's be represent d the Blokhi epening appl ) coordinate ude  is the s the angle e and  ap the cross-p , in Figure product of th portional to ly stratified a downward-pr typical horiz pagation, bo the-top boom acecraft asce ng considere ere and a no ate system an [Plotkin et a rule that lim ed in acousti ntzev invaria y. The crux system, illus elevation ang ast of the x a art. roduct of v 2-22). This is quantity an the normal tmosphere o opagating bo ontal atmosp th for analy s. For the nt and reentr d whether to n-flat Earth. d 3-D ray tr l., 2007; Plot its analysis t c form as nt. As long a of the extens trated in Figu le of a line t xis. The pie 2-16 ectors gives d the cross- ver a oms, heric sis of long y are add This acing kin et o the (5) s one ion is re 2- hat is rcing

point of t sketched i x,y that c offset to t an initial r Schu Beca There are trajectorie Schulten relation fo where q i somewhat and the ra Equa natural ch integratio derivative layered sc surface. G easy to ex he x axis is n Figure 2-2 an be related he ellipsoid). ay position a FI lten’s 3D Ra use the atmo a number s derived fro [Schulten, 19 r the ray trac dt qd  s the wave no formidable l dt dq dt dq dt dq z y x    y trajectory x  dt xd  tions (7) and oice for the n and ray tub s of wind an hemes. Thi iven a grid tend this to the prime m 3, are genera to , ) and Calculation nd vector are GURE 2-23 y Path Integ sphere is curv of 3-D ray m the eicon 97]. That f e: qaq   rmal vector, ooking, but i q z aq q y aq q x aq                is given by: w q qa    (8) are ver 3-D extens e area techn d sound spee s was set up of profiles (a include profi eridian at th lly defined in altitude h ab of EFG coor obtained, the Ellipsoidal ration ed around th tracing form al equation. ormulation is   qw   a is the soun n Cartesian c z w y w x w          y similar in ion. These iques as in th d gradients a as a simple s is available le lookup for e equator. terms of loc ove the ellips dinates from ray path ma Earth and E e Earth, a 3-D ulations in t The method surprisingly w d speed, and omponent for form to the 2 equations w e original c re required, r shell atmosph from 4D we , for a fu Aircraft loca al flat Earth, oid (or abov ,,h is exp y be compute FG coordina atmospheri he literature selected for simple, and w is the vec m it become -D relations ere impleme ode. There w ather than ju ere, with sin ather forecas ll 3-D atmosp tion and flig i.e. , (or l e the geoid, w licit and strai d in EFG co te system. c ray tracing , based on use here is provides the tor wind fiel s: used by Th nted using t ere several st z gradients gle profile n ting systems here. The a ht paramete ocal tangent hich has a k ghtforward. ordinates. scheme is ne integration o that present following v d. Equation omas, and w he same nu complexities as in horizo ormal to the ) it would be tmosphere lo 2-17 rs, as plane nown Once eded. f ray ed by ector (6) (6) is (7) (8) ere a meric : 3-D ntally local very okup

routines a are compu Prov origins an step sizes Later ellipses si the spatia Admittanc sample ra types of o 2.2.2.3 S As a signatures rather tha the local s Prop where p waveform Figu When a p become si The wave. Wh for weak shock. T lready contai ted. ision was ne d timing of t at their reflec al cutoff lim milar to the m l homogenei e ellipses ar ys at all azim ver-the-top ra ignature Agi lready mentio . Each portio n being ambi ound speed a agation spee is the acous advances pr FIG re 2-24 show ortion of th gnificant and wave propag en a shock f shocks is the his leads to a n , as argu eded for ref he four rays i tion points. its in the ho ethod used ty of a tradit e used for g uths to estab ys. ng ned one imp n of a signat ent speed 0a nd the acous d  0 aa  tic overpressu oportionally t URE 2-24 E s portions o e wave appro a shock form ation speed g orms, its pro average of th simple "area ments, and t lection of ra n a tube, to e rizontally st in TRAPS [T ional 2-D la eneral guida lish which r ortant facet o ure advances is given by tic velocity      0 1au re in the wa o its original volution and f the wave fo aches vertic s, with a bal iven by Equa pagation spe e isentropic balance" co he full 3-D g ys at the gro nsure that th ratified mode aylor, 1980] yered atmosp nce, but it w ays would re f sonic boom proportiona the local sou u:    02 1 p p   ve and 0p is pressure, as steepening lding over. al, the gradi ance between tion (9) repr ed is govern speeds corres nstruction fo radient comp und. This ey remained l were simp . Because a here, this pr as necessary ach the grou propagation lly to its amp nd speed plu the ambient sketched in F of sonic boo This does n ent is strong steepening a esents the ise ed by the Ra ponding to  r insertion o onents need required coo in phase whi le to obtain, 3-D atmosph ocess was n to use bru nd as either p is the aging litude. The p s the perturb pressure. E igure 2-24. m signature. ot, of course enough that nd losses. ntropic spee nkine-Hugon p just befor f shocks. Th ed in Equatio rdination bet le having dif using admit ere does not o longer pos te force and rimary or va and steepeni ropagation s ed increment ach element , actually ha loss mecha d of a quasi- iot equations e and just aft e aging proc 2-18 n (7) ween ferent tance have sible. trace rious ng of peed, a to ( of the ppen. nisms linear , and er the ess is 9)

2-19 implemented in two steps: an "age parameter" that defines how much the wave has steepened, and a general area balance procedure. The Age Parameter Sonic boom propagation may be summarized by the linear solution [Witham, 1956] provided in Equation (5) and repeated here as Equation (10) for convenience.   B Fpp  0 (10) The arrival time t of a linear acoustic wave element  at distance s along a ray is given by 0 0 s ads t  (11) For a weak non-linear wave, whose propagation speed is given by Equation (9), the arrival time is given by           s p pa dst 0 0 0 2 11     (12) which may be written    ss a ds p p a dst 0 000 0 2 1    (13) by virtue of 02 1 p p    being small compared to 1. Rearranging Equation (5),   B F p p   0 ,    sF a dst s    0 0 (14)      s Ba dss 0 02 1   (15) (s) is known as the age parameter [Hayes et al., 2009]. This quantity depends only on ray tracing. The advance sketched in Figure 2-24 depends only on this parameter (which is a function of s along a given ray) and the F-function of each element. Equation (14) represents isentropic propagation speed, and leads to multiple-valued regions, equivalent to characteristics crossing. That represents shock formation. As with crossing characteristics, where a shock is fit at the angle bisecting the characteristics before and after it, once Equation (14) predicts an infinite slope that point is replaced with a shock whose speed is the average of the isentropic speeds ahead of and behind it. This property (shock at the average of the isentropic speeds on either side) leads to the "area balance" rule for shock fitting. The construction sketched in Figure 2-24 is done, including the overlaps in the bottom sketch. A shock is inserted into each overlap region, positioned such that the overlap area ahead of it equals the area behind. The overlap regions are then deleted. The process can be intricate for complex signatures, but a systematic algorithm, described in the next section, is available and has been implemented in PCBoom.

Area A ge [Middleto The key to balance o handling process fo Figu integral o regions. points of o complex Equation equal that shocks wi similar m iterating t Midd integral o where  shocks fo it can evo between s intersectio coalescen Haye piecewise the differe Balance an neral signatu n et al., 196 this method f the F-funct multiple cros r its applicat re 2-25 illus f the F-funct Each region verlapping f signatures is (14), then re of the origin thin each reg ethod consis hem up and d leton and Ca f the F-functi   I is the aged rm. Because lve in the for uccessive m ns of overl ce) can occur s et al [Hay fashion. Re nce as 1 : d Middleton re folding a 5] and used i is definition ion correspon sings, is very ion. FIGURE trates the ke ion. It is ag can be determ orward runni generally tre solve multipl al. A brute f ion by comp ts of drawin own until the rlson [Middl on:     0 dF coordinate fr the coordina ward or rever axima and m apping segm , and rules w es et al., 19 taining the n -Carson-Hay nd shock fit n the ARAP of a function ds to crossi well descri 2-25 Midd y elements o ed and steepe ined to be " ng segments ated by the e-valued reg orce method uting the area g lines at th area is balan eton et al., 19 om Equation te  is strain se direction. inima of  . ents. Regio ere provided 69] noted th otation for th es Method ting method [Hayes et al that is the in ngs in the in bed by Midd leton-Carlso f signature f ned along w forward runn define shock area balance ions by notin is to identify s of the over e "un-aged" ced. 65] develope (14). Area ed according Middleton a Segments m ns with mu for the select at it was not e original  was develop ., 1969] and tegral of the tegral functio leton and Ca n aging met olding. A f ith the F-fun ing" or "bac locations. T rule: perfor g that the ar multi-valued lapping lobe slope acros d a more dire balancing d to the age p nd Carlson w ay overlap, ltiple overla ion of the pro necessary t as itself and ed by Midd TRAPS [Ta original F-fun n. The pro rlson, who g hod. unction S is ction, and w kward runnin he formation m the constr ea of the age regions, the s before and s the origin ct method by ictates that arameter and rote I in a and shocks pping segm per shock in o consider th a strained ve leton and Ca ylor, 1980] c ction, so tha cess, and rul ive a step-by computed a ill develop f g." The cro of shocks in uction defin d F-function n iteratively l after the shoc al F-function working wi I is conserv F-function v piecewise ma are located ents (represe each case. e integral I rsion represe 2-20 rlson odes. t area es for -step s the olded ssing more ed by must ocate k. A and th the (16) ed as alue, nner, at the nting in a nting

2-21     Fs1 (17) The equivalent of Equation (16) at propagation distance 0s is       dFS0 (18) where the -∞ denotes beginning the indefinite integral far enough forward so that any steepening will be covered. At propagation distance s, where 1 is given by Equation (17), Hayes et al show that the equivalent to Equation (16) is       sSS    2 2 1 01  (19) When plotted, Equation (19) yields the same result as Middleton and Carlson's procedure, but has the benefit of not having to identify the reversal points while performing the integration. The reversal points are readily found by testing the difference between successive values of 1 . Computing S as described by Hayes et al [Hayes et al., 1969] is simpler than the multiple I process of Middleton and Carlson. Hayes et al.'s subsequent description of identifying shocks is, however, more complex, relying on a graphical interpretation and not providing rules for multiple overlap regions. The original ARAP program [Hayes et al., 1969] did not fully automate the shock insertion process, but rather output the overlap information and left it to the user to insert the shocks. Middleton and Carlson provide a complete procedure:  Compute the curve given by Equation (19);  Remove backward-running segments;  Identify all intersections between the remaining segments;  Insert shocks at each intersection; and  For multiple shocks within a segment, follow a hierarchal procedure defined in “Universal Model for Underexpanded Rocket Plumes in Hypersonic Flow” to identify the proper shock(s). PCBoom implements aging by using Equations (17) through (19) and applying Middleton and Carlson's shock fitting procedure. The amplitude is then scaled via the Blokhintzev invariant. This procedure permits rapid calculation of the signature at any position along the ray. The age and amplitude parameters may optionally be written to an output file, and used to compute boom for other configurations at the same flight and atmospheric conditions. This file is exploited by the GENGS system [Plotkin, 2009] that is used for low boom shape optimization. Optimizers can cycle through several thousand configurations, and the ability to re-use the ray tracing results saves considerable computation time. 2.2.3 . Focal Zones Focal zones occur when a combination of vehicle maneuvers and atmospheric gradients give rise to concave wavefronts, and the subsequent propagation distance is sufficiently long for the focal point to be reached. Conceptually, it is similar to focusing by a lens. The lens analogy can, however, be misleading. It is a good description for a wavefront with constant curvature. Except for rather special cases, an aircraft in motion will not generate such perfectly focusing waves. Curvature will vary, so that different portions of wavefront (generated at different times at the aircraft) will have different curvature. The focus

2-22 point will thus move in space, and in two dimensions will trace out a line. This line (or surface in three dimensions) represents a focal zone. Mathematically this zone is an envelope of rays, or a caustic. The location and shape of a caustic is defined by geometrical rays. The geometrical acoustics portion of standard sonic boom computer programs is adequate for location of foci. This was well verified by the experiences of Wanner et al [Wanner et al., 1971] in planning a sonic boom focus flight test program, and by NASA [Holloway et al., 1973] in planning boom measurements from Saturn launches. Onyeonwu [Onyeonwu, 1973] developed a sonic boom prediction program which has the specific capability of identifying the location of focal zones. Determination of signatures at focus requires more geometrical information than location, however. The concept that a real focus is usually distributed over a caustic surface, rather than a point, suggests that focus amplification would depend on the extent of the caustic surface as compared to the original wavefront. A more quantitative viewpoint is that the caustic forms a boundary to the wave system within which diffraction effects play a key role in the focused signature. Caustic location and geometry are defined from both atmospheric conditions as well as vehicle motions. The curvature of the caustic surface, an important parameter in the signature itself, is described in the treatment of focus behavior. There are a number of conceivable ray configurations which can lead to focus. Within PCBoom, however, only those which can be produced by a maneuvering vehicle in a smoothly varying atmosphere are of interest. The PCBoom Technical Reference [Plotkin et al., 2010] reviews the ray configurations associated with vehicle maneuvers and normal atmospheric gradients, and identifies the kind of foci which can occur and which are implemented in PCBoom. 2.2.4 . Sonic Boom Metrics PCBoom computes full sonic boom waveforms. The footprint and signature post-processor computes Pmax, Lpk, Lflt, CSEL, ASEL and PL (Loudness). These metrics are described below. Pmax is the peak overpressure in the signature. It is presented in units of pounds per square foot (psf). This is the traditional physical metric for N-wave sonic booms. Both Pmax and Pmin are presented. Lpk is the peak overpressure level, refpk PPL max10log20 , where Pref = 20 Pa. It is just the peak overpressure in decibel format. Lflt is the unweighted energy-integrated level, in units of dB. It is defined by              dt P tP flt ref t L sec110 2 10log10 (20) where Pref = 20 Pa as above. The integration period is inclusive of the entire boom. This is essentially an unweighted sound exposure level. It is sometimes denoted ESEL. Note that the time integration is normalized by one second. CSEL is the C-weighted sound exposure level [ANSI, 1996]. ASEL is the A-weighted sound exposure level [ANSI, 1996]. PL is the loudness, in units of PLdB. This metric [Shepard et al., 1991] is based on Stevens Mark VII loudness [Stevens, 1972]. For sonic boom application, each one-third octave band is first integrated over a time inclusive of the entire boom, similar to Equation (20), but with time normalization being the human auditory response time of 70 milliseconds.

2-23 As noted, Pmax is the traditional measure of the amplitude of N-wave booms. CSEL is specified by standards [ANSI, 1996] for quantification of high energy impulsive sounds. It correlates very well with peak overpressure, with CSEL = Lpk - 25 dB to within a dB or two for most N-wave booms. It correlates poorly with subjective loudness of booms [Leatherwood et al., 2002]. PL correlates very well with subjective loudness of booms across a wide variety of shapes, while ASEL works almost as well for simpler symmetric boom shapes [Leatherwood et al., 2002].