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NASATechnical

Paper3134

November 1991

A Loudness Calculation

Procedure Applied toShaped Sonic Booms

, J / •-

-//

Kevin P. Shepherd

and Brenda M. Sullivan

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('c_S,_) 1 3 'p CSCL 20A

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https://ntrs.nasa.gov/search.jsp?R=19920002547 2020-03-09T22:10:04+00:00Z

NASATechnical

Paper3134

1991

National Aeronautics andSpace Administration

Office of Management

Scientific and Technical

Information Program

A Loudness Calculation

Procedure Applied toShaped Sonic Booms

Kevin P. Shepherd

Langley Research Center

Hampton, Virginia

Brenda M. Sullivan

Lockheed Engineering & Sciences Company

Hampton, Virginia

_ _ • _ _ _ _i_r1_ _ _

Abstract

The economic viability of a supersonic commercial transport airplane

would be much enhanced if it could fly supersonically over land. Efforts

to design an airplane to produce a minimized sonic boom at the ground

require knowledge of the impact of ,sonic booms on people. Loudness,

being a fundamental and well-understood characteristic of human he_zr-

in9, was chosen as a means of quantifying the magnitude of sonic boomimpact on people. This paper describes in detail a procedure that can be

"used to calculate the loudness of sortie booms. The procedure is applied

to a wide range of sonic booms, both classical N-waves and a variety ofother shapes of boom,s. Th.e loudness of N-waves is controlled by over-

pressure and the assoeiated rise time. The loudness of shaped boomsis highly dependent on the characteristics of the initial shock. A corrt-

par|son of the calculated lou&tess values indicates that shaped booms

may have signifieantly rrduccd loudness relative to N-waves having the

same peak oveTpressurr. This result implies that a supersonic traTts-

port designed to yield ntinimized sonic booms may be sub.stantially moreacceptable than art _tneonstrained design.

Introduction

The National Aeronautics and Spaee Administra-

tion (NASA), in conjunction with major aerospacecompanies, is investigating the technical and eco-

nomic fe_Lsibility of a new generation of supersonie

transport airplanes. The first phase of this investiga-tion is aimed primarily at addressing critical environ-

mental concerns about at nlospheric impact, airport

noise, and sotlic })()oln. The possibility that engineemissions might cause major pertm'bations to strato-

spheric chemistry, leading to det)letion of the ozoile

layer, nmst be exl)lored thoroughly and resolved be-fore serious consideration can t)e given to the devel-

ot)ment of a supersonic transport. Pul)lic acceptance

of such a transI)ort will also depend on its at)ility tomeet airport noise lev('l standards currently applied

to newly designed subsonic transt)orts. In a(ldition,this NASA/industry 1)rogram aims to establish either

tha.t sui)ersonic flight over lan(l with acceptably low

levels of sonic boom can ])e assured or that design for

sut)sonie cruise over land will not ml(tuly compromisethe economic vial)ility of the aircraft.

Consi(leral)h_ effort is t)eing devoted to an ex-

atnination of the feasibility of designing and oper-

at.ing a supersonic transport that will yield a sonicboom acceptabh_ to tim affected population. Cur-

rent supersonic airplanes pro(tuce a sonic boom at

ground level that is generally referred to as an

N-wave. Such a pressure disturbance is character-

ized by a rapid rise to a maximum positive over-

pressure, which is followed by a relatively slow de-

cay to a below-amt)ient pressure, and then an abrupt

return to atmospheric pressure. Theoretical analy-ses suggest that precise tailoring of the volume and

lift (tistributions of a supersonic airt)lane can yM(t asonic |)oom at ground level that differs from the con-

ventional N-wave. Therefore, appropriate methods

are required to assess sonic t)oom impact in or(ter to

guide the design and operating ('onditions of the air-

plane. There is no COllSensus within the regulatory

and scientific comnmnities regarding the appropriate

metric for sonic booin assessment. Loudness, beinga fundamental and well-understood characteristic of

human hearing, was chosen for this stu(ty as a means

of quantifying the magifitude of sonic booms.

Tim purpose of this paper is to describe in detaila procedure that can be used to calculate the loud-

ne.ss of sonic booms. This procedure is based largelyon an approach described by Johnson and llot)inson(ref. 1). Their apt)roach utilized a loudness calcu-

lation procedure that had a low frequency limit of

50 Hz. Since the (tominant acoustic energy of sonic

booms is well below this frequency, they proposed an

ad hoc method to incorporate this low frequency en-

ergy. The procedure proposed in this paper utilizes amore recent loudness calculation procedure that ex-

tends the frequency limit to 1 Hz. The procedure

is applied to a wide range of sonic booms, and par-tieular emphasis is given to an examination of the

loudness of sonic booms that (lifter in shape from

the classical N-wave. Finally, estiinates are made of

the loudness of sonic booms as they would be heard

indoors, hased on the measured noise reduction pro-

vided by typical dwellings.

Symbols

A,/3

D

f

i

P

P

T_, T> 7.._

first ail([ s('cOlld pr('ssur('s, r('s[)('('-

tiv('ly, shown ill figur(' 3, N/hi 2

-- 7"1 +7)

Fourier t ransfornl

frequency. [tz

l)cak ov(,rpr(,ssur(, (s(,c fig. 1), N/nl :_

l)rcssurc. N/in 2

time parameters shown in figures 1and 3

t I lille, SCC

IV4 A-weighting

11"(, ('- w(!ight ing

7" rise titlt(", SeC

_' [rc(tu(m('y, ra(tians

Loudness Calculation Procedure

Several al)l)r<)a(!hos havo ])(,(,tl |)r()t)()so(l for ass('ss -

ing the iml)act of sonic booms ()n t)('()t)le (reI_. 1 3).

The American National Stamtar(ts hlstitutc (ref. 2)

has adot)ted the C-weight('(t SOml(t t)rcssmc level.

The ('hoicc of C-weighting. which (mlpllasizes h)wfrc(tu('ncics relative to other noise inll)act metrics

such as A-weighting or Percciv(,d Noise Level, is an

attcnlpt to incorporate Ill(, cffc(ts of" structural vi-

brati()ns and the resulting; ratllin_, ()f ol)j(,cts. The

A-w('ight(,d somM pressm'e hwcl has been proI)ose(1

by Brown and tlaghmd (ref. 3) so |hat sonic 1looms

can 1)e assessc(t by the sam(' method ('(mml()nly used

for subsonic airplane flyover re)is(,. Sil,('e the l)ur-

t)(ise of this t)apcr is to exanline human r(,st)ons(, to

r(qatively small (tiflbr(mc(,s in sonic boom signatur(,s.

a more sophisticated loudness ('al('ulat ion t)roc('dur(,has been adopted. The api)(,n(tix ('(mr aills conveni('nt

methods for the calcutati(in of A- and C-weighted

S()llll(l pI'(_SSllF( ' 1CV(qS [()F SOlli(' |)()(HIIN.

The following sections des(:rit)(, the steps required

to calculate the lou(hwss of sonic t)()(mls. Tile ap-

I)roach req/fircs the pressur(, time hisi(lry (If the s(lnicboonl sigmlture as input. This tim(' history is t rans-

f'orme(t into the frequency domain and lhen convert('d

into an cfl'ective one-third (/ctavc band sound pres-

sure hwel Sl)(!('trum. Tll(' lou(hl(,ss of the sonic |loom

is then calculated 1)y using this spe('trum.

Energy Spectrum of Sonic Booms

N-wave sonic booms. S(wcral investigators

(refs. 1 and I 7) have derived expressions for the

spectruin of an N-wave sonic boom. Using Ill('

nomcn(:latm'c _4iv('n in Ill(, pressure fmlction of fig-m(' l giv(,s lh- F, mrier transform

'Jb

]"(_') : / ; ])(t) C i.,l (,_• 2

whi('h is fl):m(t t(> I)('

i21'(7) sin _'T 1 -7" 1 sill _'T2)

F(_,) - _'2TI (T2 TI)(1)

Prey;sure

. _ T 1 f_

l)

Fi_4ur(, 1. N-wave s(mi(" I)o.m signature.

1)elinin_z 7- - 7_ Tl and D = T1 + ;/'2 yields

the fl)llowin/('xl)r('ssi()n which is (,quivaleilt to 4hosederived il_ r('f('r('n('('s 1 and 6:

ill >(r sin _,I)_ cos =2-=)'+'r_ l)sin +,T_=cos @)

1+(+') ._':27(D - 7-)

For the cast' (if iIlstantan('ous rise trine (7) - 7' 1

:-0). equation (t) reduces to

i2P(sin _'7]- J/'l cos _'T1)

F '_,) ,.,.,'_TI

which is ('(tuiv;Ih'nl t() How(!s' f()rmulalion (rcf. 5)

since, in this cas_,. D = 2TI.

Tile flmetion I/P(_)[2 determin('s the energy spec-

tral density that is presented in figure 2 for an N-wave

having ;1 t)eak overt)ressure of 50 N/m 2, a total (lura,

tion (2T2) of 0.35 sec, and a rise time (r) of 0.008 sec.

The energy spectrum level in the figure is given per

Hertz, the reference t)I'essure is 2 × 105 N/m2, and

th(' reference time is 1 sec. The function cycles

rapidly between successive zeros which, after the first

few eyeles, are equispaced at intervals of 1/D in fie-

quency. Above at)out 40 Hz (indicated by the dashed

line in fig. 2 at 1/_rr tlz), the maxima are seen to be

modulated at a much sh)wer rate. Tiffs rate is (leter-

mined by the rise time of the N-wave.

100

Energy

spedral 80 [densily,dB/Hz 60

40 !

I2O

0O.i

=P-r-

;12 dB/octave

I I0 ,I

Frequency, jHz i

¢33/rtl) 1/_ r

I(X) 1000

Figure 2. l']nergy spectrum of N-way[' sonic I)()om wilh t)eak

overpr('ssur(' of 50 N/m 2. rise t ilno ()f 0.0()_, st'(', aIl(t total

duration of 0.35 sec.

As long as the rise time is Inueh shorter than

the total duration, the envelol)e ()f the st)ectrum can

be represented t)y the folh)wing thr('e straight-line

segl I leilt, s:

1. At the lowest frequeilcies, the spectrum level rises

at 6 (tB/o(:tave an(1 peaks at a frequency ()f ap-

proximately v/3/rrD Hz. The amplitude at this

frequency is controlled by the peak overt)ressure

and total duration (IF(,_,)I 2 -- p2D2/3).

2. The spectrum level then (teeays at 6 dB/octave to

a frequency equal to 1/rrr. At. this frequency, the

amplitude is contr()lled by the peak overpressure

and rise time (IF(_')[ 2 - p2r2).

3. Above this frequency, the spe(:trum level deeays

at 12 riB/octave.

Examining the effects of changes in the param-

eters defining an N-wave is instructive. For the

case of zero rise t,ime, the third straight-line segment

(12 dB/oetave) does not occur; the spectrum level

merely continues to decay at, 6 riB/octave. A dou-

t)ling of overl)ressure just raises the ('ntire spectrum

bv 6 (lB. A doubling of the total duration increases

the maximum sI)eetrum level I)y 6 (tFI and halves the

frequency at which the maximum occurs. The net

result is all ill(Tease of 3 (tB in the eltergy ('otltellt

of tile signal. The lnaxilnllnl in the st)ectrum moves

along tim line having a negative slol)e of 6 dB/o('lave

so that only the lowest frequencies are afl'eete(t l)y

a chaIlge in total dm'ati(m. A (t()ut)ling of rise time

halves the fre(luency at which the transiti(m ()(wurs

to the 12-dB/o('tave segm(,n(. Thus, th(' rise time

for constant ()v(q'pressure ('()nlr()ls the high frequ('n('y

conte]lt of t,he st)ectrunt.

Shaped booms. For a long airplane (e.g.. 100 m

long), it has been suggeste(t (refs. 8 an(t 9) that the

midfield sonic t)oom pressm'e signature may not have

fully evolved into an N-wave at ground level. Thus,

an airt)lane can t)ossit)ly be designed and operated

to yield a sonic boom that differs fronl an N-wave

in such a way that the iml)act on t)eot)le is r('(hl_e(t.(Darden's approach (ref..)) p('rmits minimizatioh of

either the initial shock or the maximmn overt)ressm'e

of the signature by means of careful t,ailoring of the

airt)lane volume aim lift (tistributiolls. These two

t)ossibilit, ies are illustrated in figure 3. A front-shock

Ininimized signature (fig. 3(a)) is characterized by

all initial shock ()f short rise time f()llowed by ;1, Hit)re

gentle rise to the t)('ak overpressure. The ('as(' of a

ttat-to I) signature (A = B in fig. 3(b)) is frequently

referred to as having minimized overpressure.

From using the nomenclature of figure 3(a), the

Fourier transform of the slmt)e(t signature was foun(l

to be

i2A (T3 sin _'7)- 7) sin ._T:I)

F(w) = cv2T'2 (Ta - 7'2)

i2 (B -,4_](T2 sin c_'Tl -TI sin _' T.2 )

+\ /

,fiTl (7"2 - T1)

Examination of this equation reveals that tim first

ext)ression is equal to the Fourier transform of an

N-wave having a peak overpressure of A and a rise

time of T 3 - T 2. The second expression corresponds

to an N-wave having a rise time of T 2 -T 1 and a

peak overpressure of B- (ATI/T2). Thus, the shaped

sonic t)oom of figure 3(a) may be considered to be

a combination of two N-waves. The first N-wave is

defined by the initial shock amplitude (A) and its

associated (short) rise time, and the second by a

3

relativelylongrisetimehavingtheamplitudedefinedpreviously.This is illustratedin figure3(c).

Pressure

B _ ]- _.- Time

(a) Front-shock minimized signalure.

-/'3 -T-_-A

(I)) Flat-top signature (,4 t3).

B - ATI/T 2

7' 1 7"2 T3

__7 -A

(c) Front-shock minimized signature shmving, combination of

1we N-waves.

Figure 3. Shaped sonic bo()m signalures.

For the case of a flat-top signature (A = B in

fig. 3(b)), the previous equation reduces to

i2A [(Ta T_) sin ,.,aTl 71(sin wT_ sit, _T,2)]F(,_,) =,.,,'2T 1(T:; "]22)

A spectrum of a. flat-top signature is presented in

figure 4 in which the total duration is 0.35 sec, the

peak overpressure is 50 N/m 2, the initial rise time

(T a - T2) is 0.008 sec, and the secondary rise time

(T2- T1) is 0.04 sec. This spectrum is virtually indis-

tinguishable from the spectrum of the N-wave (fig. 2),

which has the same duration and peak overpressure

and a rise time equal to the initial rise time of the

fiat-top signature.

4

Energyspectraldensity',dB/Hz

4O

2o0[

.1 I0 100

Frequency', Hz

1000

Figur(' 1. En('rgy spectrmn of fiat-top son!(' l)oom.

An examt)h, spectrum for a shaped sonic t)oom is

presented in figure 5. Once again, the initial rise time

was chose.n to lye 0.008 sec with a duration of 0.35 sec

and an initial shock anlplitude A of 50 N/m 2. The

secondary rise time is 0.04 sec and the peak over-

t)ressure is 10ft N/m 2. At. the low(,st frequen-

cies, the spectrmn level is 6 dB higher !halt the

N-wave spectrlnl! of figure 2 hecause of the (lou-

|)ling of overpressure. Above al)out 10 Hz. the

enveloI)es of the N-waste and shaped-boon! spec-

t,ra are remarkal)ly similar, with the latter distin-

guishe([ by more erratic modulations t)ecause of

the effect of lh(, secondary rise time. It is appar-

ent that at the lowest frequencies, the sl)ectrunl of

the shape(t boom is controlh_d })y the peak over-

t)ressure and duration. This ilhlstrates that a

shaped ho()m, with a far higher overpressure than an

N-wave, ca!| ('tmtailt silnilar acoustic energy excel)t at

the very low(!sl f!('q!!(meies where the hulnan ear is

extremely insensitive. Assuming that the secon(tary

140 r"

120 -

I00 #_

Energy' Ispectral 80density.dB/Hz 60

J4O

20i

• I 1 10 100 1000

Frequency', Hz

Figure 5. Energy speclrum of front-shock minimized sonic

boOlll.

rise time is substantially longer than the initial rise

time, the spcctrmn levels of a shaped boonl will

generally t)e controlled by tile initial shock alnt)litudcanti rise time at all but the lowest frequencies.

One-Third Octave Band Sound Pressure

Level Spectrum

The energy within any arbitrary wavefi)rm p(t) is

t)roportional to

which is equal to

1FF z --

27v. _ IF(_')I2 d_ 7r IF(_') d._

t)y Parseval's theorem an(t the fact that IF(w)l is aneven function of c_,.

The energy E within ally frequency t)and Wl to

:z2 is thus

E = IF(w)[ 2 dw' (3)7r, I

The calculation of one-third octave band energy lev-

els can thus be i)erformed by using a suitable nu-

merical integration method. This calculation can be

checked by summing the energy over all bands and

confirmiilg that the result is equal to the integral of

[p(t)] 2. This integral, which is independent of tile rise

time, is giv(,n as

F [p(t)] 2 dt = p2T 2, cy_

for the N-wave in figure 1 and is given as

F [ ][/0] 2 dt = 2. vc _ (A+B)(BT 2-AT1) +A2T3

for the shaped boom in figure 3.

In order to apply a loudness calculation pro-

cedure, it is necessary to convert the one-third

octave band energy levels into equivalent one-third

octave band sound pressure levels. Loudness calcu-

lation procedures are designed for use with soundsthat arc contiimous in time, and one-third octave

balm sound pressure levels are typically required as

input. Ill order to use the one-third octave band en-

ergy levels described previously (having dimensionsof (Pressure) 2 x Time) in a loudness calculation

t)rocedure, they nmst be converted into equivalent

one-third octave band sound pressure levels (having

dinlensions of (Pressure)2). The t)rocedure recom-

nlended t)y Johnson and Robinson (ref. 1) has been

adopted. This procedure is based on the assump-

tion that although the (turation of a sonic boom fronla comnmrcial transt)ort is 200 400 reset, the sound

energy in tile audio frequency range is contained in

the two pulses that occur at the front and rear ofthe waveform. Furthermore, it is a.ssumed that these

pulses have durations that are shorter than the criti-cal time of the auditory system. This critical time is

the time necessary to evoke a fllll auditory response,with the result that. tile loudness of acoustical stim-

uli of shorter duration is governed |)y serum energy.

A sound having a duration kruger than the critical

time will evoke no greater sensation of loudness nomatter how long it persists, and therefore loudness

is controlled by sound intensity in this instance. Ac-

cording to Johnson and Robinson (ref. 1), the vahlcof this critical t.ime is approximately 0.07 sec. Thus,

the calculated band energy levels arc divided by 0.07

and converted to sound pressure level by using the

standard reference pressure of 2 × 10 ,5 N/nl2.

As noted previously, the sound energy in the au-

dio fi'equency range is contained mostly within the

two pulses that occur at the fi'ont and rear of the

waveform. For any supersonic transI)ort the time |)e-

tween these pulses will be nmch greater than the au-

ditory critical time, and the two lmlses will t)e hear(t

as (tistinct events. However, the energy spectral den-

sity functions presented previously inclu(ted the en-

tire wavcform; thus. the sound pressure h,vels should

be reduce(t by 3 (tB I)efore calculating loudness. Thiscorrection assumes that the waveform is symmetri-

cal, both front and rear, so that the two tmlses will

be equally loud.

Calculation of Loudness Level

The loudness method selected for this investiga-

tion is Stevens' Mark VII (ref. 10). This method

is in contrast to that used by Johnson and tlobinson

(ref. 1). The equal loudness contours in the Mark VII

method extend to a very low frequency (1 Hz), and

thus they are appropriate for sonic booms that con-

tain significant low frequency energy. The calcula-

tion procedure for loudness level assumes that the

noise signal has been measured ill one-third octave

bands. The sound pressure levels in each band are

converted into a perceived value (in sones) and then

totaled according to a suinmation rule. The total isthen converted into a calculated Pcrccived Level by

IncaIls of the power function rclating perceived mag-nitu(te to sound pressure. A dout)ling of loudness in

sones is accomplished by raising the signal level t)y9 dB.

The spectrum of a one-third octave band sound

pressure level is required as input to the loudness cal-

culation procedure. This spectrum can be obtained

from the spectral density formulations and procedure

presented previously, or, for other sonic boom shapes,

it can be calculated by using widely available Fast

Fourier Transform computer routines.

Loudness of Sonic Booms

Loudness of N-Waves

Calculated loudness results are presented for a

range of those parameters that can be used to de-

scribe an N-wave. In particular, the range of values

of overpressure and rise time were selected to encom-

pass values likely, to be encountered in practice.

Tile duration of sonic boom signatures associ-

ated with any supersonic transport airplane (200

400 msec) will be nmch longer than the auditory criti-

cal time. Under these conditions, calculated loudness

is independent of the total duration of the signature.

Tile relationship of loudness level with both peak

overpressure and rise time is illustrated ill figure 6 for

Loudnesslevel,

dB

12O

110

100

90

8(1

Rise time.msec

-)

8

7(1 t I I J t12.5 25 51) 100 200

Overpressure, N/m 2

Figure 6. Loudness level of N-waves for outdoor listening

conditions.

a range of N-wave sonic boonls, each with a total

duration of 350 msee. Loudness level increases ap-

proximately 7 dB per doubling of overpressure and

decreases with increasing rise time. This latter ob-

servation reflects the relationship between tile rise

time and the high-frequency content of the sonic

boom illustrated in figure 2. A change in loudness

lcvel of 9 dB represents a halving, or doubling, of

loudness. These observations regarding the loud-

ness of N-wave sonic boonls have generally been con-

firmed by several experimental investigations (e.g.,

refs. 11 14).

6

Noisereduction,

dB

30

25

2O

15

10

5

Windows

_ Windows

.......... n.

0 10 100 1000

Frequency, Hz

Fignre 7. Noise ro, iuction for typical residential structures.

II0

IO0

Loudness 91)

level,dB

8O

70

60

Rise time,msec

1').5

24

8

I I I I I12.5 25 50 IO0 200

Overpressure, N/m"

Figure 8. Loudness level of N-waves for indoor listening con-

ditions and with windows open.

Estimates were made of the loudness of sonic

booms ms they would be heard indoors, based on mea.

sured noise reduction provided by typical dwellings.

The curves of noise reduction presented in figure 7

were derived fl'om measured data (refs. 15 and 16)

for a range of hollses, both with windows open and

closed. These noise reduction values were applied to

the sonic boom spectra before the loudness calcula-

tion was performed. Figures 8 and 9 illustrate the

relationship of indoor loudness level with both over-

pressure and rise time of N-wa_e sonic booms. In

these figures the ovcrpressure and rise time are char-

acteristics of the N-wave incident on the structure.

The trends are very similar to those obtained out-

doors, with the loudness levels approximately 10 dB

lower with windows open and 20 dB lower with win-

dows closed. This assessment of indoor levels ob-

viously makes no attempt to include the effects of

100

9O

Loudness 80

level,dB

70

60

5(_

Rise lime,msec0.5

I248

! I I I I

12.5 25 50 100 200

Overpressure, N/m 2

Figure 9. Loudness level of N-waves for indoors and windows

closed.

I00

95

Loudness 90

level,dB

85

8O

A, N/m 2

5Ot_

75 iJO i i 4I ,0 20 30 0 50

Secondary rise lime, msec

Figure 10. Loudness level of shat)ed booms for outdoor lis-

tening conditions with t)eak overt)r(!ssure of 50 N/m 2 andinitial rise time of 2 reset.

structural vibrations or secondary acoustic radiation

due to vibration-induced rattling of objects.

signature shown in fig. 3(b)), the loudness level is in-

dependent of the secondary rise time and equal to the

loudness of an N-wave having the same overpressure

and rise time as the initial shock of the flat-top sig-

nature. Reduced levels of the initial shock amplitude

lead to substantially reduced loudness, particularly

for large vahms of the secondary rise time. This ob-

servation was confirmed in an experimental investiga-

tion conducted by Niedzwiecki and Ribner (ref. 17),

which is the only study to ineort)orate shat)ed sonicbooms.

The noise reduction values in figure 7 were used

to calculate indoor loudness levels for the same range

of shaped booms as described previously. Figures 11

and 12 show the results for the conditions of windows

open and windows closed, respectively. The loudness

level of the fiat-top signatures is independent of the

secondary rise time, as was noted for outdoor loud-

ness. The effe.et of a reduced initial shock amplitude

is also very similar to that observed for the outdoor

CaSO.

9O

85

Loudness 80

level,dB

75

70

A, N/m-

5O

I j

65 0 l'0 10 30 410 50

Secondary rise lime, msec

Figure 11. Loudness level of shaped booms for iIMoor listen-

ing conditions and with windows open.

Loudness of Shaped Booms

As was previously noted for N-waves, the dura-

tion of a shaped boom has no effect on the loudness

level for the range of interest (200 400 msec). For il-

lustrative purt)oses the range of shaped sonic booms

is constrained such that the peak overpressure (/3 in

fig. 3(a)) is fixed at 50 N/m 2, the initial rise time

(Ta - T2 in fig. 3(a)) is fixed at 2 msec, and the to-

tal duration is fixed at 350 msee. Figure l0 presents

outdoor loudness levels for a range of values of the ini-

tial shock amplitude and secondary rise time (A and

T2-TI, in fig. 3(a)). For the case of A =/3' (a fiat-top

The example loudness calculations described pre-

viously were for a constant value of initial rise time

(2 reset). The range of practical interest for this

parameter is approximately 1 8 reset. Figure 13

shows outdoor loudness levels for a range of shaped

booms for which the initial rise time is 8 msec. All

other parameters remain the saine. The same trends

that were evident in the previous cases can t)e ob-

served here. The fiat-top signature is equivalent to an

N-wave having the same characteristics as the initial

shock. The curves for lower initial shock amplitudes

also exhibit a reduced loudness level for increasing

values of secondary rise time. However, compared

8O

75

Loudness70Level.

dB65

6O

A, N/m"

5O

25

55 I I I I I

0 I 0 20 30 40 50

Secondary rise lime, msec

Figm'e 12. Lou(hmss hw(q of shaped boonls fi)r indoor listen-

ing conditions and with widows closed.

95

,4, N/m _

90Loudness 85

level,dg 758(I 2_

70 I i i i ,0 l(I 20 30 40 50

Secondary rise lime. msee

Figure 13. Loudness level of shaped I)ooms for outdoor lis

tening conditions with peak ov(,rpressurc of 50 N/m 2 and

initial rise time of 8 reset.

with the previous case, the rate at. which loudness

level decays wilh increasing secondary rise time is

reduced. For small values of the initial shock ampli-

tude and for wdues of the secondary rise tixne smaller

than those of lhe initial rise time, figure 13 indicates

that the loudness of the shaped boom exceeds the

loudness of the flat-top signature. This difference is

due to the loudness contribution of the second seg-

ment exceeding the loudness contribution of the ini-

tial shock.

Concluding Remarks

A procedure for the calculation of the loudness

of sonic booms has been described. This procedure

was applie(t to a range of sonic t)ooms, both clas-

sical N-waves and shaped booms, that might be of

practical interest for a supersonic transport airplane.

It was concluded lhat the loudness of N-wave soni(:

booms is conlrolh'(t by rise time and overpressure.

The loudness ()f shaped hoolns is highly depen(tent on

the characteristics of the initial shock. A comparison

of the results of calculations for N-waves and shaped

booms indicates that shaped booms may have signif-

icantly re(luced loudness relative to N-waves having

t.h(_ sanlo peak overpressure. This was found fl)r both

in(h)or and outdoor listening conditions. This result

iml)lies that a supersonic transport designed to yield

minimized sonic l)ooms may be substantially mor(,

accei)table that, an unconstrained design. Clearly,

other studies involving many other aspects of sonic

t)oom impact are r('quired before the issue of overland

supersonic flight can be resolved.

NASA Langley l{(.sear('h Center

l|ampton. VA 23665-5225

()cto|mr 21. 1991

8

Appendix

Calculation of Other Sonic Boom ImpactMetrics

It has been proposed (ref. 2) that tile assessmentof high-energy impulsive sounds, such as a sonic

booIn, artillery firing, and quarry blasts, should be

based oil tile use of a C-weighted sound exposurelevel to describe single impulsive sounds and on

the use of a day-night average C-weighted level todescribe the cumulative effect of such sounds. Tile

sound exposure level is tile level (in decibels) of the

time integral of squared, weighted sound pressureover a given time period or event, with reference

to the square of the standard reference pressure(2 × 10 -5 N/m 2) and a referenee duration of 1 see.

Tile calculation of the C-weighted sound exposure

level can be obtained by using the energy spectral

density formulations presented previously corrected

R)r the C-weighting flmetion. A useful expression

for the C-weighting IVC as a fllnction of frequency f(given in Hertz) is

1.00715 (ff() 2

'Vc(f): [1 + (_)2] [1+ (f2) 2]

where fl = 20.598997 Hz and .f2 = 12194.22 Hz.

The weighting in decibels to be added to the

energy spectral density values is given by 20 log

I'I.'_:. The frequency-weighted energy spectral density

should then be integrated over all frequencies by us-ing equation (3). }t should be nole(t that this cah'u-

lation does not include corrections for the auditorycritical time and the double t)oon_ beeaus(_ the sound

exposure level has a reference duration of 1 see.

The most common method for the assessment of

environmental noises is by means of tile day-ifightaverage A-weighted sound pressure level. The ai)-

propriate measure for a single ev(mt is A-weighted

sound exposure level. The eal('ulation of this quan-

tity parallels the C-weighted soun(t exposure level.except that the A-weighting is used which can be

conveniently compute(t (ref. 18) by using

C,s .1

(i

1-1 (,s' + 27cpi )t--1

where C = 7.397234 x 10 !), Pl,2 = 20.6. Pa = 107.7,

P4 = 737.9, PS,(i =122(}0, and s is the complexfrequency iw. Tile weighting in decil)els 1o t)(, added

to the energy sl)ectral density values is 20 log II" A.

9

References

1. Johnson, D. R.; and Robinson, l). W.: Procedure for Cal-

culating the I,omhmss of Sonic Bangs. Ac_Lstwa. vol. 21,

i,(3. 6, 1969, pp. 307 318.

2. American Natiomd Standard Mvthod for Assessment of

High-EneT_]!l lmpulsi_ Soumt,s With R_spect to Residcn-

tml Communities. ANSI S12.1 1986 (ASA 63-1986),

Atnciican Inst. of Physics, c. 1986.

;]. Bl'OWlt, .h'ssica G.: alld l]aghmd, (]OOl'g(' T.: Sol(it BOOlll

Loudness Sludy and Airplane (!onfig_,lration l)evclop-

merit. AIAA-88-1167, Sept. 1988.

1. Y(mng, J. R.: Energy Spectral l)mlsitv of tlw Sonic

Boom. J. Acoust. Soc. America, w)l. ,l(}, m). 2, iXug. 1966,

pp. 196 198.

5. ttowes, \Valtoll I,.: Farfield SI)e('trlllll of I}lo Sollic B()olll.

.l. Aco_tst. Soc. America. vol. 11. mJ. 3, Mar. 19(i7,

pp. 716 718.

6. Onch'y. P. B.: and l)mm, D. G.: Frequency Spectrum of N

Waves With Finite Rise Times. J. Acolt,st. Eric. America,

vol. 13. no. q, Apr. 1968, pp. 88!) 890.

7. Pease, C. B.: A Note on the Sl)ectrum Analysis of Tran-

sients and the Loudness of Sonic ]_FtllgS. ,1. H,';'t.tltd *'d V/-

In'alion, voI. 6, no. 3, Nov. 1967, pp. 310 311.

8. Seebass, R.; and George. A. R.: Sonic-Boom Minimiza-

tion. ,1. Acoust. Soc. Amcrwa. vol. 51. no. 2, F.I. '3,

D'b. 1!172, pp. 686 691.

9. l)arden, (Thrisline M.: Minimizaticm _g" Sonic-Boom Pa-

ram_ tcrs i7_ Real and lsot/wrTmd A tmo_p/wr_,_, NASA TN

13-7842, 1!175.

111. Slevmls. S. S.: Po.'c('iw'd Level of Noise by Mark VII and

Decibels (E)..1. A_,m._t. Soc. Arm'rica, vol. 51.11o. 2, pt. 2.

I_)q). 1972, pp. 575 601.

11. Broadt)cm. 1). l';.; and Robinson, D. W.: Subjective

Measuremvnls of lhe Relative Annoyance of Simulated

Sonic Bangs aT'Ld Aircraft Noise. J. Sound /'_ Vibratior_,

vol. I. no. 2, Apr. 1964, pp. 162 17,1.

12. Zepler, E. E.: and Harel, .1. R. P.: The I,oudness of

S,mic Booms a,,d Other hnlmlsiw_ Sounds. ,1. Sound &'

I'ibrahon. vol. 2. no. 3, July 1965, pp. 249 256.

13. Pearsons, K. S,: aim Kryter, K. D.: Labor'atorg Tc.sts

of Sut_)ccttl,c lb,,lions to Sonic Booms. NASA CI/-lS7.

i1965].

11..h)lmson, ]). l{.: and I{obinson, D. W.: The Sul)jectivo

Evaluation of S_lli( Bangs. Acustica. vol. 18, no. 5, 1967.

pp. 2,11 25S.

15. [lo?t.s_ No_.s_-lbdrzr'tion Mvu.sltr_,ments for ['so in Studu.s

of Azrc*'oft Fll/o*_cr Nmm_. AIR 1081, Soc. of Automotive

Enginoers, Inc., ()el. 1!171.

16. Mcasltr_,rm,_ll ,,d Ev_duation of Environmental Noi.sc

F_wm Wind t:'l_ rV.t/ ('onversioT_ Sy,stem._ irt Alameda and

Ri_,crsi&' ('o,_fl,, WR 88-19 (Conlract C-87-112/Fih'

31621. Wyh, th's,arch. Oct. 1988.

17. Niodzwiecki. A.: and llibner. H. S.: Subjective Loudnoss

of "Minimiz('d'" %)hie" Ih)otn Waveforn,s..1. Ar'owstic Soc.

Am_r'ica, vol. 61, x,). 6. Dec. 1978, pp. 1622 1626.

18. Aarts, R. M.: ;\lgebraic Expression for A-Weighting Ile-

spouse. J. So_tTtd U Vzbration, vol. 115. no. 2, Y.lay 1!187,

p. 372.

10

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4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

A Loudness Calculation Proce(lur<' Applied to Shat)ed St>hi(Booms WU 537-03-21-(Y3

6. AUTHOR(S)

Kevin P. Shepherd and Brenda _I. Sullivan

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Resea]'clt (',ettt(,r

HaInl)ton, VA 23665-5225

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

Nat iomtl Aeronautics anti S[)a('c A(tntiltistration

\Vashi]lgton. DC 20546-0001

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REPORT NUMBER

L-16913

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NASA TP-3134

11. SUPPLEMENTARY NOTES

Sitepherd: Langh3 + Research Cehtcr,Hampton, VA.

Hampton, VA; Sullivan: I_o('kht'('d Engineering & Sciences Co.,

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Subject Category 71

13. ABSTRACT (Maximum 200 words)

The economic viability ()f a sttt)('r_,oni(' c<)mnmrcial transport airt)lan(_ would be IIltlC}l ellltan(:ed if it cou](t

fly supersonically over latt(]. Efforts to design an airt)lan(, t(/ pro(lu(:(, a ntinimize(t sonic boom at, tile groundrequire knowl(,dg(_ of tit(, imt)a('t of sonic booms ()n people. I_ou(ht(,ss, l)('ing a fun(lanmntal anti well-understood

character)st)(' of truman heaiing, was ('h()sen as a means of (tuantiilvin _ the nlagnit|t(h, of sonic boom impacton peol)le. This t)ap('r (h>s(_ril)es in detail a t)rocedttr(_ that can bt, ttscd to calculate the loud]ross of sonic1)()ores. The proce(htre is at)t)lie(l t_)a wide range of sonic booms, |)ot}| classical N-waves and a variety of oth('rshapes of boon]s. The lott(hmss ot: N-waves is controlh'd tly overl)rt+ssur(, and tile associated rise time. The

loudness of shat)e(l booms is highly dcI)(m(h,nt ()n the characteristics of the initial shock. A conlparison of thecalcttlated h/tttln(,ss vahtcs i:t¢iicat(,s l imt simt)e(t booms tuay have significantly reduced loudness relative to

N-waves having the same t)cal< (tv('rt)r(?sstlre. This resuh )rot)lies that a sui)ersonic transport designed to yiehtminimized sonic l/ooms may t)(, sut)stantially nt()re acc('t)tal)h, thm_ an un(:onstrained design.

14. SUBJECT TERMS

Sonic boOm; Noise: Lou(ln('ss: Air('raft noise

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NASA-Langley, 1991