A numerical study of the effects of aircraft manoeuvres on the focussing of sonic booms

July, 1973.

- 5 JUNI 1974

A NUME:RICAL STUDY OF TEE EFFECTS OF AIRCRAFT MANOElNERS ON THE FOCUSSING

OF SONIC BOOMS by Rona1d

o.

Onyeonwu VlIEGTU!~::J [;{.'u·JJi'Ur~i)E EmUOTH~:K Ktuyverweg 1 - D,ELFT

urIAS Report No. 192 AFOSR TR-74-0181

.,.

•

Submitted July, 1~73 Revised October, 1973

November, 1973

A NUMERICAL STUDY OF THE EFFECTS OF AIRCRAFT MANOEUVERS ON THE FOCUSSING

OF SONIC BOOMS

by

Rona1d O. Onyeonwu

r,

•

( ,

Acknewledgement

The author'is greatly indebted to Dr. H. S. Ribner, the originater of

this project, for his guid~ceand stimulating discussions and, above all, for

his great sense of understanding throughout the duration ef this research.

Many thanks are due to the Staff of UTIAS for ~ cooperation at

various points in time, and in particularto Dr. N. D. Ellis for free

consul-tation on computer preblems and genereus assistance 'in photographic reproduction.

This research was supported by the generous. offer of UTIAS Research

Fellowship , the Univer.sity of Toronto Open Fellowship , by Air Canada, by the

C~adian Ministry of Transport, by the National Research Council of Canada under NRC Grant No. A-2003, and by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No.

AF-AFOSR-70-1855. I extend my sincere gratitude to all these sources.

This list of acknowledgements can hardly be complete without a deserved

voteof appreciation to my wife, Winifred, who was the 'bread winner' during the

duration of this research and, for her continued tolerance of 'endless weeks' and late hours at the computer centre •

•

Sunnnary

The effects of aircraft manoeuvers on the focussing of sonic booms are investigated for a model atmosphere which is piece-wise linear 'in wind and sound speeds, and piecewise constant in wind direction. Ray-tube areas and tr~jectorie s are obtainedin closed form forareal atmosphere with winds and aircraft in

arbi trary manoeuvers • A mathematical formalism is develaped for identif'ying the focussing ray in general manoeuvers, leading to a direct and accurate determina-tion of the caustic surface ground intercept. The algorithms derived in the analysis co~lement those of Hayes to form the basis for a computer program for predicting sonic boom preperties. There appears to bea ten-fold reduction-in

computing time as compared with the Hayes-Haefeli program.

A computer printout of successive ray-cone ground intercepts at equal emission time intervals forms a kind of grid-pattern (hereafter called "acoustic lines") on the ground. Regions of high pressure are recognized by a crowding of the·acoustic-lines, implying a reduction in ray tube cross-section area. Thus the "acoustic line density" serves to provide a qualitative guide to the overall effects of manoeuvers on sonic boom intensities.

Numerical results give ray-, shock-, and caustic surface-ground inter-cepts and the variations of pressure signatures and peak overpressures along theseintercepts. Manoeuvers studied include circular turn and turn entry, accelerations, pull-ups and pushovers.

"

CHAPl'ER LO 2.0 3.0 4.0 5.0 6.0 Tab1e of Contents Acknow1edgements Summary List of Symbo1s INTRODUCT ION

1.1 Brief Review of Previous Research

1.2 Brief Review of the State-of-the-Art in Sonic Boom Prediction

103 Resumè of Present Studies RAY ACOUSTICS

2.1 Under1ying Assumptions 2.2 Imp1ications of Assumptions 2.3 Geometrica1 Considerations

204 C10sed Form Solutions of Ray Equations

MATHEMATICAL FORMALISM OF RAY FOCUSSING THEORY FOR ARBITRARILY MANOEUVERING POINT SOURCE

ii iii vi 1 1 2

3

5 5 5 6 7 12 3.1 Formulation 12

3.2 Physics of the Prob1em 14

3.3 Aircraft Manoeuver Equations 17

3.4 Condition for Grazing Incidence in General Manoeuvers 21 APPLICATION OF TEE GENERAL THEORY TO ARBITRARY AIRCRAFT

MANOEUVER IN REAL ATMOSPHERE WITH WINDS

4.1 Ray Equations ~nd Differentia1 Parameters 4.2 Special Manoeuver Conditions

4.3 Specific App1ications PRESSURE SIGNATURE CALCULATION

23

23 31

33

37

5.1 Ray Tube Area Calculation 37

5.2 Initial Conditions for Sonic Boom Calculation 40 5.3 Signa1 Distortion Due to Change in Propagation Speed 43 DISCUSSION OF RESULTS

6.1 Corridors (Carpets)

6.2 Manoeuvers in the Horizontal P1ane Level Turns

Level Acce1erations

47 47

6.3

Manoeuvers in the Vertical Plane

6.4

Pressure Signatures CONCLUSIONS

RECOMMENDATIONS FOR FURTHER STUDIES REFERENCES

APPENDICES

A - Approximating the Velocity Profile in a Layered Medium B - The Phase-Shift Integral Computation

C - The Shock Location Procedure in the Pressure Signature D - Listing of Computer Program

FIGURES 52

53

57 59

60

•

List of Symbols

a Speed of sound

K Snell's constant

t

~-direction eosine

M Mach mmlber

N Wave normal or phase velocity vect0r

n Z-direction· .cosine

n Unit vecto~ along wave normal direction

r Cylindrical coordinate

s Distance measured along the ray

S Ray or group velocity vector

St

Horizontal projection of

S

;-s

v Project of

S

on wave normal plane

t Time on the ray

T Time of theaircraft

U Wind component along ~

V Wind corrvonm t along T}

W

Wind vector

W

v Projection of

W

on wave n0rmal plane

x, y, Z Aircraft reference coordinate

X, Y, Z Ground fixed coordinate

Z

r As defined in Eq. (2.9(a))

GREEK SYMBOLS ti

a: Wind gradient

•

13

Cif_l)1/2

5 Sound gradient

~, T}

,Z

Coordinates aligned with vrave "1l0Tmal di re ction

E SUB SCRIPTS a o g x y X Y

Aircraft climb angle

Aircraft heading angle with respect to tr~e North Wind direction with respect to true North

Angle defining wave normal plane

Wave normal inclination to the horizontal Mach angle

Complement of the Mach angle

~A

a/5

Aircraft coordinate referred to the fixed axes Conditians at aircraft flight altitude

Conditions at the ground level Component along x-axis

Component along y-axis Component along X-axis Component along Y-axis

,

CHAPTER 1

100 INTRODUCTION

A natural consequence of the flight of supersonic aircraft is the

forma-tion of shock wave which, on its propagation through the intervening atmosphere,

is subjected to attenuation and refraction. On reaching a ground-based observer,

a sudden jump in pressure (above the ambient) is felt; this sudden jump in pressure

is of ten referred to as the 'Sonic Boom' (called sonic bang in Britain). Sonic

boom is an objectionable, startling acoustic phenomenon which is capable of

physical damage. As a natural phenomenon, sonic boom can never be avoided but

its magnitude, annoyance and damage capabilities could be minimized through

re-search. In this regard, the past decade has witnessed an era of the most

inten-sive research on all phases of the sonic boom phenomenon - prediction of

genera-tion and propagagenera-tion, design configuration optimization for minimization and

quantification of the signal for psychoacoustic studies.

1.1 Brief Review of Basic Research in Sonic Boom Prediction

The propagation aspect of what is now knawn as sonic boom theory has

its origin in the works done in the late nineteenth century. One of the first

theoretical accounts of the effe cts of wind and sound speed îradients on the

propagation of acoustic disturbance was rendered by Rayleigh , who then

intro-duced the concept of sound rays ~~d derived the equations for their paths.

Rayleigh obtained the law of refraction at the interface between two different

horizontal flow streams by considering the trace velocity of the wave front along

the interface. As was pointed out by Barton2 , it was not clear to Rayleigh at

the time that in the presence of wind gradients, the rays and the wave normals

were not necessarily coincident. The problem of sound propagation in an

atmos-phere in which wind speed varies linearly with height was solved by Barton who

found that the ray paths were parabolas for an initially horizontal wave front.

The corresponding problem of linearly varying temperature and sound speed was'

solved by Lamb3 , who shawed that in the first case the sound rays are cycloids

generated by circles rolling on the horizontal line of zero absolute temperature ,

ffild in the second case the sound rays are arcs of circles whose centres lie in a

vertical plane through the origin of the dis turbanc e .

Since the emergence of these basic papers on the geometric theory of

sound rays, a number of classical treatments of geometrical theory of sound

4

propagation in the atmosphere have appeared. Notabie among them are: BloL~intzev ,

who started with the Navier-Stokes equations and derived the equations of sound

propélogation in an inhomogeneous movingmedium; Keller 5, who started off "d. th a

deduction of the discontinuity conditions for curved discontinuities in any

continuous medium, and proceeded to deduce the acoustic e~uations as the

varia-tional equations of the non-linear fluid dynamics; Groves , who directly applied

the laws of geometricai acoustics to the Eikonal equation of Blokhintzev; and

Hayes7 who deduced the theory of geometric acoustics as a special case of the

geometric theory of the general linear wave propagation, to mention just a fewo

A review of previous research on sonic boom, no matter how brief, can

hardly be complete without the mention of the principal contri butions to the

basic theory of sonic booms as it evolved from the mid-twentgeth century.

Funda-mental contributionslro the basic theory were made by Landau , Friedrichs9 ,

Hayes lO , and Whitham . The use of this basic tr.eory gene rally gives quite

Continuing research has however pointed to certain inadequacies of the basic theory. First, it cannot predict pressure levels that occur in the neighbour-hood of a caustic; second, it does not take into account the effects of atmos-pheric turbulence on the sonic boom pressure signature. Due to the statistxcal nature of turbulence, quantification of the actual turbulence parameters affect-ing sonic boom signature is still as elusive as ever although there -is widespread agreement on the n_-quali tati ve understanding of the phenomenon. The effe cts

05

turbulence have been studied by many investigators, including Crow13 , Piercel , and George15 et al. Thepresent study will not include the effects of turbulence.

The caustic surface is the envelope of rays and the differential ray tube area vanishes anywhere on this surface • Because the basic theory provides that the pressure amplitude varies inversely as the square root of the ray tube area, the pressure tends to infinity as the caustic is approached. The pressure field in the neighbourhood of the caustic is usually referred to as the 'Super boom'. Considerable attention has been given to the problem of predicting tenable pressure . amplitudes near the caustic. The mathematical formulation of the ge-haviour of the pressure signature near a caustic has been given by Guiraudl

and Hayes 17 • However, satisfactory solution to the mathematical formulation remains elusive. The main difficulty in obtaining a solution lies in the fact that as the shock enters the caustic zone, it is partly reflected qud possibly diffracted thus creating a very non-linear situation. The shock is essentially a free boundary domain, requiring two independent boundary conditions governing its orientation and intensity. One Qf the most impressive attempts to solve the caustic problem was made by Seebass lö and others are continuing. While the caustic phenomenon is important, its effect will not be included in the present studies because it is highly localized and affects only a small area. Besides, since it

does not move with the aircraft as does the shock wave, its position can be placed at will for any prescribed flight plan.

1.2 Brief Review of Previous Studies

Reference 19 contains a body of information detSiling the current state of knowledge in re gard to generation, propagation, and for minimization, of sonïc boom, and theory for prediction of boom patterns, signatures, and intensities. What are reviewed here are the various applications of the baSic theory to the development of predictive computer programs currently in use and not superseded.

For the prediction of the effe cts of meteorological factors on the lateral spread of sonic boom, Kane20 et al, using the theory of Friedman21 et al based on Whitham' s22 method, computed lateral spread and overpressure distributions for steady level flight. Recently, Haglund23 et al have carried out calculations for SST class airplanes performing operational manoeuvers; the results presented show mainly the variation of maximum overpressures with selected manoeuver para-meters which were restricted to operational limits.

Dressler24 et al had studied statistical magnifications of sonic boom under steady conditions using piece-wise constant thin layers for temperature, wind magni~ude and direction. Pressure calculations were not involved.

Tarnogradzki25 carried out analytic determinationof the wave front for various atmospheric models, the most important of which is an atmosphere with piece-wise linear wind and sound speeds and constant wind direction. N~ither manoeuver effects nor pressure calculations were presented. Onyeonwu2b presented results of calculations of sonic boom corridors (carpets) for various cambinations of altitude, Mach number, wind magnitudes and directions using a model atmosphere

t

in which wind and sound speeds vary linearly with altitude. Neither manoSuver

effects nor pressure calculations were involved. Thery27 et al, Maistrè2 et al

performed geometrical studies of the wave front in mano~uvers ~d obtained ground

patterns of the wave front similar to those of Lansing3 for isolated cases. The

studies were limited to level flights and involved no pressure calculations. Perhaps themost recent comprehensive sonic boom analysis, and to

some extent synthesis, of existing theory is due to Hayes 29 et al. In that

analysis, Hayes s howed that the ray tube area could be obtained by quadra:ture

I

end developed a systematic procedure for corre-cting the shape of the pressure

signature to account for the cumulative effects bf the change in wave propaga-tion speed. The algorithm realized fromthe analysis fOrmed the basis for two

papers by Haefeli3 0 ,3l. References 30,31 were concerned mainly wi th the effects of meteorological factors and manoeuvers on overpressures and pressure signatures

of light and heavy aircraft, and provide a compendium of numerical data for

combinations of various parameters and atmospheres.

Clearly all the references cited above have their merits and their

shortcomings 0 There are important gaps yet to be filled by continued research. A major gap 'left in all the previous studies is the apsence of a self-consistent,

coherent study of the effects of variation of manoeuv~r parame~ers on thefocuss-ing of sonic booms. ,The wave front geometry, as well as its strength

distribu-tion changes with manoeuver and reception times. Consequently1any overpressure or signature computed at a gi ven ground location is valid only at that instant

and location, and is not necessarily representative of the most severe effect

of that manoeuver 'or manoeuver parameters. Further, at the present time, there is no available theory to predict tenable pressures on the caustic (lineartheory predicts infiTIlte pressure there).

Hawever, flight tests 37 have shown that pressures at the caustic could be :Bivre times higher than normal. It is clear, then, tha teven when the theory

is available to predict meaningful pressures on the caustic, the inevitable step will be to attèmpt to pre-locate the caustic. No attempt has been made so far

to study the effects of manoeuver parameters systematically varied, on the lateral

spread of the caustico otherimportant questions regarding the caustic pheno-men on which have neither been asked, nor answered, include, for instance - since the caustic, unlike the shock wave, does nat move with the 'aircraft but persists only for a fraction of the total manoeuver time - how is its duration related to

relevant parameters such as magnitude of acceleration? How is strength likely

to vary along the caustic path (on a qualitati ve basis since we cannot at present compute it), and how is tbis strength variation likely to depend on parameters such as acceleration? These omissions and limitations provide the moti:vation

for the present work.

,1.3 Objectives and Scope of Present Studies

The main objective of the present work is to study in a coherent manner

the effects of variation of manoeuver parameters on theintensification of sonic booms and the geometry of the wave patterns. The aim is to provide a sound basis for selecting where to seek quantitative information regarding sonic boom

inten-sity under any meteorological candition. Since for the most part, we shall be interested in the disturbances that reach the ground, part of this study will involve deta:iled examination of the chronological sequence of arrival and the

mode of propagation of the ground wave system for manoeuver durations long enough to cover caustic formation.

Physically, disturbances are emitted continuously along flight path, and an infinite number of rays are emitted at each manoeuver point. However, for computational convenience, disturbances are considered to be emitted at discrete intervals (e.g., 1 sec.) along a finite iiumber of rays at each point

(e.g., 100) 0 This implies long manoeuver times ani small increments in azimuthal

angles (to define rays).

To carry out this study using the Hayes/Haefeli computer program in which ray paths are obtained by quadrature would require large computer time to

obtain the ray-cone pattern for each manoeuver point. Further, to avoid in- , advertently limiting the lateral spread of the caustic, it is essential to

contain all the ra;ys reaching the ground. This implies that the computer program must be capable of selecting at each manoeuver point the'last ray that reaches the ground at grazing incidence before ray tracing is commenced - a feature not incorporated in Hayes 29 program. A recent flight test conducted in Australia39 with Concorde-002 and checked with the ARAP program (based on Ref. 29) showed that the program does not give accurate results near cut-off. That~rogram stops

computing ray path when its inclination to the horizontal is 2 or less. To achieve the desired objective therefore it is necessary to develop a computer program wi th the same capabili ty as Hayes program (compute signatures , fit shocks and plot automatically), yet posesses the added features mentioned above.

Ac cordingly , a piecewise linear atmospheric model of wind and sound speeds, and a piecewise constant model of wind direction is used to obtain ray paths and ray tube areas in analytical form in order to proYide the computational speed desired.

Of the two methods of obtaining the caustic -(~~san envelope of ray conoids (b) as the locus of cusps on the wave fronts (shock waves), the first is preferred because it does not involve interpolation of ground arrival loca-tions as does the second with attendant inaccuracy in caustic geometry. However,

a far better method of obtaining the caustic in general is to determine the

envelope analytically. To this end a general mathematical formalism of focussing theory is developed for identifying the focussing ray from a point source in general m~~oeuvers in a real atmosphere with winds. This formalism constitutes an extension of the planar treatment of Warren35 in linear atmosphere without winds to non-planar cases with winds.

In tpis study, an attempt is made to answer some of the questions posed in Section 1.2. Where it is felt that shock overpressures, pressure signatures or any other characteristics of the wave system are likely to elucidate the caustic phenomenon, computations are made accordingly. To round up the picture, results of corridor width calculations are included.

Since Haefeli30 has examined wind and Mach number effects on pressure signatures in great detail, emphasis will be minimized on this aspect of the problem.

This presentation is organized into six main chapters. The Ray Acoustics, the underlying assumptions and implications thereo~ are given in

Chapter 2" culminating in closed form soluti0ns of ray equations. The mathematical formalism for ray focussing and the general conditions for grazing incidence are presented in Chapter 3. Chapter

4

is devoted to the application of the formalism to ray focussing in a real atmosphere with winds leading to specializations for cases of interest. Pressure signature calculations and analytical expressions for ray tube areas are detailed in Chapter 5, wl1.ile numerical results are dis-cussed in Chapter

6.

For completeness, certain details associated with shock

•

CHAPTER 2

2.0 RAY ACOUSTICS

The propagation of an acoustic disturbance through an inhomogeneous

medium obeys the laws of geometrical acoustics. The laws of geometrical

acoustics state, in part, that a wave front carrying a disturbance from a

surface of arbitrary shape (in this case the aircraft body) moves such that

its normal velocity relative to the medium is the undisturbed speed of sound. The normals are the orthogonal trajectories of the successive positions of the

wave front, and the 'rays' may be thought of as carrying the disturbance. In a quiescent medium, the ray and the wave normal are coincident, in a motional

medium (such as atmosphere with winds), the ray and the wave normal are

different.

In general, the envelope of the acoustic disturbances hereafter

referred to as the wavefront (attached to the aircraft at a reference point,

say, the nose) does not have the form of a regular Mach cone except in the

special caser 6f.~ straight flight at constant speed through a uniform atmosphere.

The wave front and the rays in homogeneous and inhomogeneous atmospheres are

shown schematically in Fig. 2(a). The calculation of ray geometry (trajectory

of a point on wave front) as the disturbances prapagate along them from the

aircraft towards a ground-based observer is of ten referred to as "Ray Tracing'

and is the subject of the next section.

2.1 Underlying Assumptions

In mathematically modelling a physical problem, one must make simpli-fying assumptions. The assumptions pertinent to this investigation are:

(a)

(b)

( c)

2.2

if À, H denote respectively the characteristic wave length of the acoustic

signal, àhd the 'characteristic scale of the atrosphere, then À/H

«

l .

the propagation distance a t of acoustic wave front must not be too large

o compared to H.

the wave front propagation occurs over a relatively small region of the

earth's surface for curvature of the earth to be neglected; over this

region of space, the sound speed and the wind velocity vector are

horizontally stratified. Irrplications of Assumptions

In assumption (a) the condition À/H« 1 is necessary and sufficient

to justify the validity of geometrical acoustics (and we state this without

proof). This assumption (~H «1) in physical terms implies that the leading

and the trailing disturbances from the source are so close together relative

to a distant observer that for all intents and purposes, they may be considered as coming from the same point- a point source. The consequence-of this

assump-tion is the elimination of a phase parameter, hence the reduction of a normally

three parameter problem to two-the azimuth angle and the disturbance emissiOl

time. Since these two parameters are invariant along the raY, the assumption

~H« 1 implies a steady ray geometry.

limitation on the application of geometrie aeoustics is that diffraetion phenomena

be negligible. Put in mathematieal terms, this implies D/H2

«

1, where À, H

are as defined in (a) alld D is wave propagation distanee gi ven by a t .

o

The first part of assumption Cc) permits the deseription of wave front geometry in geometrie rather than geopotential co-ordinates, and the use of

geometrie rather than geopotential atmospherie data. The seeond part is neeessary

in order·to make the solution traetable, and is justified in part, byassumption

(b) •

2.3 Geometrieal Considerations - Co-ordinate System Definition and lnitial Wave Normal Orientation

The purpose of this seetion is to de fine the eo-ordinate system to be used throughout this analysis and to relate the initial wave normal direetions to the parameters of theproblem. It is eonvenient to de fine three eo-ordinate

systems for the purpose of this analysis. The eo-ordinate system: ~,~, Z,

posi-tioned sueh that the wave normal lies in the ~-Z plane, is employed in the

ealeulation of ray geometry; the x,y,Z system is an instantaneous eo-ordinate for refereneing the position of the aircraft and has its origin at a referenee point on the aircraft (preferably the nose); the X,Y,Z is the ground-fixed eo-ordinate. The first two co-ordinates are related to the third through a simple trans:formation.

It is eonvenient to traee the rays in a co-ordinate system moving with thewind at the aireraft altitude because it lends itself to the use of simple geometrieal relations obtainable from spherical trigonometry. Now eonsider an aireraft manoeuvering in an arbitrary manner in the positive x-axis and generating

a eone of disturbanee. At any instant of time, sueh a eone is shown in Fig.~(a).

Let us suppose that a disturbanee was emitted from a point on the aircraft

tra-jectory at time t

=

O. Then after a unit time, the disturbanee would have

prapa-gated to a terminal point alongthe ray as shown schemqtieally in Fig. 2(b). Then in terms of the parameters defined in these figures, the initial wave normal directions are, af ter Friedman21 , Hayes: 29

x-dir'n ~

=

si!lj.lcosÀ + eosllsinÀcos<p

y-dir'n m

=

cosllsin<p (2.1)

Z-dir'n n

=

COsllcosÀeos<p - si!lj.lsinÀ

The angle " defining the wave normal vertieal plane is given by

=

tan- l { t3 sincp }

' . eos À

+

t3sin7'\eos<p (2.2)

where t3

=

(if_l)1/2

and the initial wave normal inelination, e to the horizontal plane through the

o

origin of the disturbance is

2 1/2

eose _o = Cl-n )

t

•

...

,

We shall specify the wind direction such that the aircraft heading angle ~ is

measured clockwise positive from the·true North (arbitrarily defined) and winds

reported as blowing from the direction A, also measured positive clockwise from

the North - Fig. 1. With this definition,the corrponents of wind velocity along

and perpendlièular to the flight path are respecti vely: in aircraft coordinate system,

W

=

-Wcos('f-A)

x

W

=

-Wsin('f-A)

y

in the ground-fixed co-ordinate system,

W_y

=

-WcosA

W

_x

=

-Wsin..6.

and the wave normal co-ördinate system,

\

u

=

W

_s

=

-Wcos('f-A±

r)

v = W = -Wsin('f-A ± r)

Tl where W,A are functions of Z alone.

(2.4)

(2.5)

(2.6)

Having defined the wave-normal directions and wind components, we now apply these to obtain the closed form solutions to ray equations to be discussed in the next section.

2.4 Closed Form Solutions of Ray Equations

21

The pertinent ray acoustic equations as may be found in some of the

standard references already cited are:

dS/dZ

=

~ (.ea + u) . (na)-l (a)

dTl/dZ

=

+ v(na)-l (b)

dt/az

=

~

(na)-l ( c) (2.7)

a/.e + u

=

a

/.e

'

+ u

=

K, (a constant) ( d)

o 0 Ol

[ ( )2 2 '2 ( -1

na

=

a k-u - a) k-u)

Note that these equations may be readily deduced from Figs. 1,2(b), by direct

applicatio~ of Snell's law for refraction of sound at an interface of dis-continuity. If the speed of sound a, and wind velocity components, u, v are

known as functions z, then Equations (2.7) may be solved directly by q~adrature.

However, it is instructive to assume a model atmosphere which is piece-wise linear in wind and sound speeds and piecewi:..:e constant in wind direction;

from altitude Z to altitudeZ. , the wind and sound speeds become _o ~ a. = a + 5Z ~ 0 u.

=

u +

a:z

~ 0 (2.8)

where a , u are known at g , and Z

=

z.

-Z , 0:, 5 are piecewise constant gradients

.0 0 0 ~ 0

of wind and sound speeds.

Eliminating t from (2.7(d)) and substituting into (2.7 a,b) gives the ray differential equations as

/ _ -1 ( 2 2

·

·

t

dT} dZ -

_-

+ v. a. (X-u.)[ K-u.) -a. ]

~ ~ ~ ~ ~

/ ( ( ( 2 2 _1. )-1

dt dZ = + t<=-u

i ) a. [ K-u.) -a. ] 2

-

~ ~ ~

where (+) sign applies to upward propagation and (-) sign to downward propagation.

,Clearly, substitution of (2.8) into (2.9) does not lead to a directly integrable function. To transform Eqn. (2.9) to the forms permitting the use of standard integrals, we proceed as follows:

Let

then

Zl = K-u. = {K-u )

-az

~ 0 Z2

=

a_o + 5Z Z _r .= Z /Z

=

'(:K-u )/a: 1 2 i ) : E =;

a/5

Z = -[Z 'a -(K-u ) ](5Z + o:fl r o o r Z2 = (5(K-u ) + a a)(BZ + 0:)-1 o o r dZ = -( 5Z + 0:) -1 Z2 dZ r r

u./a.=;{Z (5u

-ex

a ) + K 0:] (5(K-u ) + a 0:)-1

~ ~ r 0 0 0 ·0

(2.9a)

In terms of the new variables, the first of (2.9) becomes

1

~

= - [Zr {[Zr(5u_o

-a

ao) +

ko:]

(5(K-u_o)+0: ao)-l}+

1}Z.~_+)-2

(2.10)

,

•

which integrates to

Z

.

-"

J

r i { Z2(5u -0: a )+ Z &x + 5(K-u )

d~ = r o o r 0 Z (5Z r +

a)2[Z~_1)172

+ a a } _o

_az

~ r r o Let So

=

5u -a a _o 0 th en

az

r

We sha11 make use of the fo11owing derived identities:

Substituting (2.12) in (2.11) and co11ecting terms gives

which, upon integrating yie1ds the required result

(2.11)

I: --I: =

[5-

1 {(U -€a )cOsh-1 (If;"Ui)+ (K-u + Ea

)[(K_u.)2_a~]

t

S{ .. ~o 0 0 a. 0 0 ol. l.

l.

-1 2 _l

*

(K-u.+ €a.) + E(U -Ea )(l-E ) 2

l. l. 0 0 (2.13)

*

tan-1 {( E(K-u. )+a.)

l. l.

For the component of ray propagation in 7l-direction, we make use of (2.6) and obtain the relation

v

=

u tan ('l'-A ±

r) :::

u tan( CJ-A)

Substitute this expression into the second of (2.9) and the differential equation for the transverse component

(2.14)

Following the method used for the longitudinal component of ray propagation and noting that a,5,A are constant within the layer bounded by Z ,go and that

CJ

f

f(~), we integrate (2.14) to obtain a l.

or

(2.15)

' '4

..

•

(2.16)

Ray-Ground Intercepts

The ray-ground intercepts in ground-fixed co-ordinate system is ob-tained by evaluating equations (2.13), (2.15) and (2.16) at

z.

=

o.

The inter-cepts in ground-fixed co-ordinates are then given by: 1

(2.17)

Shock-Ground Pattern

There is no sinple formula for calculating shock-greund intercepts in gener al manoeuvers. A rather complex formula may be obtained from the method of Chapter 3. However, shock-ground patterns are deri ved from ray-ground

intercepts as follows:

The co-ordinates of (2.17) and their times of arrival at the ground are saved for all the rays emitted from each manoeuver point for the entire manoeuver history. If we consider the end of the manoeuver as the time at w~ich the aircraft overflies a ground-based observer (the victim), then the first portion of the shockwave to reach the victim (standing under the flight path) was emitted from the last manoeuver position and carried by the ray lying in the vertical plane containing the flight path (eI>

=

0). Labelling the time of arrival of this· ray as t , we then retrace the aircraft trajectory and, at each manoeuver point, we se~rch the saved table of arrival positions and times for the raysthat arrived at t

=

t • This process is carried back to the starting manoeuver point. In a ho~izontal manoeuver, two rays will always be found for each manoeuver point. For manoeuvers in a vertical plane there may be two rays reaching the ground for each manoeuver point or none at all. The locus of these linearly interpolated points at t

=

t is the shock-ground pattern at t = t • For straight line manoeuvers, it is onlyOnecessary to search the tables f8r the rays arriving at one side of the track because of symmetry • Curvillinear flight paths however require search of tables for both sides of the flight track. If focussing occurs, it will naturally appear as cusp on the shock pattern. All the patterns in this study are obtained in the rnanner outlined above "except where otherwise stated.

CHAPTER 3

3.0 GENERAL THEORY OF RAY FOCUSSING FOR ARBITRARILY MANOEUVERING POINT SOURCE

3.1 Introduction

A major part of this thesis involves geometrical studies of ray

focuss-ing due to arbitrary aircraft manoeuvers. It is quite befitting therefore that

appropriate general theory be develaped for the purpose of this study.

Focussing in ray acoustics is a geometrical phenomenon which arises whenever sound rays emitted from successive points on the source trajectory merge. The envelope of the focussing ray is termed a caustic line while the

envelape of ray (Mach) conoids is termed a caustic surface. Aircraft ~~oeuvers

and/or atmospheric refraction cause successively emitted rays to have an envelope. Linear theory predicts an infinite pressure anywhere on the caustic surface

-an untenable result. Since however -an International effort is underway in

predicting tenable results in the vicinity of the focus, it is useful to accurately identify the particular ray around which the differential ray tube area vanishes when focussing occurs. At the moment, the autbor is unaware of any method of identifying this unique ray for any given combination of manoeuver parameters and atmospheric profile unless for the trivial case of cut-off condition. Once this unique ray is identified (and this can be donefor every point on source trajectory), ray tube area can be calculated along this ray only, using the analytical results of Section 5.1, hence the pressure signature along the ray.

The basic theory is formulated quite generally without recourse to specification of manoeuver parameters or the physical characteristics and properties of the medium through which propagation is taking place. For this reason, the theary has a wide range of applicability, including Hydroacoustics. 3.1 Formulation

Let x

l ,x2, ••• xn be a system of n independent variables; Let

a

_l",z2' ••• Zi

m be m unknown functions of these variables , and ;Let flj, f2j ,·· .fnj, f

nl,fn2,· •• fnm be given scalar functions of xl ,x2, ••• xn and

a

l ,z2'· •• lim• Then

in functional notation, any system can be represented as:

F(xl ,x2

,···x

n ; Zl' Z2'···Zm) :; 0

Alternatively, we may write (3.1.1) as:

Z. :; _{<P. ( xl ' x2 ' • • • ,} x )

J J n

The governing total differential equations for (3.1.2) are:

dZ.

J

(3.1.1)

(3.1.2)

'

.

t

which really is equivalent to nm distinct equations:

dZ l

êlz

l OX l = f ll , dX2

=

OZ2 ~Z f 21, . 2 d)L = _dX 2 = 1 f 12, ••• , dZ l

di:"""

_n OZ2 f 22, ••• , dX n

oZ

_m

dx

_n

=

=

=

f ln f 2n f mn - - --- - -- ---- -- - - --- - - . (3.1.4)

Let us suppose that there exist functions Zj of x

l ,x2, ••• xn satisfying these

mn

relations. We can obtain the second derivative 02Z./dX.d~ (ifk) in two

J ~ K

different ways, and since they must be identical, we must then have:

=

i

f

k

leading to the cartesian tensor identity:

df .. df. . _dfk· df kj

~+

~J f_kj = J. + f .. ~

dzj

~ ~ ~J ~ _J (3.1.5) i, k = 1,2, ••• n

,

.

j = 1,2, •• om

The equation (3.1.5) provides the necessary and sufficient condition for:Equation

(3.1.3) or its equivalent, (3.1.4), to be completely integrable. If (3.1.5) does not reduce to an identity, there can be no integrals of (3.1.3), except possibly one or more of the implicit functions defined by (3.1.5). Complete integrability of (3.1~3) implies the existence of an infinite number of integrals depending upon m arbi trary constants •

Now let Z. be position functions defining the positions of the

manoeuver-ing source x,y,Z ana the propagatmanoeuver-ing ray

s,T}

•

..

Let also the independent variables

x. denote the souree manoeuver parameters such as the heading angle ~, climb angle

~

À , Mach number, M and ray parameters such as azimuth angle ~ and the al titude Z.

We may define focussing as a geometrical condition in which the rays emitted from different source positions at different times arrive at the same space location simultaneously. The requirement that the rays from infinitesimally different spatial

orlglns arrive simu1taneous1y at a point on the p1ane defined by Z

=

constant,

is expressed mathematica11y by the-vanishing of appropriate camponents of

(3.1.3) identica11y. That is,

(3.1.6)

The prob1em of integrating system (3.1.6) is rea11y that of finding the re1ation

~ (Xi)

=

C among x

1,x2, ••• xn such that these n variables and their differentia1s

dx

1,dx2, ••• dxn satisfy the given re1ation, C being an arbitrary constant. In

general, a formal solution to (3.1.6) is obtained by deriving the apprapriate integrabi1ity condition from (3.1.$) and if (3.1.6) is comp1ete1y integrab1e, this condition yie1ds identities from which corresponding differentia1 equations are obtained and solved.

302 Physics of the Pröb1em

Let us now introduce physics into the mathematics. Let the source

of the ray be an aircraft manoeuvering in an -arbi trary manner in four

dimen-siona1 time-space. Since we can recast equation (3.101) into the form:

it is permissib1e to write the ray equation as:

x

=

r

(M,

T,~,

X ,Y ,Z

_{a a a}

,s,

~,

t,Y)

(3.2.1)

(3.2.2)

It is necessary to emphasize that M and ~ are, in the strictest sense,

inde-pendent parameters. However, for focussing consïderation, they assume the ro1es of independent variables. The Mach nurnber Mand ray emission time T are interchangeab1e as independent variables because they are simp1y re1ated,

and this fact wi11 be exp10ited in what fo11ows. In Equation (3.2.2), X, Y

deno'te the fixed co-ordinate syst.em; the Mach number M or ray emission time T (which ever proves to be more convenient) is the independent variab1e for the

aircraft manoeuver; the azimuth ang1e ~ is the indepêndènt variab1e for ray

prop~gation; Xa,ya,Za are the unknown positions of the aircraft (ray origin),

s,~ are the unknown ray positions and t, is the time a10ng the ray. The Equation

(3 02.2) is (3.2.1) syrnbo1ized in physica1 termso

The dependent variables in (302.2) are functiona11y re1ated to the

-independent variables and parameters -as fo11aws:

Xa

=

Xa(T) Y = Y (T) a a Z

=

Z (T) a a

=

"

•

,

where X _a ",Y ,Z are now aircraft co-ordinates referred to ground fixed system. _{a a} For the ray,

and since ~~= ~ (Z ep _{a' ,}

T)

Tl

=

T}(Za,ep,T) t

=

t(Z ,ep,T) a

x

=

X(X,Y ,Z

,M,

~,Tl, t,T,ep) a a a y

=

Y(X ,Y ,Z ,M, ~,Tl, t,T,ep) a a a

we can then,in view of (3.2.3) and (3.2.4), express X and Y as:

X

=

X (Z , ep,T)

a

Y

=

Y(Z , ep,T) _a

Identify the mathematical and the p~ysica1 syIDbo1s as fo11ows:

Xl ~ Z a X 2 -+ ~ X 3 ~ ep Zl ~ X a Z2 -+ Y a Z3 ~ ~ Z4 ~

_Tl

Z5 ~ t Z6 ~ X Z7 .-+ Y (3.2.4)

With this syIDbo1ic identity, we obtain from (3.1.3) and (3.1.4) the fo11owing

dx

=

a

ex

_a

dT +

ex

a dep

and dy

=

a d~

=

dT)

=

dt

=

dy

a

dZ

_a d~

dz

_a

~

a dt

TI

_z dx dX

=rz

a dY

=i-a dZ a dZ a +

dT

d~ d T

dy

a deI> deI> + d~ ~ deI> dZ _a +

~

dT +

~

_deI> dZ +

~

_dT+ dt deI> ~ a dZ + _{ex dT + dx deI>} a c;:r ~ dZ +

~

dT + dy deI> a T ~ (3.2.6)

It ean be dedueed from (3.2.3) - (3.2.5) that not all the partial derivatives

in (3.2.6) and (3.2.7) are non-zero. Also, the dependenee of ray parameters

on T, Z , is easily translated to dependenee on M. The eondition for ray

foeuss-a

ing requires that equations (3.2.7) vanishd dentieally. That is,

ex dZ + dx dT + ex deI> = 0

rz

_a a

dT

~ (3.2.8) dy dZ + dy dT +

~

=

'

deI> = 0

dz

a

dT

a

Now eonsider system (3.2.6). In any given physieal problem, X ,Y ,Z , are known

a a a

from aireraft manoeuver history as funetions of T; eI> is known, being an

inde-pendent parameter, .and other trajeetory parameters from geometrieal eonsideration

of initial ray orientation; ~,~,t are the solutions of the ray aeoustie equations

(Chap.2) in terms of the properties of the medium in whieh the propagation is taking plaee. Thus the partial derivatives are readily available. Also the partial derivatives appearing in (3.2.8) are funetions of the left hand side of system (3.2.6). Henee substituting (3.2.6) into (3.2.8) and eolleeting terms leads to two equations in two variables with dZ a ' dT, replaeed by dM:

P dM + QdeI>

=

0

..

which upon elimination of dM, d<P" leads to the desired resul t:

P

*

S - Q

*

R = 0

where P,Q,R,S are functions of

ç,T),

t, <P,T, Z •

a

Equation (3.2~lO) defines the unique candition for ray focussing. ·'Since this equation must be satisfied by the general ray, substitution of the general ray into (3.2.10) specifies the azimuth anglefor which focussing occurs for a

given Mach number and other trajectory parameters. The theory of the preceeding sections pre-supposes a knowledge of the aircraft manoeuver equations and para~

meters 0 To enable us to apply the theory to specific problems, we shall in the next section develop the appropriate manoeuver equations for an aircraft manoe u-vering in arbi trary :tna.rL'l1er in the earth 's atmosphere 0 For application to

Hydroacoustics, the manoeuver equations appropriate to the aquatic source must be used.

We may summarize the analytical results of this section as follows: (i) It will enable us to compute pressure signature only forthe focussing

ray, and permit a graduated approach to the focus.

(ii) If desired, only the ground intercept of the caustic surface for the entire manoeuver history maY be displayed. Existing computer programs will display only the ground intercept of the focussed wave front at a given instant but not on a continuous basis.

It was pointed out in Chapter 2 that there is no simple formula for calculating the shock-ground pattern in general manoe·uvers. Analytical expres-sion for the shock wave pattern onany horizontal plane defined by Z = constant may be obtained by putting dt = 0 in Equation (3.2.6), and integrating the re-sult, this time making use of (3.1.6)0

3.3 P~rcraft Manoeuver Equations

The purpose of this section is to derive the relevant equations that permit the description of aircraft position as a .function of manoeuver time,To In this ~'l1alysis, a knowledge of the aircraft trajectory is required in locating the origin of an acoustic disturbance. Because of the complicated distribution of the physical properties of the atmosphere as a function of altitude, as well as the complex behaviour of the drag funqtion ·at supersonic speeds, closed ferm solutions are not usually attainable. Consequently, numerical solutions for general manoeuvers become inescapableoOn the other hand, closed form polu-tions are possibleif certain hypotheses are accepted for ·the characteristics of the flight path. We shall state these hypotheses as the need arises •

If we denote the aircraft position co-ordinates as X , Y ,Z rela-tive to a fixed co-ordinate system, the trajectory of the airc~aftamay~e defined .by the following equatians:

dx

a

dy

dTa

=

VCOSÀCOS~ + W

y

dZ a

dT

= VsinÀ

while i ts dyna.mics is defined directly by the following equations:

T

-

D - WG'i (sinX +

1jg)

=

0

L - WG-;· (cosr\ + V

'Xjg)

=

0

(3.3.4)

WG

c

:

+ O.T

=

0

~

where V

=

velocity, À'

=

path inclination to vertical plane,

1:E

=

flight path angle, WG

=

Weight, D

=

Drag, L

=

Lift, T

=

thrust, ei

=

specific fuel consumption, g

=

acceleration due to gravity.

Alternatively, the dynamical system defined by

(3.3.4)

is derivable from

(3.3.1-3.3.3)

by differentiating the set of equations twice and definipg as follows:

~

=

(T-D)jWG

~

=

LjWG

~,~ being the normal load thrust factor and normal load lift factor respectively. This done, the dynamical equations become, in the lift direction, .

d2Z d2x d2y

(~cos~b-cosÀ)g

=

cosÀ _____ a -

sin'(sin~ ~

+

cos~~)

dT2 ' \ dT dT

(3.3.6)

in the flight direction,

iz

d2x d2

(~-sinÀ)g

=

sinÀ

~

+ cosÀ

(sin~ ~

+ cos'f

~)

dT dT dT

=

dVjdT

and in the transverse direction,

'f

<1>b being the ba.IlÄ angle.

Defining the aircraft velocity V, in terms of Mach number M, and sound speed;a, we have

v

=

Ma so that dV = a; [ dM +

if

'

."\

da ] dT dT ' Sl.n/\ élZ a

Substituting (3.3.1-3.3.3) into (3.3.6-3.3.8) and arranging terms, we obtain the desired manoeuver parametersas follows:

dÀ dT 2 da - M sinÀ élZ a

:X

.

)

a (3.3.12)

Note that the coupling of these equations requires that they be solved simul-taneously. This step is inevitable when a climb - turn - acceleration manoeuver is being executed. However, we can decouple these equations by considering individual manoeuver separately. Such individual manoeuvers include continu0us turns in horizontal plane at constant speed, constant speed horizontal turn-entry, straight horizontal accelerations, dive-pull-up, climb-pushover, etc. It is convenient to recast (3.3.10 - 3.3.12) with the aid of (2.5) into the fol1owing forms:

dÀ

dT

=

~gsin<1>b

_vcosi\

-tanÀ [sin('f'-A) : a - Wcos

('~E-.A) ~a

]

=

(~cos<1>b

-cosÀ)g/V-sin2À [COS('f'-A): + Wsin('f'-A)

~

]

a a

dM

(n.r-sinÀ)g/a - IlsinÀ :-dT

a

+ McosÀsin"à {COS('f-A) !:a + Wsin('f-4)

~a}

where W is the wind magnitude at flight altitude.

One exarnple illustrating decoupling of

(3.3.13 - 3.3.15)

will suffice. Cons.ider a continuous turn manoeuverin . a horizontal plane. For this case,

À

=

0, and

(3.3.13), (3.3.15)

become respectively:

~

=

n (T)g/a

dT T

The trajectory equations

(3.3.1-3.3.3)

then become

z

.=

z .

a 0 dx a dT dy a

dT

=

VeT) cOS'f(T) - WcosA

Then for a constant speed turn

(3.3.16)

gives immediately

~d the trajectory becames defined explicitly as follows:

z

=

Z a 0 Xa =

~

+

V2cos('fo

±

6)/~gsin~b

- WsinA( T-T ) o y _a

=

+

y + V2sin('f +

6)/rLgsin~b

0 - 0 - L ;: WcosA( T-T ) o

(3.3.18)

•

'

.

where a

=

sound speed at flight altitude

o

b,. = ~g sin<l>b (T-T 0)

IlIJa

_{o '}

the top signs denote right turn and bottam signs left turn.

3.4 Condition for Grazing Incidence in General lIJanoeuvers

Since we are interested primarily in the shock waves that reach, or at

least come within a few prescribed fe et to the ground, it is necessary (from

point of view of computer time economy) to consider only those rays that satisfy

this condition. This requirement is met by calculating the maximum azimuth angle,

<l> ,based on the known atmospheric conditions at the point of ray emission and

a~ts ultimate destination. The two ray parameters determined by <l> are "e ,

max 0

obtained through consideration of initial ray orientation. They are related to the flight path parameters and Mach number through the expressions:

tan, t3sin<l>[cosÀ + t3sinÀcos<l>]-l

1

cose

o

_2 2 2"

=

[Ml - (t3cosÀGos<l>-sinÀ)]

jM

The Snell's law of refraction may be written as:

K u + a

Icose

=

u + a

Icose

o o o g g g

(3.4.1)

Of all the rays emitted at the altitude, Z , where wind and sound speeds are u , a respectively, the last one to reachOthe ground (or an arbitrary altitude

o 0

Z , where wind and sound speeds are u ;a ) is the one that does so at grazing

g g g

incidence, Le., for which cose = 1. Hence the Snell' s law for this special

case takes the form: g

K

=

u + a

Icose

=

u + a

o o o g g

so that

cose _{o o g} = a

I(a

+ u -u ) _g

0 (3.4.2)

Equation (3.4.2) defines the initial inclination of the ray that is to reach the ground at grazing incidence. Equating (3.4.1) and (3.4.2) yields

V(a + W

cos(~,)

- W

cos(~,)

)-1

g g g 0 0

1 (3.4.3)

2 2 2"

where

cos(w-r) _Slnu .

{l_[

S

i~-SinÀ(COS~COSÀCOS~-Si~SinÀ)

_{2 1/2} J2} cosÀ(l-(cos~cosÀcos~-si~sinÀ) )

COSW si~-sinÀ cos~cosÀcos~-si~sinÀ

+

---~---~---~~---~~~~~---COSÀ(1-(cos~cosÀcos~-si~sinÀ)2)1

2

(3.4.4)

Equation

(3.4.3)

defines the maximum azimuth angle ~ for which a ray reaches

max

the ground at grazing incidence for arbitrarily manoeuvering aircraft in a windy atmosphere. I t shows also that the condition for grazing incidence depends on the characteristics of the atmosphere at the origin and destination of the ray as well as the aircraft climb angle À and Mach number M. ~he angle ~ is

ob-max

tained by solving

(3.4.3)

iteratively for a given À,~,A, and W and M. In the absence of wind gradients,

(3.403)

reduces to

cos~ =

max

sinÀ + M(l-(a ja ) ) 2 1/2

o g

•

CHAPTER

4

4.0 APPLICATION OF THE GENERAL THEORY TO A,RBITRARY AIRCRAFT MANOEUVER JN

REA!, ATMOSPHERE WITH WINDS

The purpose of this section is the application of the general theory

of Chapter

3,

to the specification of the ray azimuth angle ~ at which

focuss-ing occurs for a given Mach number and flight trajectory parameters. We state an iII:!PortallJlt hypothesis for this analysis - once a ray is emi tted from i ts origin, its subsequent propagation is unaffected by subsequent aircraft manoe

u-vers but only by the gradients of the properties and the characte.ristics of

the medium through which it is propagating. In what follows, refer to Fig. 1

for all references to angles and co-ordinate definitions unless otherwise statedo

4.1

Ray Equations and Differential Parameters

We shall re-write the ray equations derived in Chapter 2 in forms appropriate for this analysis. The intercept of a point on the ray cone with

any horizontal plane defined by Z = constant is given by:

x

= X + ssin('P-y) - ~cos~&~y)

a

Y = Y + scos ('P-y) + ~sin('P-y)

a

(4.1.1)

For the aircraft manoeuver equations we re-express equations

(3.3.1) - (j.j.3)

in the following integral forms:

where X a = J(MaOCOSÀsin'P +

WX)

f-l dM Y_a =J (MaOCOSÀcos'P + Wy)f-l dM f = dM dT (c.f.

3.3.15)

(4.1.2)

Z -Z N-l _Zi+l-Zi

~ = 1 a (F(Zl)-F(ZaO(}

+L

(F(Zi+l~-F(Zi))

a -a _ai+l-ai 1 0 i=l N-l (4.l.3) Z

-z

L

Zi+l-Zi 1 a

(G(Zl)-G(Za) ) + (G(Zi +l)-G(Zi))

T} = a -a _{ai +l -\}

1 0

i=l

The Snell' s law of refraction for a layered medium may be written as: K = u + a /cose = u + a/cose = const.

o 0 0 (4.1.4)

Consideration of initial ray orientation leads to the follawing trigonometric results:

sin8 = _o cos~cosÀcos~ - si~sinÀ

cosr = (s~-sinÀsin8 )/cosÀcos8

o 0

from which one obt a ins :

1

K = a

[1-(cos~cosÀcos~-si~sinÀ)2]-2

+ u

o 0

tanr

=

~sin~(cosÀ

+

~sin~os~)-l

1

where

~

= (Nf_l)2

(4.l.5)

(4.1.6)

We see that equations (4.1.1) are the explicit forms of (3.2.5); (4.1.2) the explicit forms of (3.2.3) and (4.1.3) the explicit forms of (3.2.4) with t omitted sinceit is not used explicitly. We now apply the requirements of

(3.208) to (4.1.1) and obtain:

dX _a + sin('P-r)d~+ [Scos('P-r) + 'Ilsin('P-r) ] d'P

(4.1.7) -cos('P-r) dT} - [~cos('P-r)

+

T}sin('P-r)] dr

=

0

dY + cos _a ('P-r) d~

+

[T}cos ('P-r)-~sin('P-r)] d'P

(4.1.8) + sin('P-r) dT} + [~sin('P-r )-T}cos ('P-r)] dr

=

0

The differential parameters appearing in (3.2.6) are defined as follows:

( -1 _(401.9)

dX

=

Ma cosÀsin'P + W

x)

f dM

a 0

..

- - - -- - - -dY = (Ma. cosÀcos'P + W:.:) f-l dM a 0 , T -1 dZ

=

Ma sinÀ.f dM a 0 d t _~

=

C dZ ₁ + 6,' dJ{ + 1'.. du a 1 ~ 0 (4.1.10) (4 .• 1.11) (4.1.12) (4.1.13) Note that dZ

a appearing in (4.1.12), (4.1.13) is used in preference to d'r in (3.2.6), and that dk, du ,d'P,dr are all functions of M and~.

o

dr

=

cos2r

[~-lMsin~cosÀdM

+

~(~sinÀ

+

cosÀcos~) d~

-~sin~(~cosÀcos~-sinÀ)dÀ](cosÀ

+

~sinÀcDs~)-2

d"K = du - E(cosj..lsinÀcos~ + sillj..lcosÀ)dÀ o

+

E(sillj..lcos~cosÀ

+ sinÀCOSj..l

)(~)-1

dM

- E COsj..lcosÀsin~~

duo

=

Wsin('P

+

r - A) d'l:" + Wsin('P

+

r-A) dr

(4.1.14)

(4.1.15)

(4.1.16)

The change in aircraft heading is related to the change in its Mach nurnber through the following equations (c.f. (3.3.13),(3.3.14»:

dl' = [ILg sin~ /Ma fcosÀ-tanÀ .g/f] dM

.L b 0 (4.1.17)

(4.1.18 )

The functions and coefficients appearing in equation (4.1.3) through (4.1.8) are defined as follows:

1

Cx; )

K-u + Ea _ f2 2 1/2

F(z) = (u -Ea )cosh- ~ + l{ 0+ 0 «K-u)c -a )

- 0 0 a -u Ea

1 2 2 2

{ (

~

(K-u + Ea ) ((K-u) -a )

G(z) ::; tan('l'+ï'-A) (u -Ea

~cosh

-1 K-u· _ E2 _.

----,,~,--~o---o 0 ,a (l-E )(K-u + Ea)

2 -1 [ E(k-u) + a ] 2 -3/2 }

+ E [(uo~Eao)(2-E )-K ]tan 2 2 2 1/2 (l-E)

[(l-E ) ((K-u) -a )]

1

2 2 2 . -2 _1. [

+ (u -u)+ E(a-a )[(K-u~ -a ] (K-u + Ea) + E(U - Ea ) 8 2 .

o 0 0 0

2 2

t

{ 2 2 }-3/2] 1

E(8((K-u )-a )] -8(E(K-u)+a](K-u) 8((K-u) -a ] A

2

8

=

1- E

2 2 2 -1

A ::; 1 + [E(K-u) + a] (8((~-u) -a ))

1

r(z) ::; cosh-1 (

~u)

_ ((X_u)2_a2 ]2

(~u

+ Ea)-l

-1 ATAN ::; tan

c =

₁ (F(Za)-F(Zl)) (a_{1-ao)-1 +}

c ::;

2 (G(Za)-G(Zl»)(a1-aO)-1 + N-1 Z -Z _{a 1} a -a ₁ 0 Z -Z _{a 1} a -a ₁ 0 Zi+1-Zi b.

=

I

(Q(Zi+1)-q(Zi) ) 3 _{ai +1-ai} d _{F(Z )}

dz

a a d G(Z )

dz

_a a

q(z)

B

= {B_€2

c*

8- 1 + € ( u -€a )(2_€2)_K ) 8- 3/2

*

D

o 0

- € 8-3/ 2

*

ATAN } tan(':I:'

+

r-A) 2 2 _.1.

(u

O- €aO) (.X-u) -a ) 2

2 2 1 . (K-u

o+ €ao)(K-u) :(uo-u) + €(a-ao)) «X-u) -a

1

2

C = - 2 2 1/2 + 2

(K-u+ Ea)( (K-u) -a ) (Ic-u + Ea)

N-1 Ll4 =

L

i=O i=O F'

=

cos~cosÀcos~-si~sinÀ E = a F'(1_F,2)-3/ 2 o da

f = (:n.:r-sinÀ)g/ao - !isinÀ dZ 0 + McosÀSinÀ.

~

a W' dW dZ a dA A' = -dZ a

1~ = W' sin(':I:'-A) - WA' cos (':I:'-A)

~ = wtcos(':I:'-À) + WA' sin(':I:'-A)

Now substitute the differentia1s dx,dy,d~,dT},dz ,d':I:',dr into (4.1.7) and (4.1.8)

a I

+

I

61sin(~-r)

-

~cos(~-r)

JdK +

[cos(~~r)(s-~)

+

~sin(~~r)

J

d~

.[ cos

(~-'l)

(S-6 5) +

T)sin(~-')')

J d'l du o + [

~

cos

(~-r)

+

63sin(~-r)

J dK -

[sin(~-r)(s-~)

-

T)cos(~-'l)

J

d~

+

[Sin(~-r) (s-~5)

-

T)cos(~-'l)

J d'l

+ [

64sin(~-r)

+

~cos(~-r)

J duo -;;:: 0

(4.1.19)

(4.1.20)

With the definitions in Tab1e 1, we recast the differential equations into the

following simpler forms:

ciK=LdM 7 dÀ

=

R 3dM

rfJ/I

=

Rl dM dUo

=

R7~ + R7d'l

o

=

_L8dM + F₁dK + F2d~-F2d'l + F 3dUo

o

=

_F4dM + _F6dK - F7d~ + F7dr + F 9duo (4.1.21) (4.1.22) (4.1.23) (4.1.24) (4.1.25) (4.1.26) (4.1.27)

Ll

=

(cosÀ +

~sinÀcos~)-2

L2

=

L1MSinPCOS~~ L3 = ~(~sinÀ

+

cosÀcos~)

*

Ll L4 = ~sinP(~cosÀcos~-sinÀ)

*

Ll L5

=

E( cosl-lsinÀcos~ + sin!-lcosÀ) . :L6

= E

COSl-lcosÀsi~ TABLE 1 L7

=

E(si!lllcosÀcos~

+

sinÀcOsl-l)(.~M)-l

L8 = l/f [M aocosÀsin'f' + W~ic+ M aosinÀ (clsin('f'-r)-c₂cos(î-r))] Fl

=

t.lsin('f' .... r) - ~cos('f':...r)

F

2

=

cos('f'-r) [~-t.5] + Tjsin('f'-r) F 3

= t.

₂sin('f'-r) - t.₄ cos ('f'-r) F

4 = l/i'[M aocosÀcos'f' +

wy

-

+ M aosinÀ (c1cos('f'-r)+ c2sin('f'-r)) ] F 6 == t.

1 cos ('f'-r) +t.3sin('f'-r) F 7 == sin('f'-r) (~-t.5) - Tjcos ('f'-r) F 9 == t.₄ sin('f'-7)

+

~COs('f'-r)

Rl == ~gsinPb/M aofcosÀ - 'g tan~f

R3 ==

(~cos~b-cosÀ)g/M

aof - gg

sin2~f

R7 == Wsin('f'

+

r-A)

e1iminating dM, ~, dÄ, ~, dk, dr, du from equations (4.1.21) to (4.1.27).

o

The eliminant of dM,

ds:P,

from

is (8 1 + 82)dM + 83 ~= 0 (8

₄

+ 8

5

)dM + 8

6

~ = 0 . (4.1.28) (4.1.29 )

where 8₁ ~hrough 8₆ are definedin Tab1e 2. Observe that (4.1.28) is simp~Y

(3.209) W1. th P

=

8

1+ 82, Q

=

8₃, R

=

84+ 85 and 8

=

86, and that (4.1.29) lS simp1y (302.10). With the aid of Tables 1 and 2 we expand (4.1.29) to obtain the fo11owing result:,

2 F 7[COS r[L8L3+ 2R1R~3(F1+ F3) + F1L3(L7- L5R3) 2 + F1L6 (L2- L4R3)] + F1R1L6]

+

[F2cos

r

[F4L3 + 2R1R7L3(F6+ F~). + F6L3(L7-L5R3) + F6L6(L2- L4R3)

1

2 2 + F6R1L6] + cos

r

[R7L8L3F9 + R1RTF9F3L3 + R~lF9L3(L7-L5R3) 2 2 2 + R1Rr1F9L3 + Rr1F9L3fL2- L4R3) cos

r

+

R~8F6L3

+

R1R~3F6L3

+

R~lF6L3(L7-

L5R3) +

R1R~lF6L3

+

R~lF6L3cos2r

_{(L2- L4R3)}

~

R~lF6L6(L2-

L4R3) 2 2 + RTF3L3(L2- L4R3) (F6 + F9)COS

r -

R~3F6L6(L2- L4R3) 2 - R~4L3(F1+ F_{3) - R1R7L3(F1 + F3) ( F6+ F9)} 2 2 - R~6L3(F1+ ~3)(L7- L5R3) ~ _{R7L3(F1+ F3)(F6+ F9)(L2- L4R3})COS

r

+ R~lL6(F6+

:9)

_{(L2-L4R3)]- L8F6L6- Rl R~3F6L6} - F1F6L6(L7- L5R3) - R1R~lF6L6 + F1F4L6 + R1R~lL6(F6+ F 9) + F1F6L6(L7- L5R3)

=

0

..

Using the relation

F 2

=

S

CO&('f'-r)

+

Tlsin(1'-r) - ~cos (1'-r) F 7 = .~ sin(1'-r) - Tlcos (1'-r) - ~sin(1'~r)

in (4.1.30) and simplifying gives the equation of the focussing ray as

where 2 Xl

=

_{cos r [LSL3+ 2Rl R7L3(Fl + F3) + FlL3(L7-L5R3)} + Fl L6 (L2- L4R3) ) + FIL6Rl 2 X₂

=

_{cos r [F4L3+} 2R_IR7L3(F₆+ F 9) + F6L3(L7- L5R3) + F6L6 (L2- L4R3)) + F6L6Rl X 3

= R7 {LSL3cos2r(F6+ F

9)+ (FlF9-F6F3)[L3(L7-L5R3)cos2r + RIL6+ L6(L2-L4R3)cos2r) - F4L3(Fl + !3)cos2r}

+ L6 (F lF4 -LSF 6)

(4.1.31)

(4.1.32)

(4 .. 1.33)

(4.1.34 )

The equation (401.31) provides the general condition for focussing for an air-craft manoeuvering in arbi trary manner in real atmosphere wi th winds. The model atmosphere is assumed to possess piece-wise linear structure. The expression is, to say the least, rather carrplicated as would be expected for a general condition. Further, except during "aerobatic displays" which, in any case, is unlikely at

supersonic speeds, an aircraft may never perform multiple manoeuvers simultaneously. Consequently, in the next section, (4.1.31) will be reduced to special cases.

4.2 Special Manoeuver Conditions

It is convenient to classify manoeuvers into two main categories: (a) planar manoeuvers: - (i) Horizontal plane - turns and straight acceleration

(ii) Vertical plane - dive-pull-up and climb-pushaver (b) non-planarmanoeuvers: - combinations of (i) and (ii) •

Special Cases:

1. For arbitrary manoeuver in a Quiescent Atmosphere (Case (b)),