70-26,382
VOLTMER, David Russell, 1939- DIFFRACTION BY DOUBLY CURVED CONVEX SURFACES.
The Ohio State University, Ph.D., 1970 Engineering, electrical
University Microfilms, A XEROX Company, Ann Arbor, Michigan DIFFRACTION BY DOUBLY CURVED
CONVEX SURFACES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
by
David Russell Voltmer, B.S., M.S.E.E.
******
The Ohio State University 1970
Approved by
6 & k v t Adviser ^Adviser Department of Electrical Engineering
t ACKNOWLEDGMENTS
This work, though written in my name, reflects the efforts of many others; I take this opportunity to credit them and to offer my gratitude to them. The continued guidance, encourage ment, and assistance of Professor Robert G. Kouyoumjian, my adviser, has been instrumental in the successful completion of this study. Professors Leon Peters, Jr. and Roger C. Rudduck, as members of the reading committee, aided in the editing of the manuscript with their suggestions. I especially thank my wife, Joan, who provided understanding, moral support, and love through a difficult time.
The majority of this work was accomplished while I was under the sponsorship of Hughes Aircraft Company. Additional assistance was provided by the stimulation and encouragement of the Plasma Physics Laboratory of the USAF Aerospace Research
Laboratories.
ii VITA
June 4, 1939 Born - Ottumwa, Iowa
1961 Be Sc, Iowa State University, Ames, Iowa
1961-1963 Hughes Work-Study Masters Fellow, Hughes Aircraft Company, Fullerton, California
1963 MoS.EcEc, University of Southern California, Los Angeles, California
1963-1966 Hughes Staff Doctoral Fellow, The Ohio State University, Columbus, Ohio
1966-1969 Research Engineer, Plasma Physics Laboratory, USAF Aerospace Research Laboratories, Wright-Patterson AFB, Ohio
1969-1970 Assistant Professor, Department of Electrical Engineering, Pennsylvania State University, University Park, Pennsylvania
FIELDS OF STUDY
Major Field: Electrical Engineering
Studies in Electromagnetism. Professor Robert G. Kouyoumjian
Studies in Antennas. Professor Carleton H. Walter
Studies in Applied Mathematics. Professor Robert C. Fisher TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS e . . , . <■ ...... ii
VITA...... j. H
TABLE OF CONTENTS . iv
LIST OF TABLES...... vii
LIST OF FIGUR.ES .o.ro,o.ofto,oooe..ooo. Vlll
Chapter
I. INTRODUCTION. . 1
A. Background...... 1 B, A Review of the Geometrical Theory of Diffraction ...... 4 1. Development,...... 4 2. Example o...... 9 3. Limitations...... 15
II. ASYMPTOTIC SOLUTIONS OF CANONICAL PROBLEMS. . . . 17
A. Introduction, .c...... 17 B. Two Dimensional Diffraction Problems. .... 21 1. Eigenfunction Solutions...... 21 2. Watson Transformation and Asymptotic Evaluation ...... 26 C. Three Dimensional Problems...... 35 1. Acoustic Problems...... 36 2. Electromagnetic Problems ...... 44 D. Illuminated Region, ...... 53 1. Two Dimensional Diffraction...... 54 2. Three Dimensional Diffraction...... 58 Chapter Page
III. DIFFRACTION COEFFICIENTS AND ATTENUATION CONSTANTS 64
A. The GTD Field...... 64 1. Soft Acoustic Cylinder...... 64 2. Hard Acoustic Cylinder...... 67 3. Soft Acoustic Sphere...... 68 4. Hard Acoustic Sphere...... 69 5. EM Sphere ...... 69 B. The Asymptotic Field ...... 71 1. Soft Acoustic Cylinder...... 71 2. Hard Acoustic Cylinder...... 72 3. Soft Acoustic Sphere...... 72 4. Hard Acoustic Sphere...... 73 5. EM Sphere ...... 74 C. Evaluation of Diffraction Coefficients and Attenuation Constants...... 75 D. Generalization and Interpretation...... 75 E. Backscatter Aspect ...... 85
IV. APPLICATIONS AND NUMERICAL RESULTS ...... 89
A. Two Dimensional Scattering ...... 90 B. Three Dimensional Scattering ...... 96 C. Exponential Factor ...... 110
V. CONCLUSIONS...... 124
Appendix
A. COMPLEX ZEROS OF THE HANKEL AND RELATED FUNCTIONS 127
/ ™QH. CD V (X) B. ASYMPTOTIC EVALUATION OF • O 0 137 (2 ) l3v QHv (x)I x-ka m
C. AIRY FUNCTIONS, DERIVATIVES AND ZEROS . 146 Appendix Page
D. ASYMPTOTIC EVALUATION OF P ^(-cos 6 ) AND
3P U (-cos 6) V ...... o . . 150 3v Eo HUYGENS-KIRCHHOFF INTEGRAL FORMULATION OF THE DIFFRACTED FIELD AT BACKSCATTERo ...... 159
REFERENCES CITED ...... 173
vi LIST OF TABLES
Table Page
1 DIFFRACTION COEFFICIENTS AND ATTENUATION CONSTANTS FOR CIRCULAR CYLINDERS AND SPHERES, 76
GENERALIZED DIFFRACTION COEFFICIENTS AND ATTENUATION CONSTANTS ...... , 80
3 Re(aA) FOR SOFT SURFACES. 113
4 Re(aA) FOR HARD SURFACES, 113
5 v/c FOR SOFT SURFACES . . 115
6 v/c FOR HARD SURFACES 0 t 0 115
7 HANKEL FUNCTION ZEROS ...... 136
8 HANKEL FUNCTION QUOTIENT...... 145
9 AIRY FUNCTIONS, DERIVATIVES, AND ZEROS. 149
10 AIRY FUNCTIONS, . O C 0 0 0 © 0 • • 149
vii LIST OF FIGURES
Figure Page
1 DIFFRACTION BY A SMOOTH CONVEX SURFACE...... 6
2 GTD RAY So ...... ooo...... 22
3 CIRCULAR CYLINDER COORDINATE SYSTEM ...... 23
4 V-PLANE CONTOURS OF INTEGRATION FOR THE WATSON TRANSFORMATION OF THE CIRCULAR CYLINDER ...... 27
5 SPHERICAL COORDINATE SYSTEM ...... „ , . . . 37
6 V-PLANE CONTOURS OF INTEGRATION FOR THE WATSON TRANSFORMATION OF THE SPHERE...... 40
7 DIFFRACTED RAY GEOMETRY FOR THE CIRCULAR CYLINDER AND THE SPHERE...... 65
8 NORMALIZED BACKSCATTER CROSS SECTION FOR THE CIRCULAR CYLINDER ...... 91
9 DIFFRACTED FIELDS FROM THE SEMI-INFINITE, CIRCULAR CYLINDER TIPPED HALF-PLANE...... 93
10 NORMALIZED BACKSCATTER CROSS SECTION FOR THE ELLIPTIC CYLINDER ...... 95
11 NORMALIZED BACKSCATTER CROSS SECTION FOR THE SOFT ACOUSTIC SPHERE ...... 98
12 NORMALIZED BACKSCATTER CROSS SECTION FOR THE HARD ACOUSTIC SPHERE ...... 99
13 NORMALIZED BACKSCATTER CROSS SECTION FOR THE PERFECTLY CONDUCTING SPHERE ...... 101
14 NORMALIZED BI-STATIC CROSS SECTION FOR THE PERFECTLY CONDUCTING SPHERE, ka = 2.9 ...... 103
viii Figure Page
15 NORMALIZED BI-STATIC CROSS SECTION FOR THE PERFECTLY CONDUCTING SPHERE, ka = 8 .9...... 104
16 NORMALIZED AXIAL BACKSCATTER CROSS SECTION FOR THE HARD ACOUSTIC 2:1 PROLATE SPHEROID ...... 107
17 NORMALIZED AXIAL BACKSCATTER CROSS SECTION FOR THE PERFECTLY CONDUCTING 2:1 PROLATE SPHEROID. . . 109
18 SURFACE RAY ATTENUATION...... 116
19 SURFACE RAY VELOCITY 118
20 VARIABLE CURVATURE EFFECTS ON THE SOFT SURFACE
RAY oeccice e.cooooo..oco..».o 120
21 HARD EM SURFACE RAYS WITH DIFFERENT ka ...... 121 0 22 TOTAL ATTENUATION, Re / adt...... 123 0 23 AIRY FUNCTION CONTOURS OF INTEGRATION...... 147
24 HUYGENS SURFACE FOR SPHERICAL SCATTERER...... 161
25 TOPOLOGICAL VIEW OF u(Q) ...... 166
ix CHAPTER I
INTRODUCTION
A. Background
Diffraction and scattering phenomena when an acoustic or electromegnetic wave is incident upon an arbitrary body have received a great deal of attention. Attempts to completely determine the resulting field have met with varying degrees of success. As expected, a diverse selection of methods is available to solve such problems (see Bouwkamp [1] and Corriher and Pyron [2]); each method lends itself especially well to some problems, but is usually quite useless on others. Formal mathematical or numerical solutions may give little or no clue to the associated physical phenomena. Indeed, they may even serve to mask the physics of the situation. A method which is general in scope, has few limitations, lends itself readily to the calculation of the field, and provides insight into the physical processes is highly desired. The Geometrical Theory of
Diffraction (GTD) is one such method.
The primary goal of the investigation reported here is to improve the accuracy of the GTD solution, to obtain a better understanding and explanation of the high frequency diffraction mechanisms associated with curved surfaces and to express the
1 2
results within the framework of the GTD. By including higher order
asymptotic terms in the diffraction coefficients and the attenuation
constants, the detailed effects of local surface properties on the
diffraction processes are obtained. This study includes the effects
of surface curvature transverse to the ray, changes of the surface
curvature in the direction of the ray, boundary conditions, and
polarization. This more complete description of the diffraction
processes should lead to better high frequency approximations which
in turn can be extended to lower frequencies than the existing GTD
solutions.
The GTD, introduced by Keller [3], as a means of determining
the diffraction from edges, vertices, and curved surfaces, is an
extension of ray optics to include not only the geometrical optics, but also the effects of diffraction. The following postulates,
quite analagous to the postulates of the geometrical optics, form
the basis of the GTD:
1) Fermat's principle applies to diffracted rays.
2) Power in a tube of diffracted rays is conserved.
3) The phase variation along a diffracted ray is that of a
plane wave.
4) Diffraction is a local phenomenon.
These postulates provide a reasonable foundation which can be justified by the agreement of GTD solutions with experiment and exact mathematical solutions. With the GTD, otherwise insoluble 3
problems can be handled; the resulting solutions are in the form of elementary functions which lend themselves to simple computations.
In addition, the GTD solution can be physically interpreted, allowing an identification of the diffraction mechanisms. The GTD cannot be applied at caustics and it must be modified at shadow boundaries and near the diffracting surface. Despite these limitations, numerous examples have shown it to be widely applicable with a surprising degree of accuracy.
Quite closely related to the GTD is the "creeping wave" theory advanced by Franz and Depperman [4] which utilizes a rigorous asymptotic expansion of Green's functions for cylinders and spheres to obtain a similar representation for the fields not only far from the scatterer, but also, on and near its surface. Kazarinoff and
Ritt [5], [6] have given a "creeping wave" interpretation to similar results for an elliptic cylinder and a prolate spheroid. Alter natively, Fock [7] applies ray optical methods to fields on the surface and in the vicinity of the shadow boundary with good success.
However, neither the early forms of the GTD nor these other theories are able to adequately account for variations in the surface curvature or for two finite radii of surface curvature. To some extent these shortcomings have been eliminated by the extensions of
Keller and Levy [8] and Franz and Klante [9] to include the effects of non-constant curvature in the case of cylindrical surfaces.
However, more general GTD solutions which are applicable for 4
non-cylindrical surfaces have not been obtained. In a recent paper,
Hong [10] derives an asymptotic solution for the field diffracted by smooth surfaces which includes the effects of the radius of curva ture and its variations along the ray path and the radius of curvature transverse to the ray path. This solution has a form very similar to the GTD solution, and it serves to supplement and check the results of the present investigation.
B. A Review of the Geometrical Theory of Diffraction
The Geometrical Theory of Diffraction (GTD) is a method for solving high frequency diffraction problems. Though not without limitations, the GTD has been successfully applied to many antenna and scattering problems, due to the broad nature of its assumptions.
This section is devoted to explaining more fully the development, presenting an example, and then describing the limitations of the
GTD. This discussion is restricted to the steady-state diffraction of monochromatic radiation from smooth, convex, three dimensional, impenetrable objects in free space, though the principles are quite similar in other media.
1. Development
The Luneberg-Kline series is a formal asymptotic solution to the radiation problem. The first order term in this series, which establishes the ray coordinate system, consists of the classical 5
geometrical optics field; it incorporates a phase variation along the ray of e ~ ^ s and an amplitude variation which conserves the power within a tube of rays. The higher order terms of the series are merely corrections to the geometrical optics field; they do not account for diffraction effects. The diffraction contribution is obtained from the GTD.
The GTD for a smooth, convex surface is based upon a general ization of Fermat's principle which includes points on the diffracting surface in the ray trajectory. As shown in Figure 1, these points are determined by the condition that the distance between the source point at 0 and the field point at P be a minimum. The part of the ray path on the surface is a geodesic of the surface. The incident ray strikes the surface tangetially at and a portion of the associated power is "attached" to the surface, propagating into the shadow region via the surface ray along a geodesic of the sur face in the direction £. Diffracted rays (and hence, power) are launched tangentially from the surface by a continuous dividing of the surface rays at all points along their trajectory, e.g., point These launch points appear as caustics to the observer at P. A diffraction coefficient, D(Q^), is defined which relates the incident field to the surface ray field; by reciprocity a similar relationship between the surface ray field and the diffracted ray field, DCQ^), is also obtained. The principle of locality advanced by Fock [7] asserts that the diffraction phenomenon is a 7
S\D&-^— ’ oV x sH*>0* BOONOM'
C Y S T I C - ^ C v j ., ■^7 I > O ' I * 'trfjjlljj*''
Q i ^ e F R O M T eyjRFAkCE k / ^ oip^ ^ c T'w
SU’E^at'e" 0tb Convex , , n StB° by a
, B i f ^ act:LOtl ?igute 7
function of the local properties of the surface material and geometry at the point of diffraction. Accordingly, D(Q^) and ^CQ^) possess a dependence upon local surface properties. The loss of energy from the surface ray field via the diffracted rays is described by an attenuation constant, a(t), which is also determined by the local surface properties, where t is the coordinate position along the surface ray. The phase variation is assumed to retain the form of geometrical optics, e . If deviations from this occur, they are observed as the imaginary portions of a. Each diffracted ray launched from the surface continues to propagate as a geometrical optics ray with a phase variation of e . As in the geometrical optics, a phase jump occurs whenever a ray passes through a caustic.
The two common types of caustics, cylindrical and spherical, exhibit phase advances of tt/2 and tt radians, respectively.
The incident field, if not scaler in nature, is assumed to be entirely transverse to the direction of the incident ray,
Each component of the surface ray field induced by an incident vector field has a diffraction coefficient and an attenuation con stant associated with it. Accordingly, the normal, h-j_, and the binormal, b^, components propagate independently along the surface retaining their perpendicular or parallel orientation to the surface.
Upon leaving the surface at launch point Q by a diffracted ray in the direction each is assumed to retain its orientation, n^ or b2> respectively. Each component of the incident field satisfies the conservation of power within a tube of rays, just as in
geometrical optics, thus relating the component amplitudes of the
diffracted field to the geometry of the ray system.
It has been found that each component of the incident ray
excites an infinite number of modes on the surface, each with its
own diffraction coefficient and attenuation constant. If the surface
is closed, each of these modes makes an infinite number of encircle ments; usually though, the amplitude is so greatly reduced by one
encirclement that all others are negligible. Whenever more than
one ray passes through a point, the total field is given by the sum
of all the rays through the point. However, the special cases of
caustics, i.e., the convergence of adjacent rays at a point, line,
or surface, require separate consideration. (A particular case of
interest is the backscatter aspect of the sphere which is discussed
in Chapter III.)
The surface ray field is not the actual field on or near the
surface. (It may, in fact, represent a field which vanishes on
the surface.) Rather, it may be viewed as a transfer function relating the field of the incident ray to the field of the diffracted
ray in much the same way as the transfer function of a network relates its input and output signals.
Thus far the appearance of diffracted rays has only been discussed in the shadow region. However, the launching of diffracted rays occurs everywhere along the trajectory of a surface ray. 9
Consequently, diffracted rays are present in the illuminated region
for a closed surface.
2. Example
To clarify the details of the GTD, the following example is
presented and discussed; reference is made throughout this example
to Figure 1. The emphasis here is on the vector electromagnetic
field. Hoxvever, a scaler field may be handled in the same manner
as one of the components of the vector field,
A given source radiates energy in accordance with geometrical
optics such that the field of the tangential ray incident at is given by
(1) E1 (Q1) = Eq .
In general, the incident field is composed of two components. The normal component, n^'E1, induces a normal component of the surface ^ — X ray field while quite independently, the binormal component, b^*E , induces a binormal component of the surface ray field. For each component at the point of incidence, the surface ray field is related to the incident field by the diffraction coefficient,
Dn (Q1) = h1 *Es/n1 *Ei or D]-, (Q^) = b1 *ES/b1 'Ei . Then the surface ray field can be expressed as 10
(2) E3 ^ ) = E1 (Q1 ).D(Q1)
= ‘V [W n (Ql) + ^ i V ^
where D(Q^) is a compact dyadic representation of both the normal and binormal diffraction coefficients. With t as the coordinate position along a geodesic, the phase shift and attenuation of the Q2 field are accounted for by a factor of exp[-/ (jk + a(t'))dt']. Qi To allow for the possibility that the attenuation constant, a(t),
is not a constant, an integration along the surface ray path is necessary. The conservation of power between adjacent rays on the surface is satisfied if the amplitudes at two successive points on the ray, t and t^, are related by
- Q2 — ? (3) |Es(t1) | 2 dn(t,) exp [-2/ a(t’)dt’] = |Es (t2)|2 dn(t,) Q 1
where dri is the length of a cross section between two adjacent
rays. Accordingly, the surface ray field at point is
expressed as 11
Q2 (4) Is (Q2) = EQ *(n1n2Dn (Q1)exp [-/ an (t') dt'] Ql
q 2 + b1b2 Db (Q1) exp [-/ o^Ct') dt’])
d n ( Q 1 ) ,Q 1 exp [-jkf rii- ’ 1
As the surface ray passes it launches a diffracted ray whose field varies in accordance with power conservation from the launch point to the observation point P which is at a distance s from Q2 „ Each component of the diffracted field at the observa tion point is related to the corresponding component of the diffracted field at a reference point Q, where Q is not on the surface, by
in which and p2 are distances to caustics of the ray system from the reference point Q. As Q~*Q2> P2_>’PC anc^ remains finite as shown in Figure 1, but p^"K). The field at any point is 12
J independent of the point of reference, so that E (P) must remain unchanged as p^0„ Accordingly, it is useful to define
(6 ) lim Spl E^Q) = E® (Q2)*D(Q2). p ^ O
The resulting diffracted field at P is given by
j Qo (7) FCP) = E0 ‘(n1n2Dn (Q1 )Dn (Q2) exp [-/ a ^ t ’) dt’] Q1
q2 + b- ^ D b C Q i ) ! ^ ^ ) exp [-/ OL(t') dt']) Qi
[ ~ P ^ ~ /dnCQi) Q2
• J n i + p j / d n c o p - exp [“jk( ' dt + 8)3 •
As mentioned earlier, an infinite multiplicity of modes
(denoted by subscript m) is excited by each component, and, if the body is closed, each surface ray makes an infinite number of encirclements (denoted by subscript Z). Thus the expression for the field becomes 13
00 CQ —d Y~ — Q2+^L (8 ) E^P) = 2 _ E -(ft n D (Q )D (Q )exp[-/ a (f) dt'] Ji-0 m=l 1 2 nm 1 nm 2 nm
q 2+£l + b b D (Q )D (Q ) exp [-/ a (t') dt']) 1 2 bm 1 bm 2 q_^ bm
P c / d n ( Qt ) s(s + p ) J d n ( Q 2 )
^2 exp [-jk(/ dt' + £L + s) + jty + jij; (Q) ] ii
where L =(pdt', ^C(Q) is the phase shift due to passage of the surface ray through caustics from to Q2 , and ipQ is the total caustic phase shift in one encirclement. This can be simplified to
Pc (9) A p ) = E1 (Q1 )'I where ^ 2Dnm(Ql)Dnm((^2 ) exP ^ ^nm^') dt'] <*1 (10) T(Q1 ,Q2) = m=l (1 - exp [-/ a (t1) dtT — jkLH-jl/j ]) 0 n m c blb2Dbm(Ql)Dbm(Q2 ) exP abm(t,) dt’] + (1 - exp [-/ a (t1) dt’ -jkL+jil ]) 0 bm c /dn (Q-,) q 9 exp [-j(k/ 1 dt’+ijj (Q)] d n ( Q 2 ) Q x ° T(Qi,Q2 ) is a transfer function describing the surface ray behavior where the first and second terms in Eq, (10) are associated with the hard and soft solutions, respectively„ This is the field associated with just one diffracted ray; the total diffracted field is represented by the sum of the fields associated with all the rays which pass through the observation point, 15 All of the terms in the expression for the diffracted ray field except for the diffraction coefficients and the attenuation constants are evaluated from the geometry of the ray system. When properly arranged, the asymptotic solutions of certain canonical problems assume a form similar to the GTD solution. The unknown terms can be determined by a direct comparison of the two solutions. These terms are evaluated for specific geometries. Nevertheless, when they are applied to general surfaces, which may differ greatly from the canonical surfaces, the numerical results have been found to be in good agreement with experimental or rigorously calculated values. The greater the surface information included in the unknown terms, the more closely the GTD calculations agree with these results. 3. Limitations In its early history, the GTD was primarily limited to high frequency applications. However, a recent study [11] indicates that the GTD solution in some cases is valid for wavelengths as long as 6 times the surface radius of curvature, and that the reflected component of the scattered field is the major source of error for such long wavelengths. Consequently, the greatest error in the use of the GTD at lower frequencies may be in the reflected field rather than the diffracted field. 16 While the GTD fails to correctly describe the field at a caustic, i.e., a convergence of adjacent rays, this inconvenience is handled by well-known methods [1 2 ] which involve a supplementary solution. The GTD, developed as a means of describing far zone field behavior, may be used successfully in regions near the body surface and the shadow boundary, also. Ray paths may still be used. However, the functional form of the field must be altered. These cases are not treated here. As mentioned earlier, the GTD is a heuristic theory based on certain postulates. Despite the fact that no proof has been given to show that it is an asymptotic solution of the wave equation, it has been successfully applied to many diverse problems. Detailed discussions of the GTD and its applications to curved surfaces are found in Levy and Keller [12] and Kouyoumjian [13]. CHAPTER II ASYMPTOTIC SOLUTIONS OF CANONICAL PROBLEMS A. Introduction The form of the diffracted field has been established on the basis of the GTD. Though the diffraction coefficients and atten uation constants are undetermined, as noted earlier, they may be ascertained by a direct comparison of the GTD fields with the corresponding fields of certain canonical problems. Consequently, the asymptotic solutions of these canonical problems in forms similar to the GTD fields are essential to the complete develop ment of the theory. This study considers a number of canonical problems in order to determine the higher order dependence of the diffraction coef ficients and the attenuation constants on transverse curvature, the scaler or vector nature of the field, and the boundary condi tions. Keller [3] has shown that the first order approximations do not exhibit such a dependence. Other studies of more general t surfaces [14], [15], and [16] have been no more revealing. 17 18 The canonical problems chosen are: 1 ) a soft acoustic infinite circular cylinder, 2) a hard acoustic infinite circular cylinder, 3) a soft acoustic sphere, 4) a hard acoustic sphere, and 5) a perfectly-conducting electromagnetic sphere. In a manner parallel to that of Levy and Keller [12], the GTD field is formulated. By a method similar to Senior and Goodrich [17], the asymptotic evaluation of the exact solutions is made. Several of these solutions already exist, but the results are independently rederived here to check the earlier solutions, to obtain higher order terms, and to put them into the context of the GTD. A comparison of the solutions allows a determination of the new diffraction coefficients and attenuation, constants. These results are examined in close conjunction with other recent works.to obtain a more general form. Finally, numerical examples show the improved accuracy and increased frequency range which can be obtained with these new results. In addition, numerical data provides the basis of a quantitative study of the effects of surface parameters upon the attenuation constants. The chosen canonical problems share in common such features as harmonic time dependence, plane wave illumination, location in free space and dimensions large in terms of a wavelength. Addi tional features such as the geometry of the body and the boundary conditions distinguish each problem from the others. 19 The two dimensional diffraction phenomenon accompanying an arbitrary vector field illumination is completely described by a linear combination of the solutions of the acoustic soft and hard problems. These cases correspond to the linearly-polarized TE (incident electric field parallel to cylinder axis and perpendicular to direction of propagation) and TM (incident magnetic field parallel to cylinder axis and perpendicular to direction of propa gation) electromagnetic cases respectively. Unfortunately, no such correspondence exists between the three dimensional acoustic and electromagnetic problems so that separate solutions for each case are necessary. However, the acoustic soft and hard solutions are related to the electromagnetic problem at a perfectly con ducting surface where the incident electric and magnetic fields, respectively, are tangent to the surface. Accordingly the surface wave induced by a binormal component of the electric field is referred to as the "soft" EM solution while that of a normal component of the electric field is called the "hard" EM solution. The well-known eigenfunction solutions to the canonical problems converge and are valid for all space surrounding the scatterer. However, these series representations converge very slowly for large values of ka. Furthermore, they are not descrip tive of the basic diffraction phenomenon, and hence , cannot be compared with the formal GTD solutions. Fortunately, the Watson Transform recasts these slowly convergent series into asymptotic 20 high frequency representations which are semi-convergent. The first few terms of an asymptotic solution usually provide suffi cient accuracy. Furthermore, careful arrangement and interpreta tion of the asymptotic series within several spatial regions enables one to compare the results with the GTD solutions. Usually, the total field is that field which exists with the scatterer present. It is composed of the incident field, i.e., the field of the source which exists in the absence of the H scatterer, and the scattered field. These fields exist in all regions and are particularly suited to the eigenfunction method. For convenience, the ray optical fields differ slightly. Though the total field remains unchanged, its components are somewhat different. The incident ray field is similar to the incident field except that it exists only in the geometrical optics illuminated region; it does not exist in the shadow region. The scattered field of eigenfunction solutions is replaced by a reflected ray field and a diffracted ray field. The reflected ray field results from Fresnel type reflections of the incident ray field and exists only in the illuminated region. On the other hand, the diffracted ray field, which is initiated by incident rays striking the surface at the shadow boundary, exists in both the illuminated and the shadow regions. Consequently, the total GTD field in the illuminated region is composed of the incident ray field, the reflected ray field 21 and the diffracted ray field. In the shadow region, the total GTD field has only one component, the diffracted ray field. Figure 2 illustrates the valid regions for each type of ray and the com ponents of the total field in each region. In addition to these fields, the surface ray field is a useful concept in the GTD. This field is useful in interpreting the transport of diffracted power in the immediate vicinity of the surface. The surface ray field is a boundary layer field near the surface. The actual boundary layer field may vanish on the surface, though the surface ray field does not. These surface ray fields are not to be confused with surface waves as defined by Barlow and Brown [18] or Jones [19]. The two phenomenon are related, but they can be distinguished from one another in the limit of an infinite radius of curvature along the direction of propagation where only the surface wave exists. B. Two Dimensional Diffraction Problems 1. Eigenfunction Solutions A plane wave of unit amplitude propagating in the direction of the negative x-axis illuminates an infinite circular cylinder of radius a whose symmetry axis coincides with the z-axis as shown in Figure 3. (The harmonic time dependence of e"*"^Wt is assumed and hereafter suppressed in this development.) The circular ILLUMINATED REGION DIFFRACTED RAY SHADOW INCIDENT REFLECTED REGION RAY AY SURFACE RAY DIFFRACTED ^ RAY SOURCE Figure 2. GTD Rays 23 z Figure 3. Circular Cylinder Coordinate System. 24 cylindrical coordinate system is introduced and the incident plane wave is readily expanded in cylindrical harmonics as given by Harrington [20] as (1 1 ) p1 = ejkx = 2 jnJn (kp)e“jn(*\ n =-co In the case of the acoustic soft cylinder (TE), the illumi nating plane wave is interpreted as an acoustic pressure or as the transverse component of an electric field. For such a case the total field satisfies the boundary condition (12> Ps =0 . p=a The well-known eigenfunction solution to this problem is given by Harrington [20] as 00 • A J (ka) (13) p = 2 jne ^[j (kp) -_i- H (2 )(kp)]. s n=-°° n H (ka) n n ' On the other hand, in the case of the acoustic hard cylinder (TM), the illumination represents a pressure or the transverse 25 component of a magnetic field. The total field satisfies the boundary condition (14) =0 9p p=a which gives V (ka) (2) (15) p = Z Jn (kp) — n V ; (kp) . n=-oo Hn (ka) n I A unified treatment o. the two dimensional problems is possible if,the similarity of the two forms, Eqs. (13) and (15), is utilized to write QJn where 1 for soft boundary Q = 3 — for hard boundary. 3X 26 2. Watson Transformation and Asymptotic Evaluation The Cauchy Residue Theorem of complex variables states that _1_ (17) — (Df(V)dV = Z Residues f(V), j J poles f (V) where the poles are enclosed on the left by the directed contour C. The function jVe“jV(J)[Jv (kP)- {QJV (X) /QHV (2) (X) }x=kaHV (kp) ] • (sin Vtt) —^, which is analytic along the real axis except at V=n, n=0 , + 1 » + 2 , 000J where it has simple poles, has residues propor tional to the summand of Eq. (16). Since (18) sin VTT = sin nTT + (v-n)TT cos n'fT + ... - (v-n)7r(-i)n it follows that if QJV (X) (19) f(V) = j V j V4> (jv (kP) - Hv (2 ) (kP)j , s m VTT > v QHv (2 ) (X)' _ X=ka then Eq. (17) can be made equal to Eq. (16) by the proper choice of contour C. Figure 4 shows the paths C = + C£ which allow this to be done. Hence, from Eq. (16) 27 Im V c , A A A A A A A A 'J I' X A A A 1,1 -3rr-2rf-tr 0 Tt ZTf 311 4tr v R @ > I \ y P x ' ^ 2 x I \ x ' \ x, / V \ / ^ ___ Figure 4. v-Plane Contours of Integration for the Watson Trans formation of the Circular Cylinder. 28 (2 0 ) p = -- 0 — ---- 27Tj J s m VTT follows and an alternative integral formulation of the total field is obtained. Within the shadow region the total field consists only of the diffracted field. Hence, the diffracted field may be found from the expression for the total field in the region where ir/2 < (j) <3ir/2 and p sin -“H-j e oo-j£ (21) pd = / f(v)dv + S f (V) dV . 2ttj “H-j e 2irj -00-j e If the replacement of V by -V is made in the first integral, the result is given by oo-je (22) pd / [f(v)-f(-v)]dv . 2 irj -oo-j £ Substitution of 29 je~^V7T JV (Z) = 1/2 [Hv (1 >(z) + Hv <2 )(z)] and ------= ■ 2 sin VTT 1-e into the integrand of Eq, (22) yields QHV (1 ) (X) (23) f (v) = (-l)V7TjV e"jVt,> ( (l) (kp) - QH (2 ) (X)' V X=ka -jVTT Je ll-e“j2v7r- and with the aid of = ejV1THv (1) (z) and H_v ^2 ) (z) = e_:jV7rHv (2) (z) one obtains QH v (24) f(-v) = -C-l)‘v |h ^ (1) (kp) Hv (2 ) (kp)| QHv (2 ) (X) X=ka -jVTT Je l_e-j2V7T 30 Thus, the integral assumes the form (25) pd -oo-je • [(-l)VjV^+ (-1) VejVC|)]dV . The above procedure, which transforms Eq. (16) to Eq. (25), is referred to as the Watson Transformation [21]. The evaluation of the integral in Eq. (25) is accomplished by closing the path along the infinite semicircle in the lower half plane, contour in Figure 4, and calculating the residues of the enclosed poles. The contribution to the integral due to the contour C3 is not readily apparent. However, if the substitution of -1 = e+J77 is made and a careful analysis of the integrand (similar to those of Pflumm [22] and Nussenzveig [23]) is carried out, it is seen that this contribution approaches zero as v _>00 for *both boundary conditions, i.e., Q=1 and Q=3/3x» Since the Hankel func tion of the first kind, Hv^(kp), is analytic in the complex V-plane for large positive values of kp, this term will contribute QHV (1)(X) (2) nothing to the integral. However, the term (kp) QHv (2 ) (x) X=ka 31 has simple poles where QH^^(X) =0. If these roots are X~ka circumscribed as shown in Figure 4, residue theorem gives the desired results. The residues of the integrand occur at the roots of Q H [V f h x ) =0. A Taylor series expansion about these roots, m X=ka gives (2 ) 3QHV (2 ) (X) (26) q h v (2)(X) + (v-[v ])+ = Q H [V] (x) m X=ka mJ X=ka 9v X=ka v=[vm ] 3QHV (2 ) (X) 3v X=ka v=[vm ] m and allows a simple determination of the coefficient of the term (vtv ]) Then, by the residue theorem, the diffracted field m is given by 32 00 . -j [V 3 (4>—7T/2) + -j [vm ] (3ir/2-^) (e 111______e ______) (27) pd = jTT Z m=l Q h / 1 } (x) m V=[V ] m subject to the restrictions tt/2 < tj> < 3tt/2 and p sin cj) < a. When written as above, x = ka is fixed prior to differentiation with respect to V, The diffracted field given by Eq. (27) is exact. However, in order to easily evaluate the series (and to put in a form similar to the GTD) an asymptotic approximation of the terms is QHV(1 ) (X) required. This evaluation of [v ] and . w V x - a , m is given in Appendixes A and B, respectively. The total field within the shadow region, which is composed entirely of the diffracted field, is now readily expressed as an anymptotic series. This asymptotic expansion utilizes the well- known and well-tabulated Airy functions and their zeros (see Appendix C) . As described in Appendixes A and B, the forms of [v ] 33 QH (X) V and of are different for each of the two 8> v C2)(X) ■ x ,k a v=[v ] L m J boundary conditions. Since the canonical problem solutions are to be compared with the GTD forms, the far zone approximation 2 2 1/2 (28) H ( j (2) (kp) - I r i / 2 e'j k< P -a ) LV 7Tk(p -a ) -[v ] cos ^ a/p -tt/4} m is used. If the field near the surface is desired, an approximation where [vm ]~kp»l should be employed. This alternative approximation takes the form H [v2 ] (kP) " " 2 e ”j7T/3 ( ^ 1 1 Ai'(-qm )(p-a) m . e"27T/3 f M ' _2/3 U 15 6 1 2 for soft surfaces and ,-1/3 (2 ) /v.n _ o J * / 3 ka Hr',[V ]i • (^f)'2 / 3 + ...) for hard surfaces. Note that the field vanishes on the surface for soft bodies as the boundary conditions require. When Eq. (28) is substituted into Eq. (27) and the results in Appendixes A and B are used, the diffracted field for the soft acoustic cylinder is expressed as oo m (29) psd ~ E m!1=1 ^2ik (Ai* (~qm ) )2 Likewise, the diffracted field for the hard acoustic cylinder is given by 35 Eqs. (29) and (30) represent the diffracted field within the shadow region, i.e., tt/2 < (j) < 3tt/2 and p sin cj) < a. These results are identical to those of Franz and Galle [24]. C. Three Dimensional Problems The characteristics of three dimensional diffraction phenomenon, though similar to two dimensional diffraction in many respects, have significant differences. Most notably, the vector nature of electro magnetic problems cannot be simplified into two scaler problems which correspond identically to the related acoustic problems. More over, the solutions depend upon the two finite radii of curvature. These features lead to more complicated eigenfunction solutions which in turn require a more careful application of the Watson Transfor mation. Nevertheless, many of the previous techniques are still applicable and the resulting solutions bear a resemblance to the two dimensional solutions. The acoustic problems are solved first and then followed by the electromagnetic problem. The electromagnetic problem can be simplified to a certain extent by recognizing the similarity between the EM solution and the acoustic solutions and by introducing the "soft" EM and "hard" EM solutions as discussed in Chapter II, Section A. 36 1. Acoustic Problems A scaler plane wave of unit amplitude propagating in the direction of the negative z-axis illuminates a sphere of radius a whose center coincides with the origin as shown in Figure 5„ The incident plane wave, is expanded in terms of spherical harmonics (Harrington [20]) to give (31) p1 = eJkr cos 6 = Z jn (2n+l)Pn (cos 6)j (kr). n=0 An acoustic pressure wave incident upon a soft acoustic sphere results in a total acoustic field which satisfies the boundary condition (32) ps =0. r=a The eigenfunction solution to this problem is given by co j (kci) (33) p = Z jn(2n+l)P (cos 0) (j (kr)-— h (2 ) (kr)| . s n n i n n < n=0 h C2 )(ka) n On the other hand, the total field of the hard acoustic sphere illuminated by an acoustic pressure wave satisfies the boundary condition A y Figure 5. Spherical Coordinate System. 38 3p (34) = 0 . r-a The. hard boundry solution. ' 3 . y v .T“J rt(x) oX ‘ h (35) 'P. = Z jn (2n+l)P (cos 6) jn (kr) - h ^ 2 ) (kr)J, h « n - n=0 9X h n (2)(X)KX) ' x=ka is quite similar to the soft boundary results. A unified represen tation of the two solutions is written as (35) P = jn (2n+l)Pn (oos 6 ) y ^ QJn+l/2 ^ Jn+l/2 (kr) - Hn + l / 2 ^ ^ QV l / 2 (2>()<'" x=ka where for soft boundary Q - _ 3 1 j for hard boundary. V 2x 9X 2X (Use is made of z^/^Cx) = n/tt/2x Z (X) where (x) is n ^ + 1/2 ww n any of the solutions to the spherical Bessel’s equation and 39 ( i) Zn+l/2 ^ t^e cori'esPonc*ing solution to the cylindrical Bessel’s equation.) In a manner similar to the circular cylinder, the residue theorem is employed to obtain an integral representation of the eigenfunction solution. However, in the spherical case the summation extends only from zero to plus infinity. Accordingly, the contour, , is chosen as shewn in Figure 6 . The choice of the real axis crossing at v = -1/2 will be apparent later. As an aid in evaluating the resulting integral, the substitution of Pnm (~cos 0) = (-l)m+nP^m (cos 0) is made into Eq. (36). The integral expression for the total field is given by (37) p = 1 § p (-cos 8 ) \ l ~ ~ 2^J p ,p s m VTT v V 2kr C 1 C 2 „ x ,QJv41./2 (x) | „ (2)„ J v+1/2 “ --- v+1 / 2 (kr) dV QHv+1/2 (x) x=ka Within the shadow region, i.e. , 0 > it/2 and r sin G < a, this expression represents the diffracted field. The integral may be simplified by the use of V z> = \ Hv(1)<2> + hv (2)w and jvir/2 j tt/ 2 — j VTT / 2 e______e_____e______to obtain 2 sin vtt .. -i2vir 1-e J 40 -\) 0 £ 2 * y t 4Tf Figure 6. V-Plane Contours of Integration for the Watson Trans formation of the Sphere. 41 , r 3v/2 -jv ~/2 <3 8 > P - 2 ^ c ; c ", e-j2v, (2v+l) P^(-cos 0) L1 L 2 (1)(X) QHv+l/2 (2) H, (1) (kr) (kr) dv„ 2kr m /2 Hv +1/2 ^ Hv+ 1 /2 <2)(X) X = k a An examination of the integrand with the aid of H v (1 ) (z) = ejVTrH (1 ) (z);, H y (2 ) (z) - e jV7rH (2 ) (z), — V v — V V and P . (z) = P m (z) —V— 1 V reveals that f(-v-l) = -f(v). Accordingly, the integral is given as r -1 /2+je «-je (39) ’ 2i J / £ M d v + 2^ J I £ -1 /2+je °°-j e 7T—t [ f (v)dV + "r-—r f f(v)dv = J f (v)dv 2vj J 2nj J v 2 ttj c .+c -°°-j£ -1 /2-je 1 2 42 It is now clear that the integrand is symmetric about the point v = -1/2, the point where the contour crosses the real axis. The path of integration is closed by the addition of the contour as shown in Figure 6, and in a manner analogous to the cylindrical case, the integral is evaluated as M0) P = „ -j2[v ]r [V P[v ]-l/2(‘cos 6) m=l 1 + e J 1 1 m J QHv(1) (x) (2) \ / 3 l H, (kr) 2kr “ [v ] V 2k: m X=ka v=[v ] m (2) where [v 1 satisfies QHr , = 0 as given in Appendix A. 1 m J [v ] (X) m X=ka QHv(1) (x) The asymptotic expansions for [v^] and 3^ Q H v<2) (X)' X=ka v=[vm ] depend upon the boundary conditions; they are given in Appendixes A and B respectively. On the other hand, to the required order of the expansion, P, ]-i/2^-cos exhibits only a dependency upon m the Airy function roots; the asymptotic expansion of P. , . , (-cos0) m is developed in Appendix Ei. Upon substitution of these results and the far zone approxi- mation for H r | (kr) into Eq. (40), the diffracted field for 1V the soft acoustic sphere is found to be .5/6 jv cos ^a/r . - fcf e m / /1 \ d ^ * 2 * (41) ps S, , ./— ;— - ~ --- 772 m=“n=l k \fit sin 6 (Ai1 C-^))* . ^ n h ) in + ...) i , ~j2V TT 1 + e J m ~jv (0— i t / 2 ) -jv (3ir/2 - 0), -jk\/r2-a2 m ' . “ m ' e e + je 2 2 1/2 (r, -a ). while the hard acoustic sphere gives / 44 Eqs. (41) and (42) represent the diffracted field within the shadow region, i.e., r sin 0 < a, tt/2 < 0 < tt, and for ka sin 0 » 1. The results of Levy and Keller [12] are identical to the first term in these expansions, while Senior and Goodrich [17] obtain both terms. 2. Electromagnetic Problem The electromagnetic or "vector" diffraction problem remains to be solved. Though the original expressions for the fields are much more complicated by the vector nature of the problem, the far-zone field results are quite analogous to those of the scalar problems. A vector plane wave of unit amplitude propagating in the direction of the negative z-axis illuminates a perfectly con ducting sphere of radius a whose center coincides with the origin as shown in Figure 6. The wave is linearly polarized in the x-direction. Within the spherical coordinate system the vector wave equation may be separated and the incident plane wave expanded in terms of the resulting vector wave functions of Stratton [25] to give (43) * - *** ~ T j-» seln<1> n®! 45 where . *•\ /•\ p Ccos 9) (44a) m oln = zn (l)(kr)------— : s--- cos (p 0 sxn 0 0P 1 (cos 0) - z ^ (kr) gg sin tf> <(> and (44b) n n ^ z ^ (kr) P ^ (cos 0) cos [krz (1)(kr)]’ 9P 1 (cos 0) , n n , n + ------g j ------^ ---- C O S * 9 [krz ^(kr)]' P 1 (cos 0) ------— ------— — :— -— sin The symbols zn ^^(kr) denote solutions to the spherical Bessel's equation as described just below Eq. (36) where the i indicates the solutions of the first through the fourth kind respectively, the primes in Eqs. (44a) and (44b) denote differentiation with A A A respect to the agrument, and the vectors r, 0, and ip are the unit vectors in the coordinate direction of r, 0, and The total electromagnetic field satisfies the. boundry condition A______(45) r x e I = 0 . 1 r=a The eigenfunction solution is expressed in terms of the vector wave functions as <46> E = i J" $3 1 ) '“oln^ -J-.ln1’ n=l j n(x) [xjn(x)]' n - ]. oln J eln h < ) (x)/ n 2 X=ka [xe„ (2> Cx)31/ x=ka For values in the far zone field of the sphere, i.e., r » a, -2 the r term has a dependence of (kr) while all other terms are of (kr) henceforth, it is neglected. The remaining components are evaluated more easily if treated separately. Accordingly, the functions (47) 5 • -E - EM + Een = ^ cos + Jn (X) , (2)., .1 Pn [krjn (kr)]» [xjn(x)]' (krhn (2)(kr)]’ -j kr kr [Xhn(2) (x)]’ 47 and (48) J . E « En + E+1I - ^ jn sin * jn (x) fo\ \ 3Pn (cos 9) Jn [krjn (kr>]1 txjn(x)]' [krhn (2)(kr)]’ + j kr [xhn(2) (x)]'ix=ka kr P (cos 0) . _n______sin 0 are employed. The radial dependence of each of these functions has a form similar to that of the acoustic cases. Just as before, the residue theorem is used to transform the eigenfunction series solution to an integral. Since Pq^(cos 0) = 0 and 3Pq^(cos 0)/30 = 0, the summation is over zero and positive integer values of n, so that the path of integration is the contour + C2 shown in Figure 6. Each term of each of the components is evaluated separately. The details of this procedure are included for the first term of the 0-component, but only the results are included for the remaining terms. 48 The evaluation is aided by the substitution of P^10 (-cos 0) = (-l)m+nPnm (cos 0) and - Pn^(cos 0) = n(n+l)Pn^(cos 0) into the original series. The resulting integral is given as ^ _ § D - V < - c e> ' 01 2irj sin vjt sin 0 1 2 Jv+l/2(x) . cos 4 (-1)V IT J v+1^2 (kr) - 2 Hv+1/2 (kr)jdV- H ( ) v+1/2 (x), X=ka The same techniques used in the acoustic cases are applied here to give the residue series representation of Eg^ as -jv tt/2 ” Ta ■» 2e33i/4 (50) E 01 - \f^kr cos ♦ 1 -j2v it V 2k m=l . m (1 + e )sin 0 -1 , I Hv (1)(x) H ^ ( k r ) . •VmP v -1/2 <_COS 0) I 3 (2), . v v ' m m W Hv (X) X=ka v=v m The evaluation is completed with the substitution of the asymptotic H v (1)(X ) - 1 , expansions of vm> P^ (-cos ®) an(^ from m X=ka v=v m Appendixes A, B, and D respectively, the result being 49 ka 5/6 jv^cos - — -1 a/r * e-j7TT/12(— , - m cos 4> e (51) EQt ~ 2 m=l (ka sin 0) k \/tt sin 0 (Ai'(-qm ))2 13qm — j7T/3 |ka‘ -2/3 e 1— . ( 1 - 6 0 I 2 I + ... ) -j 2V TT . , m 1 + e -jv (0-U/2) -jv (3TT/2-0) -jk/r2 -a2 /in . . m \ e je / 2 2.1/2 (r -a ) The same methods are used on the remaining terms to give 5/6 -1 . jv cos a/r 00 -jir/12fe) . j V e ____ \ 21 cos (j) e - i (52> Een m=l k /tt sin 0 q^Ai(-q^ ) 2 <*+ i^ + - n e- ^ r 2/3+ ...) ______2s______-j 20 TT 1 + e -j\) (0-TT/2) -j\) (3tt/2-0) -jk\/r2-a2 / m . . m N e ( e (r / 2 -a 2.)1/2 50 5/6 jv cos ^ a/r Jtt/12 J^j J sin (j) e m (53) E - - Z m=l k/u sin 9 (Ai'C-q^))2 17 \ -jir /3lka \~2/3 (1+60 I 2 1 + ...) -j2v IT 1 + e -jv (0-7T/2) -jv (3tt/2-0) -jk*^2-a2 / i n . . m <. e ______(e + Je * ~ 2 2 .1/2 (r -a ) and oo nff/i o lk a |5 ^6 J^mcos ^ a//r - e~j 7tt/12 j j gin d> e (54) EflTT , Z m“l o a sin 9) k /tt sin 0 qm (Ai(-qm ))2 13qm _ 3 Q- j T T / 3 | M r 2/3j 2°V -j2V TT , , J m 1 + e -jO (0-TT/2) -jv (3tt/2-0) -jk/£2-a2 / ® . j «n . e (e + je ) , 2 2.1/2 (r -a ) 51 It is readily seen from the factor of ka sin 0 in the denominator of Eqs. (51), (52), (53), and (54) that when ka sin 0 » 1 only the terms E.,_and E, T are of significance to the desired 011 (pi order of approximation. These two components of the diffracted field are denoted as the hard EM and soft EM sphere solutions respectively, due to their close resemblance to the hard acoustic and soft acoustic results. These solutions are alternatively referred to as H- and E-waves, respectively. As before, the diffracted fields can be unified by employing a Q operator. The Q operator is the same for the soft EM component as it is for the other soft boundaries, i.e., Q = 1. However, the hard EM component is unlike its hard acoustic counterparts; its form is Q = /rrx/2 (9/3x + l/2x}» The general form of the Watson Transformation for the EM sphere then becomes 3P~[v ]-l/2 ("cos 6) m QHV U ) (X) j Hr* j (kr) $ m V=[v ] 52 where Q = 1 and $ = A Q ° /ttx/2 (9/3x + l/2x) and $ = 0 cos (J) for the hard EM boundary. A combination of these terms leads to the electromagnetic field diffracted by a perfectly conducting sphere as -jTl/12 |ka|5/6 J V 06'1 a/r , e -=• e (56) ET ~ - <|> sin (j) S ^ m=l k/7T sin 0 (Ai’C-q^))' -2/3 + ... ) -j2V 7T i i m 1 + e -jv (e-u/2) -jv (31T/2-0) 2 n ^ a (e + je ) (r -a ) e - a/r e cos | i ------— ------ AZ? + 3 1 e-J"/3|kar2/3 . (1 + eo^m + on- 2 6 I 2 1 * * *' ______2Qqm I______ -j 20 TT , . J m 1 + e -j0 (0-ir/2) -jVm (3TT/2-0))(r2_a 2rl/2 e-jk*/r-a2 (e + je 53 As in the previous cases, these results are valid for the shadow region, i.e., r sin 0 < a and tt/2 < 0 < tt and for ka sin 0 » 1. Nevertheless, it will be shown later that all of these representa tions of.the diffracted field are also valid in the illuminated regionc D. Illuminated Region The previous analysis is valid only for the shadow region. Though the general form of the residue series, Eqs. (27), (40), and (55), does converge in the illuminated regions of the left half space, i«e., x < 0 or z < 0 for cylinders or spheres, respec tively, their utility is severely limited (see Nussenzveig [23]). The asymptotic approximations which lead to rapid convergence of the first few terms of the series in the shadow region are not valid in the illuminated region, and in its general form, the series is difficult to evaluate. A reformulation of the transform which is valid for other angles is suggested, and the substitution of equivalent forms for the angular dependence of the series appears advantageous. Indeed, the method used by Franz and Galle [24] and others employs such a separation of the integral and a substitution to recast the resulting integrals into forms which yield, in addition to the diffracted field residue series, integral representations for the incident and reflected fields. Though this study is primarily concerned with the diffracted 54 fields, the results of a steepest descent asymptotic evaluation of these integrals is presented here for the sake of completeness„ As expected the diffracted fields in the illuminated region are of the same form as those in the shadow region- However, a comparison of the illuminated and the shadow solutions reveals that a discontinuity in the diffracted fields exists at the shadow boundary- Alternative representations exist which are continuous in the shadow boundary region, but they are not included here- 1- Two Dimensional Diffraction The integral representation for the total field resulting from a plane wave incident upon an infinite circular cylinder, Eq- (20), remains valid in the illuminated region- With the substitution of J (z) = 1/2 H ^ ( z ) + 1/2H ^ ( z ) into the V V V second term of Eq- (20), it is readily separated to give § (-D V7rejV7T/'2e_jV^ sin VTT Hv (2 )(kp) ’ ~---- dV where and C2 are shown in Figure 4. The first term in braces leads to an integral which is readily evaluated as the incident field, 55 (58) eJkx - ejkp cos * = 2 j1 J The second term is evaluated by closing paths and in the upper and lower half-planes respectively, with a contour at +17T infinity. In addition the substitutions of -1 “ e J for both ejV7r/2 1V7T/2 e-J3VTT/2 contours and • eJ + ------for contour 1 - e " j 2 w 1 - e“J2V7T C2 along with the Debye expansion for the Hankel function are employed. Cauchy's Residue Theorem is used to show that no contribution results from this term. For the third term, substi- +17T +1TT tutions of -1 *» e J and v = -V on contour and -1 = e J , e3'm / 2 1W /2 , e^ 3 w / 2 and ------e + on contour C_ readily 1 . e-J2 w 1 - e'j 2 w lead to 56 (b (-l)VJV1r/V JVl11 fQHv(1)(X) 59 Hv (2)(kp) dV ( ) - 27Tj 2 sin w iQH (2)(x) X=ka Cl+ C 2 (1) je 7TejTT/2Q-j3VTT/2JV(t) QH. V (X) / Hv (2)(kp) dV 2lTj -j2VTT 1 - e 'QHv(2) (x) Xnka 1 ^ TOjn/2e-j3w/2e-jv4. QHv(1) (x) Hv (2)(kp) dV -j2w 27Tj -0 -je 1 - e QHv(2) (x) X=ka oo-je 1 r jTT/2 ivtt/2 -jvcb QHv(1) (x) Hv(2)(kp) dV . 2iJ ' e e -oo-je QHv (2)(x)'X=ka The two former integrals in Eq. (59) are evaluated by the same technique employed in the shadow region to yield a similar residue series which is recognized as the diffracted field. Saddle point methods, e.g., method of steepest descents, may be used to 57 evaluate the latter integralc This shows the integral to yield the reflected field, which can be determined alternatively by the Luneburg-Kline seriesWhen these results are all gathered together, the total field in the illuminated region is given by Incident Ray (60) p = e Field I Reflected Ray jVIT/2 -jV«)> (2 ) - f / e (kP) d7 Field — 00 QHv (2 )(x) I x=ka v w -j[vm ](3TT/2- Field QHV(1) (X) H [v}] (kp> 9 ^ Hv (2 )(x)/ X=ka L m V=[V ] m For completeness, the far zone reflected field as determined by Franz and Galle [24] is given by rriv r la cos cf)/2 |1/2 -jk(p-2a cos (jj/2) r_ (3-8/cos2p/2) (61) ps ~ -I 2 p ^ I e (1 + j16ka cos p/2 + 1 512 (ka cos cp / 2)2 and 2 -jk(p-2a cos (p/2).. . .(3+8/cos p/2) U J16ka cos cp/2 1 512(ka cos p/2)2 The description of the field is now complete with the evaluation of the diffracted fields As can be seen from Eq„ (60), the diffracted field in the illuminated region is the same as Eq„ (27) except that 3tt/2 4- p is substituted for p - tt/2 in the field expression for the surface ray which is propagating in the positive p- directionc 2c Three Dimensional Diffraction The methods employed in the two dimensional analysis are, in the main, applicable to the three dimensional cases„ The integral representation of the total scaler (acoustic) field, Eq» (37), is rewritten as 59 +jlt/2 -jVTT/2 7Te (2V+1) (63) p p^(-cos 0 ) 2tt j -j2VTT x - e Cl+ C 2 QHV+-1 /2 ^X^ 1 (2 ) ,, A 2JV+l/'2(kr^ “ HV+l/2('kr) \>+l/2 I The first term is seen to yield . 'A 3/2 jvrr/2 (64) I5J J sin vn pv<-cos 0> W (kr> cl+ c 2 0° “ 2 jn (2n+l)P (cos 0)j (kr) = e^kr C0S ® = e“^kz , n=0 n n the incident fields With the aid of a series of rather tedious but straightforward substitutions and algebraic manipulations the last two terms give the reflected and diffracted fields„ Crucial to the evaluation are the substitutions Pv^(-cos 0) = P ^ ( c o s 0) cos [TT(v+y)] - ^ (cos 0) sin[Tr(v+y)] 60 and (1 + e-J2 w )/(l - e"j2 w ) - 2e-J2V7T/(l - e"j 2 w ) + 1. The resulting expression for the total field is given by I Incident jkz (65) p = e / Field o+je - j f e ^ ^ v e ^ 7^ 2 tan vtt P .. ,_(cos 0) L v-x/z Reflected Field QHV (1)(X) f H Hv (2)(kr) dv V 2kr QHV (2)(X)/ X“ka 13TT/4 -j3[v ]tt/2 °° 2e [v ]e m + JU m=l -j2[v ]tt P [v 1-1/2(C0S 0) « . m in 1 + e Diffracted i Field T I H r?,\ (kr) 2kr l^QHv(2)(x)/X“ka [Vm] v-[vm ] which is valid for 0 < 0 < tt. The far zone reflected field has been evaluated by Depperman and Franz [26] via the method of steepest descents and shown to be (66) p e exp[-jk(r-2a cos 0/2)] [1 - 2r 2ka cos 0/2 61 and (67) p,r ~ — exp[-jk(r-2a cos 6 /2 )] [1 + ----— r--- n lT 2ka cos 0/2 The diffracted field aggres with Eq„ (40) except that P r , ,/9(-cos 6 ) is replaced by P- ..(cos 8 ) and an extra m L m J U factor of j is present in Eq. (65) due to additional caustic traversals by the raysc These differences effectively replace the surface ray of Eq. (40) which is propagating in the positive 6- direction by one in Eq. (65) which has an additional traversal around the sphere„ The latter representations include asymptotic expansions for ^CCS * anc^ eonsecluent:ly» they are valid only for 0 < 8 < tt, though the incident and reflected field representations are valid for 0 = 0 , also,, The vector (EM) problem is solved in a manner quite analagous to the scaler three dimensional problem. The far zone vector components of the field exhibit, independently of each other, the characteristics>of a corresponding scaler field. As expected, the results for the EM sphere, in addition to the incident field, — i ~ jkz , , E = xe , are given by 62 a -jk(r-2a cos9/2) (6 8 ) E = . r, , , (1-2 sin20/2) , 7 sin2 0/2 -2 sin4 0/2 , * L -L T J -J + O C + . 0 0 J 2ka cos 0/2 4(ka)2 cos6 0/2 and (69) Ebr - 0 • Er ~ - cos * - g e-jk(r"2a C0S 6/2) .[l+j 1 7 sin2 6/2 2ka cos^0/2 4 (ka)2 cos^0/2 for the reflected field, while the diffracted field is in agreement with Eq0 (55) in the same way as Eq„ (65) agrees with Eq. (40), The reflected field components, Eqs„ (6 8 ) and (69) are given by Logan [27]. 63 With the important exception of the backscatter aspect, this completes the asymptotic determination of the fields from exact eigenfunction solutions„ These canonical problem solutions are used in Chapter III to determine the diffraction coefficients and attenuation constants of the GTD. The backscatter aspect is treated in Chapter III, also*, CHAPTER III DIFFRACTION COEFFICIENTS AND ATTENUATION CONSTANTS The GTD formulation of the diffracted field in Chapter I is complete except for the evaluation of the diffraction coefficients and the attenuation constants. In this chapter these unknowns are found by a direct comparison of the GTD solution with the asymptotic expansion of the corresponding exact solution as found in Chapter II. The GTD formulation is described in detail for the soft acoustic cylinder. However, only the results and special points of interest are presented for the remaining cases. A. The GTD Field 1. Soft Acoustic Cylinder A plane wave described by p i = e *i lex is incident upon a soft acoustic infinite circular cylinder of radius a as shown in Figure 7. As previously discussed, the total field is identically equal to the diffracted field in the shadow region, and the comparison between the GTD and the exact solutions is most easily made there. According to Eq. (8 ) the diffracted field at point P due to the ray incident on a circular cylinder at Q-^ is given by 64 % > • Y O V 2 it 1 v>*e X>e •?}-& .te * s?)\V&' m=l 2Tra (1-exp [- f a, (t')dt'-jk27Ta]) \/s 0 a[ while that due to the ray incident at is given by (71) p d (P)~ °°S ______Vbm_3__bm_4_ ((V Dk (Q/) m==l 2Tta (1-exp [- / a, (t')dt'-jk27Ta]) v's~ 0 a[3TT/'2-cj>-cos a/p] exp [-/ otbm(t') dt ,-jk(s+a(37r/2- The asymptotic nature of the GTD formulation is emphasized by the use of ~ rather than =. As a consequence of pc= 00 for cylindrical geometry, /dn (C^)/dp (Q2) /pc/s(s+pc) =/dp(Q3 )/dp(Q^) /pc/s(s+pc) =l//s. A further consequence of the geometry is the absence of any caustics and, hence, the caustic phase shift, iJj , equals zero. 67 For convenience the incident ray field is chosen to have a zero phase reference at the origin. These forms are simplified by noting that the diffraction coefficients at all four points are identical, i.e., D ^ C q p = Dbm 2 (D, )2e (Jk+abm)a cos la/p e-jk,/P ^ (72) p„d (P)~ E S ' ' m=l (1-exp [-27Ta(jk+a )]) (p2-a2)1/4 bm ( e- ^ k+abm ^ ^ -Tr//2^a + e"^jk+abm) (37T/2-cj))a 2. Hard Acoustic Cylinder There are no essential differences in form between the hard acoustic and the soft acoustic cylinders; the diffraction coeffi cients and attenuation constants of the hard surface are merely substituted for those of the soft surface. The result is 68 (d f e ^ k+0lnm^a cos~la/Pe'"jkv^ ~ ^ , 00 \ nmj (73) p,d (P)- Z ■ ~ m=l (l~exp [-2ira(jk+a )])(p -a ) nm . e-(jk+anm) (tj)-Tr/2)a +e-(jk+anm) (3ir/2-cj))a 3. Soft Acoustic Sphere A plane wave described by p^ = e^kz is incident upon a soft acoustic sphere of radius a as shown in Figure 7. Though pc = 00 in the cylindrical case, it has a finite value in the spherical case of pc = a(a sin 0 - s cos 0)/(a cos 0 + s sin 0). Likewise, the area elements, dr), are affected by the finite value of p . The result of these effects is that /dr) (CL ) /dr) (Q„) /p /s(s+p ) i 2 c c = /a/r sin 0 (r^- a^)--*-/^. A further effect of a finite pc is the existence of caustics and the necessary phase correction. The cylindrical wave nature of the surface ray field gives rise to a phase advance of ^C(Q) = tt/2 every time a ray passes through a caustic. With these changes incorporated into the previous cylindrical results, the soft acoustic sphere has a GTD field given by 69 (D )2e (J'k+abm ) 3 cos"la/r A °° bltl (74) p d(P)~ Z m-1 (1+exp [-2ira(jk+a ) ]) (r^a2)3-^ s^n b m r 4. Hard Acoustic Sphere The only change from the soft acoustic sphere is the sub stitution of the hard diffraction coefficients and attenuation constants for the corresponding soft ones. The diffracted field is then (D )2e (:jk'H)'nm) a cos”la/r e~jk/r2-a2 , 00 nm (75) p d(P)~ Z h m=l (1+exp [-27Ta(jk+a ) ]) (r2-a2 )1/4 V r s i n 6 nm • ( e-(jk+anm) (0-7r/2)a +je-(jk+anm) (37r/2-0)a 5. EM Sphere The EM sphere solution contains both a binormal and a normal component corresponding to a soft and a hard solution, respectively. The incident field, described by E7 = xe^kz, and the appropriate 70 surface vectors are shown in Figure 7. An evaluation of the surface vectors in terms of the coordinates of the field point P shows that n^ = -n^ = x cos <|> + y sin 4>, = -b^ - -b^ = -$» =*• -9, A and n^ - 0. Subscripts 1 and 2 are associated with surface ray propagation in the positive 9-direction while 3 and 4 indicate propagation in the negative 0-direction. The corresponding dif fracted field is expressed as .2 (jk+a ) a cos-1 a/r 00 (Dbm) e (76) E^P)- - -Jk^ 2 • (e_^ k"*"01bm^ +je“ ^ k+0ibm) (3Tr/2-0)a) e / 2 2.1/4 (r -a ) .2 (jk+a ) a cos a/r » nm e nm /—a--- -0 COS -jk/r2-a2 e J • (e- (jk+“nm)(e-,r/2)a +ja-(3 k+anm)<3,t/2-8)a) (r -a ) B . The Asymptotic Field The desired asymptotic solutions for the fields are found in Chapter II and are repeated here for convenience. (The equation number of each solution from Chapter II is included.) 1. Soft Acoustic Cylinder —j7T/12 kaj 2 „ “ 1 J 72 2. Hard Acoustic Cylinder -jTT/12/kaj1/3 00 jV cos ^ a/p (30) z e m h m=l /2rrk 1 \ . . /ka\2/3 m 1 + ' + ... \ 30 . . -j2v IT 1 - e J m -jkv/p2-a2 -jV ( e-jTT/12|ka} ejvmcos 1a/r Pc ^ | 2 ' ra-1 k /tt sin 0 * (— qm ) ) ^ : m ~ j tt/3 /ka j 2/3 , ^ + g o e 2 ! + • ■ - j 1i j.+ e ~j2v J m TT ! 2 2 j e-JVm (6-W2) +je-:V]ii(3W2-e); e - ^ ( A a 2)1'2 Hard Acoustic Sphere —jtt/ 12/ka j v cos ^a/r ^ " 6 1 "2 j 6 m 74 5 , EM Sphere ka, I5/6 iv • cos “1 a/r / 3—j TT/12 — I e m (56) - 17q . , -2/3 ka (1 + e~ " 60 2! -j2v TT m 1 + e -jv (0-7T/2) -jv (37T/2-6) / m . . m v , 2- 2-1/2 -jk/r2-a2 (e + je ) (r -a ) e J 5/6 -1 , ka iv cos a/r 00 -jir/12 m 21 0 cos cj) Z 0 m=l k /tt sin 0 q (Ai(-q ))2 -2/3 ka 2 20q m -j2\) TT m 1 + e jO (0 tt/2) jOm (37T/2-0) 2 2 1/2 . k1 / ~ 2 2 ik/r -a (e + je ) (r -a ) e 75 C. Ey-alua.tlQ.n_.of Diffraction Coefficients, and Attenuation Constants The diffraction coefficients and attenuation constants are found by a direct comparison of the GTD and the asymptotic solutions for each boundary condition. These results are summarized in Table 1. Column A of Table 1 gives the square of the diffraction coefficients as found by Keller. The square of the higher order diffraction coefficients for surfaces of constant curvature are obtained by the product of Column A and Column B, the correction terms. Column C gives Keller's original attenuation constants while higher order correction terms of the attenuation constants are given in Column D. The product of Column C and Column D yields the complete attenuation constants for surfaces of constant cur vature . D. Generalization and Interpretation Though the diffraction coefficients and attenuation constants obtained in the preceeding sections lead to increased accuracy in describing diffraction from surfaces with constant radii of curvature such as circular cylinders and spheres, this is not necessarily true for more general surfaces where the curvature varies along the ray path or where there is an arbitrary curvature transverse to the ray path. Fortunately, however, if the earlier works of Keller and Levy [8] and Franz and Klante [9] and the TABLE 1 DIFFRACTION COEFFICIENTS AND ATTENUATION CONSTANTS FOR CIRCULAR CYLINDERS AND SPHERES SQUARE OF DIFFRACTION COEFFICIENT ATTENUATION CONSTANT 2 S u r f a c e D • (Column A) • (Column B) °m ™ (Column C) • (Column D) A. Keller's Result B. Correction Terms C. Keller's Result D. Correction Terns AK-q^) - 0 qj^ - 2.33811 S o f t • , / 2 \ 2 / 3 1 -jir/3 C y l i n d e r 1 k a ) qm 30 q2 - 4.08795 _-l/22-5/6al/3e-j-/12 Ai*(~qx) - .70121 ifn h So f t ^1/6(Ai'(-qm ))2 V /6(*!)1/3 M e A c o u s tic Ai’(-q2) » -.80311 and , ^ ( 2\2/3 -3"/3 S o f t E M I k a ) qm l 3 0 4 ) S p h e r e H a r d i J _ J t \ e- . W 3 M el)2/3(l C y l i n d e r \ ka / \ 30 - 2 10 / **» + - L . - 2 10 I qm Ai’t-qJ - 0 , 2 \213 1 qm H a r d H e0 ^ 4 ) ee A c o u s t i c M ) (iff q x = 1 . 0 1 8 7 9 _-l/2 -5/6 1/3 -Jtt/12 S p h e r e I a e q2 - 3.24820 ( A i(-q ))2 qm m a Ai(-qx) - .53566 Ai(-i2) = -.41902 H a r d V ka / ^ 60 EM S p h e r e - 2 (.10 4 I] qm ra 77 recent work by Hong [10] are considered, generalized diffraction coefficients and attenuation constants can be obtained. A description of the surface geometry is given below using the notation of Hong, The radius of surface curvature along the ray trajectory is called the longitudinal radius of curvature, p^. (This corresponds to the radius of curvature a for circular cylinders and spheres.) The first and second derivatives of the longitudinal radius of curvature with respect to arc length along the ray trajectory are denoted as p and p , respectively. (These S S terms are identically zero for circular cylinders and spheres.) The normal component of the radius of curvature in the direction of the binormal is called the transverse radius of curvature, ptn. (For cylinders, ptn = °°, while for spheres, pfcn = a.) Finally, with the introduction of these terms, it is useful to classify surfaces as either singly-curved surfaces, i.e. Ptn = °°, or doubly-curved surfaces, i.e. P =! °°. Attenuation constants associated with the hard acoustic and hard EM boundary conditions can be obtained directly from Hong's work. Specifically, these attenuation constants follow from a GTD interpretation of Hong's Eqs. (3,49) and (4.17). It is noted that Hong's results are consistent with those of Keller and Levy and Franz and Klante. An examination of the exponential factor in Hong's doubly-curved hard acoustic solutions, Hong's Eq. (3.49), shows that it reduces to the hard acoustic attenuation constants 78 of Table 1 whenever a circular cylinder or a sphere is considered. If P = “ is substituted into Hong’s solution, the singly-curved tn solution obtained by Keller and Levy and by Franz and Klante results. With this agreement, Hong's exponential factor is assumed to be the doubly-curved hard acoustic form of the atten uation constant. Through a similar comparison of Hong’s Eq. (4,17), the doubly-curved hard EM attenuation constant is also obtained. As a further confirmation, these two general forms are identical in the case of the cylinder. This is to be expected since the hard cylindrical solution can represent either an acoustic or an EM field. The soft surface attenuation constants are obtained by a slightly different approach. Though Keller and Levy do have a singly-curved soft surface solution which reduces to Table 1 for a circular cylinder, no doubly-curved soft surface solutions are available. However, the solution can be determined readily by first observing that the doubly-curved soft surface field in the 3ps 3G 3ps shadow region satisfies the expression —— =2/ — ~— r ds f * 3n 3n on u This corresponds to the expression for the doubly-curved hard 3G surface field, p, “ 2 / — p. ds ' , and it has a solution of the rh 3n ^h same form. Then, by using the Taylor series expansion for the field in the vicinity of the surface, 79 9pg (0) 3ps (0) p (n) = p (0) + — --- An => — r An, where n is the coordinate rs *s 9n 9n * normal to the surface, it is apparent that p very close to the s 9ps surface is proportional to . Hence, pg has the same form as p^ which is available from Hong, An examination of Hong's work reveals that the soft boundary condition requires a different solution of his Eq, (3=41) which modifies his Eq, (3.47) to o eliminate all of the 1/q terms in the soft attenuation constant. Til The resulting general form is given in Table 2. This form agrees with Table 1 for a circular cylinder or a sphere. Moreover, the absence of a transverse curvature dependence for soft boundary attenuation constants is also evident in Table 1. Consequently, the doubly-curved soft surface results of Table 2, are found to be identical to Keller and Levy's singly-curved soft surface results = The general form of the diffraction coefficients is not obtained as readily. Neither Keller and Levy nor Franz and Klante provide any insight into the generalized behavior of the diffraction coefficients. (Keller and Levy obtain a more general diffraction coefficient, but it no longer has the same interpre tation as the diffraction coefficient originally defined.) A GTD form of Hong's solutions is obtained by substituting the far zone form for the near zone surface ray fields. Hong's repre sentations make use of the near zone approximation of the TABLE 2 GENERALIZED DIFFRACTION COEFFICIENTS AND ATTENUATION CONSTANTS SQUARE OF DIFFRACTION COEFFICIENT ATTENUATION CONSTANT 2 S u r f a c e * (Column A) • (Column B) * (Column C) • (Column D) A. K e l l e r ’s R e s u l t B. Correction Terms C. Keller’s Result D. Correction Terms Soft -1/2 ,-3/o i/3 -1-/12 A c o u s t i c — 2 : _ e J a k c 1/3 ! + ( ^ - ' 2 / 3 0 + -1&- + e - J - / 3 a n d 1 Ike ) ^m\ 30 40 180 J S o f t EM k1,0(Ai'(-^))“ fi tn g 1 + ( i J “ ) + 45s" + “ho) g \ tn H a r d ? ' A c o u s t i c 1 IJ. + e - j - /3 - 2 V10 40 60 J + - L - ( - i + A . - ^ y \ e " J * /3 q tn ' - 2 V 10 40. 60 90 J I m q tn / Tn _-l/2,-5/6 l/3e-j-/12 j. e^ ( ^ ) /3 g / -) \2^3 / / I 0 C 0 v 1 + (tr) ( ’m l 30 + 4Bfi_ + _ 1 8 o ) - ( ^ ) 2/3( ^ - z f V e + lt5 V ) g ' tn H a r d EM . j _/ j . . !s!s'L-^/3 - 2 \ 10 4 c 6 0 i: q t n ' ^ ( l ! - l t - ^ + ^ ) ) e'W 3 ra 1= t n 7 81 Hankel function which is valid when kr - kP .i.e., - 8 ■— /> m -jir/3 • H (2)(kr) ~ 2e:)7T/3 _g ) V v Qm lT On the other hand, the GTD solutions require the far zone (2).. fl~ ” ^ k r 2 approximation, ^kr^~varkr e * which is valid for kr » kPg >> 1. When the latter Hankel function approximation is substituted for the former in Hong's solutions, the GTD form of the field results . This modified form may now be compared term by term with the GTD solutions to obtain the doubly-curved hard acoustic and hard EM diffraction coefficients. These results agree with Table 1 when applied to circular cylinders and to spheres. The soft surface diffraction coefficients are obtained by a similar procedure. By adapting Hong's method to the soft boundary condition as described above, the soft surface solution is found. Then this solution, valid near the surface, is also modified to exhibit far zone behavior. A term by term comparison with GTD solution readily yields the results in Table 2, As before, these results agree with Table 1 for circular cylinders and spheres. An examination of Tables 1 and 2 reveal several similarities which further affirm the validity of Table 2. The coefficients of q and q of Column D of Table 2 are identical, i.e., ^m m 82 2/3 2 1 2 .. 4 , 2| — j tt/3 £ , , , , kP 60 “ 4JPgpg 13lPg * a feature shared by the g 2 ^2/3 e -j7r/3 corresponding coefficients of Table 1, which equal ka 60 The coefficients of q and q of Column B of the tables are related in a similar way. While both tables exhibit an absence of 1/q^ terms in Columns B and Columns D for soft surfaces, the coefficients of the 1/q— 9 terms for the hard surfaces in Column B m are the negative of the corresponding coefficients of Column D, (Terms involving p„ are an apparent exception in Column B of © Table 2. However, since Hong assumed the surface to be symmetric at the shadow boundary, £> = 0 in the diffraction coefficient.) O Finally, as mentioned earlier, entries of Table 2 reduce to those of Table 1 for circular cylinders and spheres. Each of the hard boundary attenuation constants contains all of the terms of the soft boundary attenuation constants, i.e., the q (or q ) terms, but in addition, each has terms which make mi m * it distinct, i.e., the l/qm2 terms. The qm (or q^ terms, common to all boundaries, are independent of transverse curvature; these terms depend only upon longitudinal curvature properties. On the other hand, the additional l/qm2 terms in each hard boundary form do contain additional factors of longitudinal curvature dependence. If the boundary conditions of the corresponding cylinder and sphere solutions are considered, this latter behavior is not unexpected. In Appendix A it is seen that those additional terms directly 83 follow from the more complex boundary conditions associated with hard surfaces. A similar boundary condition dependence for the general forms logically follows, Though there is a similarity of forms of attenuation constants between the soft and hard boundary conditions, the attenuation is strikingly different. For soft surfaces the attenuation is related to = 2.33811, while for hard surfaces it varies with = 1.01879. This difference is related to the fact that the soft surface boundary conditions require a non-zero field to vanish at the surface. The resulting effect is that the power radiates rapidly from the region of the surface; the rate of attenuation of this field near the surface is great. On the other hand, the hard boundary condition does not impose such severe constraints on the field, and con sequently, there is a lesser rate of attenuation. Included for all types of boundaries are terms involving p 8 and p . To aid in their interpretation it is important to recall that surface ray modes are established according to the radius of longitudinal curvature at the point of diffraction. If the surface does not change along the ray, the higher order modes are soon negligibly small due to their high rate of attenuation. However, if the surface changes along the ray, then, as pointed out by Franz and Klante, each surface ray mode will continuously excite an infinite series of surface ray modes at each point. Consequently, the higher order, lossier modes are continuously re-excited along 84 a surface of varying curvature, resulting in a greater rate of attenuation. This increased attenuation does not depend upon whether the curvature is increasing or decreasing, only whether it is changing, i.e, p ^ 0, It is expected that if p < 0 , the surface is tending 8 8 toward a condition of lesser radius of curvature and hence, in creased attenuation. On the other hand, if p >0, the attenuation 8 can be expected to decrease. The general forms predict this behavior. As a result of transverse curvature, each hard boundary exhibits distinct behavior which differs from the others. For singly-curved surfaces the hard acoustic and hard EM solutions are identical. However, whenever doubly-curved surfaces are encountered, the hard acoustic solution is more greatly attenuated, whereas the hard EM solution is less attenuated than their singly-curved counterpart. The magnitude of this transverse curvature effect is th*. ,-jame for the two cases; the sign is opposite. The diffraction coefficients relate actual fields (either incident or diffracted) to the fictitious surface ray fields, and their physical interpretation is not as apparent as that of the attenuation constants. However, several observations as to their behavior are in order. 85 All diffraction' coefficients exhibit transverse curvature dependence, though this dependence is not the same for all surfaces. All diffraction coefficients also exhibit a dependence upon variable curvature through terms involving p . The p terms are not present 8 S in the diffraction coefficients since p is assumed to be zero at g the shadow boundary in Hong's work. The presence of additional correction terms in the hard boundary diffraction coefficients compared to the soft boundary diffraction coefficients is related to the more complex nature of the boundary conditions just as for the attenuation constants. . E. Backscatter Aspect The GTD field from doubly-curved bodies may be found from Tables 1 and 2 for all regions remote from the body, shadow boundaries, and caustics. Though the backscatter aspect, i.e., (j) = 0 or 0 = 0 for circular cylinders and spheres respectively, is of primary concern in many problems, the GTD as formulated thus far will provide backscatter results only for the cylindrical cases, i.e., singly-curved surfaces. The presence of the axial caustic at 8 = 0 invalidates the GTD for backscatter calculations for the sphere. More generally, any surface of revolution will be simi larly invalid. To overcome this limitation, Levy and Keller [12] obtain a correction factor by comparing a known field containing an axial caustic with its asymptotic expansion. This correction 86 factor is multiplied times the GTD solution to yield the proper form of the GTD field in the vicinity of the caustic. An alterna tive method employed by Senior [28] yields the backscattered field by setting 0 = 0 in the eigenfunction solution and then making the Watson Transformation. This representation can be interpreted readily in terms of the GTD, but is difficult to apply to arbitrary bodies. Kouyoumjian [29] utilizes a third technique, a steepest descents method of evaluating a Huygens-Kirchhoff integral. In this method the Huygens sources on an arbitrary spherical surface which encloses the body are determined by the GTD. The Kirchhoff integral description of the radiation from these sources is then evaluated by the method of steepest descents, subject to the approximations of the GTD. Since the Huygens-Kirchhoff integral employs the GTD directly and is valid for general surfaces, it is used to determine the GTD field at backscatter. This technique, generalized to consider higher-order contributions, is discussed in detail in Appendix E. A single mode of the GTD field near the backscatter aspect of a sphere assumes the form -nk/rM /2 -a 2 e lim (77) in 0-K) /sin 0 87 Not only does this field exhibit a simgularity at 6 = 0,.. but an infinite number of such rays contribute to the backscatter field. By the method of Appendix E, these difficulties are overcome and a finite (and correct) asymptotic value of the field is obtained in the following form . i 2 . . ,w 2 2.1/2 d jE Dm (a) e g-jk(r -a ) (78) Sn ~ -j2ir[v ] [ 2 2 (1 + e ) /r -a -2/3 tm o -jTr/3 ka • e ^ 4 v/2ik [1 + 1 1 1/2 1 where 1/2 indicates the upper case for scalers and the lower case for vectors. It readily follows that the backscatter field is obtained if each mode of the GTD field as given by Eq. (77) is multiplied by the correction factor -2/3 -jTT/3 ka 1 i^17^4 v^TTk sin 0 [1 + 1/2 1/3 This new representation exhibits a k dependence compared to the — 1/6 k behavior of the usual GTD field. A generalization of this method to arbitrary surfaces, though quite desirable, is not attempted in this work. The same principles 88 would apply, but the complexity of the mathematics for non-spherical surfaces makes each body a detailed study in itself. CHAPTER IV APPLICATIONS AND NUMERICAL RESULTS The theory developed in the preceeding chapters is now applied to several bodies in order to demonstrate the usefulness, accuracy, and range of validity of the GTD. These examples have been chosen to correspond with measured or exactly calculated data already at hand or in the literature, and consequently, a varied mixture of units and parameters are involved. The scattered field is calculated for several different geometries. In particular these calculations include the following bodies: 1) the infinite circular cylinder, 2) the semi-infinite plane tipped by a circular cylinder, 3) the infinite elliptic cylinder, 4) the sphere, and 5) the prolate spheroid. Most calculations are for backscatter cross sections, but a few bistatic cross section calculations are included. Moreover, both acoustic and electromagnetic calculations are made in most cases. 89 90 Since the scattered field includes the reflected field, inaccuracies quite unrelated to the diffracted field may be included. More will be said of this later. Additional calculations of the attenuation and velocity of propagation of the surface rays are made for several different geometries. Comparisons between these results allow the observa tion of the effects of variable curvature, transverse curvature, and vector fields on the exponential factors. A. Two Dimensional Scattering Two dimensional geometries offer the simplest calculations for the GTD; in particular, the infinite circular cylinder is the easiest of these. The GTD calculations of the normalized cross section per unit length for a circular cylinder are compared with similar exact solutions by Senior and Boynton [30] in Figure 8. In the GTD calculations the diffraction coefficients and attenuation constants as given in Table 1 are substituted into Eqs. (72) and (73) to obtain the diffracted field for soft and hard surfaces, respec tively. Only the lowest order mode and only the first encirclement of the surface ray are found to be significant. The reflected field is found from the Luneberg-Kline series which is identical to Eq. (61) for soft surfaces or Eq, (62) for hard surfaces. Three reflected field terms are included for both the soft and the hard surfaces. As can be seen, quite accurate results are obtained down to ka - 2, This is well below the formal limits set by the theory, 1.5 n o 0^0 0.5 Figure 8. Normalized Backscatter Cross Section for the Circular Cylinder (----- Exact Hard, ooo GTD Hard, Exact Soft, ••• GTD Soft). 92 i.e., -ka >> 1. Even so, the resulting inaccuracy may be in the reflected field rather than the diffracted field. A very easy means of examining the diffracted field quite separately from the reflected field is available with the cylinder tipped, semi-infinite plane as shown in Figure 9. The lower right quadrant, x > 0 and y < 0, has only a diffracted field with no reflected field present. From Keller [31], both an exact and a first order GTD solution are available which are compared with the GTD solution given by Kouyoumjian and Burnside [11] in Figure 9. This improved version of the GTD includes the first four modes calculated by either Eq. (72) or (73) for rays 1, 2, 3, and 4 as shown in Figure 9, while Keller includes only rays 1 and 2. Excellent agreement between the improved GTD and the exact solution is observed for ka > .1 for soft surfaces and ka > .3 for hard surfaces, a significant improvement over Keller’s original results. It is clear that the reflected field is the source of most of the inaccuracy for the backscatter from the circular cylinder. A similar limitation resulting from the reflected field has been noted by Senior [28] in the case of spherical scatterers. Variable surface curvature effects may be observed in the backscatter from an infinite elliptic cylinder. However, the higher order terms of the reflected field are unavailable and 93 (3) (2) (4) 0.9 0.8- HARD SURFACE 0.6- 2 0 .9 - ^ 0.4 OS' 0.2' SOFT SURFACE 0.1" ka Figure 9. Diffracted Fields from the Semi-Infinite, Circular Cylinder Tipped Half-Plane, u^ ~ F(0,ka) e^ ^//p (----- Exact, Keller, ••• Kouyoumjian and Burnside). 94 they are obtained only by a tedious evaluation of the Luneberg- Kline series. In addition, the GTD results cannot be compared with the exact solution since it is also unavailable. Consequently, only the surface ray effects, i.e., the.oscillatory behavior of the backscatter field, can be observed. The backscatter cross section of an infinite elliptic cylinder with a 2:1 axial ratio and an interfocal distance of 2a is compared in Figure 10 with similar data from a circular cylinder. The diffracted field is calculated by using the results of Table 2 in either Eq. (72) or (73) for soft or hard surfaces, respectively. The reflected field is obtained from the first term of the Luneberg-Kline series with the radius of curvature of the specular point, i.e., the tip, substituted for the cylinder radius a in Eq. (61) or (62). The soft circular cylinder data is scaled so that its radius a is equal to the radius of curvature of the elliptic cylinder tip. The similarity of the two soft surface behaviors indicates the reason for the scaling; the backscatter from soft surfaces exhibits very little diffracted field dependence, but is dominated by the reflected field which is primarily determined by the specular point radius of curvature. The surface ray effects are apparent in the pronounced oscillatory behavior of the hard surface backscatter cross sections. The two sets of data have a ratio of the average oo Oo oo Oo 0 2 4 6 8 10 ka Figure 10. Normalized Backscatter Cross Section for the Elliptic Cylinder ( Exact Hard Circular Cylinder, ooo GTD Hard Elliptic Cylinder, Exact Soft Circular Cylinder Scaled, ••• GTD Soft Elliptic vo Cylinder). 01 96 period of oscillation of 0-93 while the ratio of the associated path lengths is 1,13, This difference indicates that in addition to path length differences there are variable curvature effects which result in a slightly greater average surface ray velocity for the elliptic cylinder. In addition, the elliptic cylinder data exhibits a greater maximum to minimum ratio than the circular cylinder data suggesting that the two oppositely directed surface rays are more nearly equal for the elliptic cylinder. As shown in Section C of this chapter, these observations are correct. The surface ray propagates with a velocity near the speed of light and very little attenuation over most of the surface, However, in the vicinity of the tip the velocity decreases significantly while the attenuation increases. B. Three Dimensional Scattering Though three dimensional scattering requires more compli cated calculations than the two dimensional cases, the additional effects of transverse curvature and vector fields can be observed. In addition, numerous results are available for comparison in the literature, A special caution should be noted for backscatter calculations from rotationally symmetric bodies; the caustic correction discussed in Chapter III, Section E must be employed. Otherwise, the GTD theory is applied just as before. 97 The simplest three dimensional body is the sphere; its high degree of symmetry greatly simplifies the calculations. The effects of a finite radius of transverse curvature may be observed, but variable curvature effects are totally absent. Figure 11 compares the GTD normalized backscatter cross section for an acoustic soft sphere with exactly calculated data from Senior [28]. The results of Table 1 are substituted into Eq. (74) to calculate the diffracted field; only the first mode and one encirclement are required. In addition, the axial caustic correction of Appendix E must be employed. The reflected field is obtained from the first three terms of (79) p / - -!-ij e-jk as given by Senior [28] which is a special case of Eq. (66) for backscatter. As before the high rate of attenuation quickly damps out the surface ray. Reasonable agreement is possible down to ka - 2, but again, the reflected component dominates and appears to be the major source of error. The GTD results for the acoustic hard sphere are similarly compared with exact calculations by Senior [28] in Figure 12 and good agreement is obtained to ka - 1.5. Substitution of the results of Table 1 into Eq. (75) readily yields the diffracted field. Again, only the first mode and first encirclement are used and the axial 4.0 s.o 2.0 1.0 Figure 11. Normalized Backscatter Cross Section for the Soft Acoustic Sphere vo (----- Exact, ••• GTD) oo 0.8 0.4 Figure 12. Normalized Backscatter Cross Section for the Hard Acoustic Sphere VO (----- Exact, • • • GTD) . vo 100 caustic correction of Appendix E is required. The reflected field is obtained from the first three terms of r (80) ph as given by Senior [28]. This is a special case of Eq. (67) at backscatter. Since the attenuation of the hard surface ray is much less than the soft, phase errors are more apparent. Con sequently, the next higher order term in the attenuation constant is necessary in order to obtain good agreement with the exact data. This term is found to contribute only a phase shift. As Senior has pointed out, the reflected field is greatly in error and dominates the backscatter field for ka < 2. Exact calculations of the normalized backscatter cross section of a perfectly conducting sphere are shown in Figure 13 along with the GTD calculations. The diffracted field is obtained by substituting the results of Table 1 into Eq. (76), retaining only the first mode and first encirclement, and using the axial caustic correction of Appendix E. The reflected field comes from the first three terms of Eq, (68) and (69), The resulting GTD data provides excellent agreement down to ka - .7. Agreement for such a small value of ka is primarily due to the vanishing of the third terms in Eqs, (68) and (69) at backscatter, i.e., 0=0. Without these terms, the reflected field remains accurate to values of ka - 1, Since the surface ray phenomenon RAYLEIGH LAW .54 0.1 ka Figure 13. Normalized Backscatter Cross Section for the Perfectly Conducting Sphere (----- Exact, • • • GTD) 102 has already been shown to be accurate for small ka, the GTD solution is valid for much smaller spheres than would be expected from the original constraints of the theory. Bistatic cross sections are also considered for the perfectly conducting sphere. Two representative results are shown in Figures 14 and 15 where they are compared with exact calculations of King and Wu [32]. The calculation of the diffracted field is accomplished by the substitution of the results of Table 1 into Eq. (76). Again, only the first mode and first encirclement are required. However, no caustic correction is necessary for bistatic calculations. The reflected field is obtained from Eqs. (68) and (69); the first two terms of Eq. (68) comprise E^ 1l while the first term of Eq. (69) is EQr . O Unfortunately, two difficulties, not present for the spherical backscatter calculations, are encountered here. First, the axial caustic unavoidably contributes to errors for 0-0. Secondly, the reflected field includes third and fourth order terms which were zero for the backscatter case. These terms exhibit _ 3 singularities of the form (tt-0) which predominate and lead to unbounded errors for values of 0 - tt. By neglecting these higher order terms reasonable agreement is obtained for 20°< 0 < 110°-130° (the upper limit is a function of ka) for the E-plane or hard EM surface rays. The soft EM surface rays which contribute to 10.0 3 s 5.0 // ©yT '> / © / ® 0 me) 2 g. i i i1 / s TTQ V® \ ® / ^ <5 Qr gi o J © n <> o \ O |. 7“ ** —4H-. i 1.0- 5 ^ o o/ K \© s' 1r* \ 11 ® / / 0 20 40 60 ©0 IG►0 120 14O 1C>0 18 e - Figure 14. Normalized Bi-Static Cross Section for the Perfectly Conducting Sphere, ka=2.9 (----- Exact E-Plane, ••• GTD E-Plane, Exact H-Plane, ooo GTD H-Plane). 10.0 5.0 (7(0) TT0 J) —0 ^ 0 o 20 4 0 6 0 8 0 100 120 1 4 0 160 ISO © -Degrees Figure 15. Normalized Bi-Static Cross Section for the Perfectly Conducting Sphere, ka=8.9 (----- Exact E-Plane, ••• GTD E-Plane, -- Exact H-Plane, ooo GTD H-Plane). 105 the H-plane diffracted fields are nearly masked by the reflected field as in many of the backscatter results for soft surfaces. Nevertheless, the H-plane calculations are in reasonable agree ment over the same range of 0 as the E-plane data. If ka is increased, the agreement of all of the data is better and for a larger range of 0 as seen by comparing Figure 14 where ka = 2.9 and Figure 15 where ka = 8.9. However, the deficiencies of the reflected field again limit the accuracy of the GTD. An additional factor, variable longitudinal curvature, is included in the calculation of spheroid backscatter cross sec tions. The axial backscatter cross sections for acoustic hard and perfectly conducting prolate spheroids with an axial ratio of 2:1 are shown in Figures 16 and 17, respectively. The acoustic case, which has an interfocal distance of 2a, is com pared with exact calculations by Senior [33]. The perfectly conducting spheroid, which has a total length of 2a or an inter focal length of 1.732a, is compared with measured data and impulse theory data from Kennaugh and Moffat [34]. The GTD diffracted fields are calculated by substituting the results of Table 2 into Eq. (75) or (76) for a hard or a perfectly conducting surface, respectively. Only the first mode and first encirclement are significant and the caustic correction of Appendix E is required. The reflected field for each 106 spheroid is obtained differently; their calculation is included with each discussion of results. The first two terms of the hard spheroid reflected field can be found by the technique of Schensted [35] as (81) p r . f-- a— -]e-Jk(r-2”309a) (1 + 1 — + h <6.928r j u 3 ka However, the GTD backscatter results using Eq. (81) for the reflected field are in great disagreement with the exact data. Nevertheless, the peaks and troughs of the GTD response are aligned with their exact data counterparts indicating reasonable agreement of the diffracted fields. Hence, the reflected field is the likely source of error and in particular, the second term of Eq. (81) is suspect due to its large coefficient. An exami nation of the results of Senior [28] for spheres suggests that an emperically determined reflected field similar to the "actual optics" field for the sphere may provide more accurate results. The reflected field determined in this manner is given by (82) (1 + j (.6-.019ka)). e i u C f = ka cosh rj Figure 16. Normalized Axial Backscatter Cross Section for the Hard Acoustic 2:1 Prolate Spheroid (----- Exact, ••• GTD). 108 The good agreement of the results in Figure 16 illustrates the usefulness of this approximate reflected field. Again, the Luneberg-Kline series for the reflected field gives inaccurate results, thus limiting the utility of the GTD„ The perfectly conducting prolate spheroid reflected field is calculated by Schensted's method [35] to give (83) F - - X |jf) e-jk The results show remarkable agreement, especially for ka > 2 where the GTD data typically lies in the midst of the measured data. With such good agreement, it seems likely that at least the next higher order term, i.e. third order term, in the reflected field is zero as it is for the perfectly conducting sphere. Though Hong [10] imposes a formal restriction of pg/Ptn prolate spheroids violate this condition. Nevertheless, the GTD prediction of the oscillatory behavior of the backscatter responses is correct for axial ratios of 2:1 where Pg/Ptni. 4. Just as the formal requirements on longitudinal curvature are more restrictive than necessary in practice, so is this limitation on transverse curvature. The degree to which this limitation can be relaxed is unknown, however. ka Figure 17. Normalized Axial Backscatter Cross Section for the Perfectly 109 Conducting 2:1 Prolate Spheroid (----- Impulse Theory, ••• GTD, xx and oo Measured Data). 110 G. Exponential Factors The detailed behavior of the generalized attenuation constants is not readily apparent from the functional forms of Table 2. How ever, when they are applied to selected surfaces the various effects are more easily seen through numerical examples. In particular, the exponential factor, jk + a, is evaluated for ray trajectories on elliptic cylinders and prolate spheroids which have a 2:1 axial ratio and an interfocal distance of 2a. (k is the usual free space propagation constant while a is the attenuation constant calculated from Table 2.) For a value of ka=10, these surfaces have a minimum kp of 2,88 at the specular point, i.e., the tip, and a maximum of O 23.04 at the shadow boundary. The results for the elliptic cylinder are obtained as a special case of the prolate spheroid when the transverse radius of curvature, p , is infinite. Calculations ’ 'tn’ are made for soft, hard singly-curved, hard acoustic doubly-curved, and hard EM surface rays. Both the real part (attenuation) and the imaginary part (phase shift) of the exponential factor are calculated. More useful forms of these terms are Re (aA) , the attenuation in nepers which occurs within one wavelength, and v/c, the ratio of the surface ray velocity to free space velocity. A starting point for studying these effects is shown in Tables 3, 4, 5, and 6 where Re(otX) and v/c are presented for several surfaces which have the same longitudinal radius of Ill curvature at the point under consideration. A circular cylinder with ka = 2.88, a sphere with ka = 2.88, an oblate spheroid with kp =2.88 and an elliptic cylinder obtained from the oblate § spheroid results for pfc = 00 are all examined along with the prolate spheroid and its accompanying elliptic cylinder. The point of consideration for the spheroids is on the axis of rotation while the corresponding point on the associated elliptic cylinders is used. Within the tables, the simplest surface, a circular cylinder, is in the upper left position; the complexity of the surface increases toward the right with transverse curvature and downward with variable curvature. The following abbreviations are used to simplify the tables: SCC - Soft Circular Cylinder, SEC - Soft Elliptic Cylinder, SS - Soft Sphere, SPS - Soft Prolate Spheroid, SOS - Soft Oblate Spheroid, HCC - Hard Circular Cylinder, HEC - Hard Elliptic Cylinder, HAS - Hard Acoustic Sphere, HAPS - Hard Acoustic Prolate Spheroid, HAOS - Hard Acoustic Oblate Spheroid, HEMS - Hard Electromagnetic Sphere, HEMPS - Hard Electromagnetic Prolate Spheroid, and HEMOS - Hard Electromagnetic Oblate Spheroid. 112 Several interesting features of the ray behavior are observed. First consider Tables 3 and 4. All soft surfaces are lossier than their hard counterparts because qm is larger than q^. The surface ray field of the soft surface is less tightly bound to the surface than that of the hard surface. In addition, their attenuation constants exhibit no transverse curvature dependence; all soft surfaces impose the same boundary conditions on the field whether there is transverse curvature or not. However, soft attenuation constants do show a dependence upon variable longitudinal curvature which tends to increase the rate of attenuation whenever the surface curvature decreases (p p < 0) and conversely, as illustrated by the prolate 8 8 and oblate spheroids and their elliptic cylinder counterparts. The increased complexity in the case of the hard surface is due to the difference in the boundary conditions for different hard surfaces and fields. Most notable is the effect of finite values of the transverse radius of curvature which requires two distinct descriptions, scaler and vector. A hard acoustic (scaler) surface ray experiences greater attenuation on a doubly- curved surface, i.e., p ^ 00, than on a cylinder, whereas a hard EM (vector) surface ray is less attenuated on a doubly-curved surface. Each of the three hard attenuation constants, i.e., the attenuation constants for the hard singly-curved surface, 113 TABLE 3 Re(aA) FOR SOFT SURFACES kp =2.88 Absence of Transverse g Transverse Curvature Curvature P..tn =P D =°° tn Absence of Variable see SS Curvature 5.14 5.14 p =0, p =0 g g Variable SEC SPS p p =9/4 Curvature g g 4.22 4.22 p =0 g P P =-9 SEC SOS g g 8.79 8.79 TABLE 4 Re(aA) FOR HARD SURFACES kp =2.88 Absence of Transverse g Transverse Curvature Curvature "D =P rt 3 g Htn Scaler Vector Absence of Variable HCC HAS HEMS Curvature 2.36 2.77 1.95 p =0, p =0 g g Variable HEC HAPS HEMPS p p =9/4 Curvature g g 2.13 2.54 1.72 p =0 g P„P HEC HAOS HEMOS g g 3.30 3.71 2.89 114 the hard acoustic doubly-curved surface, and the hard EM surface, exhibit a dependence upon variable longitudinal curvature which is similar to that of the soft rays. The v/c data presented in Tables 5 and 6 exhibit features similar to Tables 3 and 4. The ray velocities in the case of soft surfaces show no transverse curvature dependence, but are affected by variable longitudinal curvature. In the case of hard surfaces, ray velocities depend upon transverse curvature in addition to variable longitudinal curvature. The comparison of surface ray behavior for similar surfaces at one point is only one method of viewing the phenomenon. Alter natively, the variation of Ee(aA) and v/c along a geodesic of a surface can illustrate the interplay of the various parameters. Figures 18 through 22 show the effects of boundary conditions, transverse curvature, variable longitudinal curvature, body size and the type of incident field. As before, the surfaces are elliptic cylinders and prolate spheroids with a 2:1 axial ratio and an interfocal distance of 2a so that for ka = 10 the minimum of kp is 2.88 while the maximum is 23.04. g Shown in Figure 18 is the real part of the exponential factor, Re(aA), for the four types of surface rays propagating along a prolate spheroid, each including the effects of variable curvature. The attenuation for all values of 0 is similar to the 115 TABLE 5 v/c FOR SOFT SURFACES kp =2.88 Absence of Transverse 8 Transverse Curvature Curvature ptn=pg p =°° tn Absence of Variable see ss Curvature .692 .692 p =0, p =0 g g Variable SEC SPS p p =9/4 Curvatureg g o 655 .655 V ° P c A =“9 SEC SOS g g .902 .902 TABLE 6 v/c FOR HARD SURFACES kp =2.88 Absence of Transverse g Transverse Curvature Curvature Ptn=pg o =°° tn Scaler Vector Absence of ' Variable HCC HAS HEMS Curvature .846 .875 .820 p =0 p =0 g g Variable HEC HAPS HEMPS p p =9/4 Curvature g g .831 .857 .806 p =0 g P Pa=—9 HEC HAOS HEMOS g g .941 .943 .884 i j 1- 20 30 4 0 50 @ 0 TO 8 0 90 0 - Degrees 116 Figure 18. Surface Ray Attenuation (1 - Soft, 2 - Hard Acoustic Doubly-Curved, 3 - Hard Singly-Curved, 4 - Hard EM). 117 behavior in Tables 3 and 4; the order from most lossy to least lossy is soft, hard acoustic doubly-curved, hard singly-curved, and hard EM. Moreover, the values of Re(aA) seem to have the same relative magnitudes for all values of 0. An exception is noted for 0 - 80°, i.e., near the region of the tip, where the soft surface value exhibits a maximum. This maximum results when the p and p' Kg terms combine so as to suppress the effect of the decreasing p § term. This also occurs for the hard surfaces, but an additional transverse curvature dependence prevents a maximum from occurring. However, such maximuma do occur for hard surfaces at smaller values of ka. Transverse curvature effects are also apparent in Figure 18. In the case of soft surfaces Re(aA) exhibits no transverse curva ture dependence. With the hard cylinder, Ptn= 00, taken as a reference for the hard surface rays, the finite values of trans verse curvature are seen to increase the attenuation for the hard acoustic doubly-curved surface while it decreases for the hard EM surface. This holds for all positions on a prolate spheroid with the effect accentuated at smaller values of Ptn, i.e., 0 - 90°, as is expected from the form of iPg/Ptn Table 2. The results of Figure 19 show that v/c for all four surfaces varies in a similar manner. As with Figure 18, the results agree at 0 = 0 with the tabular data, i.e. with Tables 5 and 6, and the 20 3 0 4 0 5 0 6 0 7 0 @ 0 © - Degrees Figure 19. Surface Ray Velocity (1 - Soft, 2 - Hard Acoustic Doubly-Curved, 3 - Hard Singly-Curved, 4 - Hard EM). 119 same relative behavior holds for all other values of 0, It is interesting to note the very small change of v/c for small values of 0 (probably due to the preponderance of the p term) followed by a rather rapid decrease and then another region of small change • (| around 0 - 90° (probably due to the dominance of p and p terms). O O The variations due to variable longitudinal curvature are viewed in Figure 20 for the soft surface ray. Its behavior with variable curvature is typical of all surfaces, but it also exhibits no transverse curvature effects which might be confused with variable curvature. Without variable curvature effects, the attenuation increases monotonically as 0 increases, i.e., as p g decreases. Heretofore, the greatest attenuation was predicted to be at the smallest value of p , i.e., 0 = 90°. However, when g variable curvature effects are included they alter the behavior and bring about a maximum in the attenuation at 0 - 80°. The primary effect of variable curvature on v/c is to increase its variation on a prolate spheroid. A decrease in ka scales down the body size while main taining the same ratio of all radii of curvature. The results of such a scale change are shown in Figure 21 for a hard EM surface ray. The same general behavior is observed for all positions on the surface, but the effects are accentuated near the tip where 0 - 90°. As expected, with a decrease in all radii of curvature, the attenuation increases and the phase velocity decreases. 8.0 4.0 0- 20 30 40 30 60 70 80 90 ©-Degrees Figure 20. Variable Curvature Effects on the Soft Surface Ray (1 - Re(aA) with Variable Curvature, 2 - Re(aX) without Variable Curvature, 3 - v/c with Variable Curvature, 4 - v/c without Variable Curva ture) . 3.0 2.0 Re —r 0 20 30 4 0 50 60 70 80 90 ©-Degrees Figure 21. Hard EM Surface Rays with Different ka (1 - Re(aX) with ka = 5, 121 2 - Re(aX) with ka = 10, 3 - v/c with ka = i, 4 - v/c with ka = 10). 122 The previous data gives only the local surface behavior; the integrated behavior of the attenuation constant is also of interest. This is descriptive of the total attenuation of the surface ray. Figure 22 illustrates the total attenuation of a surface ray along a geodesic of a prolate spheroid. For reference, the results for a sphere with the same ka are also shown. At small values of 0 the attenuation is less for a prolate spheroid due to the much larger p , but it increases with increasing 0, i.e., as p decreases. The effects of variable curvature also cause a g wider spread in values among the hard surface rays on a spheroid than on a sphere or cylinder. Many more variations of surface conditions can be examined, but the salient features of the general behavior have been illus trated here. When a specific surface is of interest, it can be studied in detail in the manner just described. 6 .0 t 5.0 - 4.0 Re locdt o 3.0 2.0 20 30 40 5060 70 80 90 0 - Degrees ,0 Figure 22. Total Attenuation, Re fDadt (1 - Soft Sphere, 2 - Soft Spheroid, 3 - Hard Acoustic Doubly-Curved Spheroid, 4 - Hard Acoustic Sphere, 5 - Hard Circular Cylinder, 6 - Hard Elliptic Cylinder, 7 - Hard EM Sphere, 8 - Hard EM Spheroid). CHAPTER V CONCLUSIONS When high-frequency diffraction from smooth curved surfaces is described in terms of the GTD, two parameters of importance are the diffraction coefficients and attenuation constants of the surface ray modes, Keller has given first-order expressions for these diffraction coefficients and attenuation constants. In this study, the second-order terms have been derived for the soft and hard surfaces of acoustics and the perfectly-conducting surface of electromagnetics; the importance of these terms is illustrated in a number of examples. Higher-order terms are retained in the asymptotic solutions of the diffraction of plane waves by cylinders and spheres to obtain the diffraction coefficients and attenuation constants for surfaces of constant curvature, see Table 1, Then the plane wave diffraction by a more general convex surface is considered, and the work of Hong is used and extended to obtain the diffraction coefficients and attenuation constants for soft, hard, and perfectly- conducting surfaces, see Table 2. These are shown to reduce to the previously derived results in Table 1 for the cylindrical and spherical surfaces, which provides a check on the various asymptotic forms o 124 125 In terms of the surface geometry, the first-order expressions depend only upon the local radius of curvature in the plane of the ray, p , whereas the second-order terms also depend upon the S derivatives of p with respect to distance along the surface ray, S • »• p and p , and for some surfaces, the component of the radius of surface curvature in the direction of the binormal, p. . In tn the case of the soft EM boundary, i.e., the incident electric field is parallel to the surface, and the soft acoustic boundary, all of the attenuation constants are seen to have the same functional dependence on p , p , and p , and furthermore to second-order, they S S 8 are independent of Ptn* ln the case of the hard EM boundary, where the incident electric field is perpendicular to the surface, and the hard acoustic boundary, all of the attenuation constants also have the same dependence upon p , p , and p . In addition, a 8 8 8 dependence on curvature transverse to the ray path is present in one second order term in each expression, which is +Pg/^Ptn for the acoustic boundary and -Pg/^Ptn f°r the EM boundary. These results are not unexpected since the boundary conditions are the same for the soft acoustic and electromagnetic problems and different for the hard acoustic and electromagnetic problems. The variable curvature described by p and p usually, but 8 8 not always, increases the attenuation of the surface ray modes. On the other hand, the curvature transverse to the ray path increases 126 the attenuation of the surface ray inodes in the case of the hard acoustic boundary and decreases the attenuation in the case of the hard EM boundary. The diffraction coefficients depend upon p^, p^, and Ptn> • o they do not exhibit a dependence upon p because p is assumed to S S be zero at the point of diffraction, as explained in Chapter III. As with the attenuation constants, the diffraction coefficients are all the same for the soft acoustic and EM boundaries, while they differ for the hard acoustic and EM boundaries. The diffraction by a cylinder-tipped half-plane shows that diffraction at a surface, like edge diffraction, is a highly local phenomenon. In this example, it is found that when second-order terms are included in the diffraction coefficients and attenuation constants, the GTD is accurate when p is as small as 0.1X. In the g other examples, the GTD contains a reflected component, and even though additional terms are retained in the expression for the reflected field, it is reasonable to conclude that the low frequency limit is determined by the reflected component. Nevertheless, the second-order terms in the diffraction coefficients and attenuation constants were found to contribute significantly to the increased accuracy of the calculated fields in all of the examples treated here; this is particularly true for the hard acoustic scatterers and the perfectly-conducting scatterers, where the diffracted component is relatively large. APPENDIX A COMPLEX ZEROS OF THE HANKEL AND RELATED FUNCTIONS An evaluation of the integral resulting from the Watson transformation requires the calculation of the roots of the equation (84) (QHv (2)(X)) =0 X= k a where Soft acoustic sylinder, soft acoustic sphere, and soft EM sphere _3 Hard acoustic cylinder Q= 3X tT J_ _ Hard acoustic sphere 2X 3x 2X nrx J. + _I Hard EM sphere. 3X 2x A closed-form evaluation of these zeros is not possible. However, an asymptotic representation of the roots may be obtained for large values of ka. The procedure, first outlined by Franz and Galle [24], is quite straightforward, but rather tedious. 127 128 (2) The cylindrical Hankel function of the second kind, (ka), (9) ' and its first derivative, (ka), are expanded in terms of an infinite set of Airy functions of the first kind and their first derivatives„ Schobe [36] gives _(2u+l) 00 u I ka\ 3 (85) Jv (ka) ~ 2 (-D N [P (5)Ai(?) + Q (C)Ai'(€)], li=0 _ ( 2H+1 00 /kaV 3 (8 6 ) -Y (ka) - 2 (-l)y — ] [P (?)Bi(5) + Q (5)Bi’(5)] y=0 p M _|2]i±2) °° u + 1 1 ka v 3 — ______(87) J '(ka) ~ 2 (.-if [ - j ] [P (OAi(C) + Q(5)Ai'(5)] y=0 M ^ and _[2]j+2] 00 u+1 fkaV 3 - (8 8 ) -Y ’(ka) ~ 2 (-1)U [P (?)Bitf) + Qu (C)Bi'(?)] ]i=0 P where /ka\- ^ ^ 3 5 = (v-ka) ( — ) P0(O = 1 Q 0 ( O - 0 Px (?) = 5/15 Qj^a) = 52/60 P2 (?) = ?5/7200 + 1352/1260 Q2 (?) = 53/420 + 1/140 129 and P0 ( O - 0 Q0 (C) = 1 = S3/60 - 1/10 <^(5) = -5/15 P2 (s) = C4/3360 - 5/60 Q2 (0 = 55/7200 - 195" / 2520 a combination of these results with (1 ) H '2') (ka) = J (ka) ± jY (ka) and Ai(ze J2tt/3^ _ 2/2e [Ai (z) + jBi(z)j (2 ) • (2 )* gives useful representations for H ' (ka) and (ka) as (89) H ^ ( k a ) ~ 2e+j7^ 3 v ka -1 gAi(ge J2^ ) + e~j2^ 352 Ai(Ce-j2^/3) 2 15 60 -5/3 ka i 7200 1260 2,1/3 (tfe + ife) Ai'Ke-^3, + 130 and (90) Hv ^ ' ( k a ) ... -2e+37r/3 ^ \ 2'3 e‘J2,r/3 Ai'(4e-J2,,/3) -n\in (is!- i) -■3 321 2 T / 3 I f A i - « e ~ J2,,/3) -2 Al(Se"J2ir/3) + ft 13360I'"— ~ 60 -j2TT/3 |_ rL _ l i d I AJ . ( 0-j 2tt/3 172 00 2 5 2 0 1 A1 { “ ' I A first approximation to the roots of each equation is obtained by setting the coefficient of the highest power of ka equal to zero* This results in either Ai(SQe J27!/3) = o or Ai'; (£Qe'j 271/3) = 0 th for the soft or hard boundaries, respectively„ VJith the m root of the Airy function and its derivative denoted as q^ and q^, respectively, a first approximation of 1/3 i . -jir/3/kal vm0 “ ^ + qm e (— I 131 and Vn,0 ' ka + Sr3'n l^f) ±S ,“ade- The exact roots are quite close to the approximate solutions above. Consequently5 the Airy function and its derivative are expanded in a Taylor series about the approximate roots. Moreover, a solution for the exact root is assumed to be of the form - -+ < . *-3’/3 n i 1/3 -'W>+el B ) ’1/3 +C2 IH)’1* - • The resulting series for the Hankel function and its derivative equal zero by virtue of the boundary conditions and, hence, they yeild a solution if the coefficients of each of the powers of ka is equal to zero. A series of equations is obtained and succes sively solved for the unknown e^'s. Since each boundary condition produces a different result, from this point on they are treated individually. To illustrate the procedure, the details of the soft boundary solution are shown, but only the results for the hard boundaries are included. Soft Acoustic Cylinder, Soft Acoustic Sphere„ and Soft EM SphereQ=1 The asymptotic representation for the Hankel fauction is expanded term-by-term in a Taylor series about the approximate root which is determined by A i ^ ,r^) = 0. In this expansion CnAi(^e is expanded as Al(ee'J2!7/3)) 5nA i a e-32’,/3) . L d‘((y]2'°/3 J6”0 d(?e-32’/3) -jrj (S--n) e 11 where ka | - 2 / 3 + M - 4 / 3 . and 2,1/3 0 &1 21 + 2 It ) + • —j 2 m+ 1 1 This leads to a series in |—|j ’ 3 I given as 133 For the equation to hold for more than a finite set of values of ka, each coefficient of (kaa/ 2 )"(2H /3 must equal zero. This results in 2 60 and = _ x “2 140 10 Similar successive solutions lead to the higher order coefficients. Hard Acoustic Cylinder; Q - 8/9x Similar calculations involving the derivative of the Hankel function leads to 1 + 60 and - 1 (qm 14 7 e2 140 10 5 3 \ 10q m Hard Acoustic Sphere: Q = V'ir/2x{3/9y ~ l/2x) The coefficients for this case are nearly the same as for the previous case; they include an additional factor to account for -H ^(x)/2x* The results are \ 2e^ /3 21 1 + 60 and 1 Sn , 63 . 343 140 l 0“ + 10 + T 7 r 3 40V Hard EM Sphere: Q = /ttx/2 {3/9x + l/2x) This case differs from the preceeding only in the sign of the additional term with the results 135 The results, summarized below in Table 7, are in complete agree ment with the earlier work of Franz and Galle [24]. Several printing errors in their work have been corrected in Table 7. TABLE 7 HANKEL FUNCTION ZEROS, (QH (2 ^(x)) , =0 s v X = k a Surface Zeros-v i , -jn/3lka 1/3 Sn ju/3|ka| 1/3 1 I’m llkal 1 Soft v ka + v It " 60 e I 2l + U 0 I 10 ' 1 (1 2 l + Hard — . ^ — -iir/3 (kal3 ^3 Sa Cylinder V = ka + q e J -n m m 1 2 / 60 \ 1 ' Sn Hard - 3 1/3 qmm 2 l.i,ka ,"1/3 1 i qm , 63 . 343 1 lka\ Acoustic V = ika + , —q e —3J tt/3 —ka] 1 1 21 Sphere m m 1 2 60 1 + - 3 1 21 140 10 10 , n— 3/1 2 1 qm qm ' Hard - 3 EM t , — -jtt/3 ka 1/3 qm2 _ 7 63 1/kal £ v = ka + q e m 1 9 M'U! 3 1 m m 1 2 60 21 140 10 10 ' 3 I 2 + '•* Sphere - 3 \ qm qm APPENDIX B (1 ) Qh / i ; (X> ASYMPTOTIC EVALUATION OF i - ^ Q H v (2 )(X)/X=ta v=v m The asymptotic expansion of the diffracted field as obtained via the Watson transformation includes the factor QHV (1)(X) 1 As in the case of the zeros, V (see 3 (2 )/ n , ‘ — "5v ^ v (x)/x=ka v=v m Appendix A), an asymptotic form can be obtained which depends on ka and the boundary conditions in a relatively simple way. Since the procedure to obtain this form employs some of the techniques used in calculating the zeros, the method is correspondingly tedious. The asymptotic expansions of the Hankel functions and their derivatives are formulated with the aid of Schobe's [36] corres ponding expansions for the Bessel and Neumann functions as given by Eqs. (85) and (8 6 ) . Asymptotic expansions for both (QHv^ (X))x=ka and (QHV^ (X) )^=ka are readily obtained. Since X = ka is a constant with regard to variations in V, then -!/3 3 ka - 2 at £ £ JL = 9£ , and an expansion for (-^ QHV ^ 0() ] x=ka 9v - 9v 9? 2 137 138 also can be found. The numerator and denominator are separately expanded in a Taylor series about the approximate solution 5„e _ __ in a manner similar to that used in Appendix A, (J m and the results are simplified by use of the Wronskian. Both the numerator and denominator are properly behaved so that the limit of the quotient is equal to the quotient of the limits as V -»■ v . The resulting expansion asymptotically expresses the I QH (1)(X) behavior of — 5 txt for large values of ka. \ ^ q v M x : vka m To illustrate this procedure, the soft boundary case will be treated. The corresponding development for the hard boundary cases will not be included, only the results will be given. Soft Acoustic Cylinder, Soft Acoustic Sphere, and Soft EM Sphere; Q=1 The asymptotic expansion of the numerator, (QI^ ^ is given as « 139 -1/3 -jir/3 ka (91) 2e Ai(?ej27T/3) X=ka J2TT/3r2 ,-1 J27T/3 j27T/3> k a Ai (£e ) + Ai'(£e 2 15 60 ,-5/3 + 13i i | A i < ^ 2,r/3) 7200 1260 I U ' j 2tt / 3 +e A i ’ (?ej27T/3) 420 140 and the denominator, as 3^ X=ka M ' 2 eJ2ir/3|_3U4 _7£. M ( re-J2ir/3. 2 110080 2521 A1^ e J 141 1/3 where £ = (V-ka)(ka/2) - For the soft boundary cases the value of the root v may be determined approximately by Ai(£Le m U Since the exact value of the root is very close to this, a Taylor series of each coefficient may be taken about this approximate value. It is useful to simplify this expansion with AiC-q^e _ e j5tt/6/2A±t(—q ) which is found from the Wronskian. With the m collection of corresponding powers of ka Eqs. (91) and (92) may be rewritten as -1/3 e-j 71T/6 [^| (93) iH (1 ) (ka) v=v TTA i ’^ ) m ^1 - — -jlT/3 |ka 2/3 29 2 j-rr/3 /ka\ 4/3 15 - 3 l 5 0 qm 6 IT) + and /8 Hv (2 ) (ka)' -2/3 -jir/3 I ka (94) 2e 9v A i 1(-qm ) fv-v m q . . -2/3 37q -4/3 i _ -JF/3 /kai _ m j ir/3 I ka 10 6 12/ 2520 2'! 142 The leading terms of the resulting quotient are given by (1 ) ka\1/3 QH, (X) \ e-j5TT/6 (— [ ( 9 5 ) fiH (2)^ / 3 V ^ V <'X)/V=:X=ka 2TT[Ai’(-qm ) r v=v m -2/3 _ 0 . .. , -4/3 ! + _E\ e- ^ /3(ka + ?■ q 2 e — + 3 0 1 4 0 0 m ' 2' Hard Acoustic Cylinder: Q = 3X This case is quite similar to the previous one except that the derivatives of the Hankel functions are employed and that the asymptotic series for V involves different coefficients. The result is 143 1/3 QHV (1) (X) (96) l-i. qh (2) (X) I 27Tq [Ai(-q )]‘ '9v v X=^a m m v=v m -2/3 1 0 q. m 3q 2 -4/3 / m 3 jlt/3 |ka 11400 200 4 ?m TT | 3 Hard Acoustic Sphere: Q |3)( ~ 2x This case involves both the Hankel functions and their derivatives, and as before, the coefficients of the series for v. The result is I QH a ) (X) 0-j5*/6(^f)1/3 (97) (2) Q V (X)/v.voX=ka 2 Tlqm[Ai(-qmm )] v=v m -2/3 m 7 ' — jtt/3 /ka (1 + 30 ‘ 20? 2 1 6 ’ 2 m vT 2 -4/3 [ m 147 \ 3tt/3 [ka ) • ll400 ' 800? 4 ' " 1 2 m 144 Hard EM Sphere; Q ~ \ J ^ 2 3x + 2x This case also involves both the Hankel functions and their derivatives and the coefficients of the series for v. The result xs 1 >1 /3 ' QHv (1)(x) (98) (2) . 2TTq [Ai(-q ) ] 2 MSv QHV (x) /v=X=ka m v m J /\ V=V m -2/3 m -jTr/3jkaj (1 + + 30 2 0 q 2 nm 2 -4/3 m 27 jrr/3 I ka 1400 ) . 800q 4 m A summary of the four cases is presented below in Table 8 and as before, they are in agreement with Franz and Galle [24] and Senior and Goodrich [17]. Additional terms may be obtained by extending the method. TABLE 8 QHv(1)(x> ' HANKEL FUNCTION QUOTIENT, X=ka Surface Q-Operator (QH ^1 \ x ) / ^ Q H v ^ v A x=ka (kal1/3 -JSir/6Ff , 2 ,2/3 q , ,, Soft Surfaces 1 * ' ,2 <1+ ta 30 e J + ••• > 27T(Ai' (-qm )) Hard Acoustic Sn 1 1 a e-J,r/3+ ... > Cylinder • - f « • m ” 30 - 2 10 3X a^/AK-^)) lka| , ,1/3 1 a i\ e-35^ 6 !^] n , , 2]2'3I%> 17 I e-jTT/3+ _ } Hard Acoustic Sphere nr v 2x [ax " 2XJ ,-,,.,-,,2 1 ikaj 30 - 2,„ 1 \ 10’ .kal1/3 J m IJ. + -k a - ^ b l (1 + | 2 ^ 3 . 3 Hard EM Sphere V 2 2X| 145 30 ?m 22 0 , APPENDIX G AIRY FUNCTIONS, DERIVATIVES AND ZEROS The. Airy functions are defined in many different ways, and unfortunately, no standard definition and notation have been adopted. In this appendix, the form used throughout this study is defined and related to several other definitions. Also included are tables of useful numerical values of the functions, their derivatives, and their zeros, The Airy differential equation, d^w(z) (99) --- — - zw(z) = 0, dz yields two independent solutions which assume a variety of forms. The form chosen for use in this work is defined by Miller [37] and Jeffreys and Jeffreys [38] as 3 (100) Ai(z) / eZt_t /3 dt 27Tj p where z may be complex and F^ is shown in Figure 23. A second independent solution is giv i.< as 146 147 l m f Re t Figure 23. Airy Function Contours of Integration. 148 ^ ze—J2tT/31 _ t^/3 (101) Ai(ze ^ = "2^ J / e dt ri °i2*n 3 „ ------/ e**- ' /3 dt 21TJ r 2 while the Wronskian relation satisfied by these solutions is (102 ) ------Ai(z)AL ’ (ze“j 2Tr/3 )e“:i27T/3-Ai(ze":)27T/3)Ai'(z). 2tt The derivatives can be obtained in a straight-forward manner by differentiation under the integral of Eqs. (100) and (101). The roots of the Airy function and its derivative which satisfy Ai(z) = 0 and Ai'(z) = 0 are real and negative and are denoted as -q and ~q^» respec tively, where m denotes the number of the root. Thefirst two roots, sufficient for most calculations, are presentedbelow in Table 9 along with values of the corresponding Ai(-qm) and Ai^-q^). 149 TABLE 9 AIRY FUNCTIONS, DERIVATIVES, AND ZEROS m qm qm A i ’(-q ) Ai<-qm) 1 +2.33811 +1.01879 +0.70121 +0.53566 2 +4.08795 +3.24820 -0.80311 -0.41902 In order to relate these functions to other common forms, Table 10 includes the definition and origin of several common forms. TABLE 10 AIRY FUNCTIONS Function Definition Conto ur Reference Miller [37] .., . 1 , zt-t^/3 , Ai(z)=— : / e dt N Jeffreys & Z F J r Jeffreys [38] L1 00 Levy & A(z) = / cos (zt-t^) dt -e> o Keller [12] 0 W(z)= — / ezt C ^ dt Fock [7] ^ r3 APPENDIX D 3P y (-cos 9) ASYMPTOTIC EVALUATION OF P (-cos 6 ) and ■V v ' 39 The Watson transform for spherical scatterers yields representations for the acoustic field which contain Pv (“Oos 0) and for the electromagnetic field which contain m Pv _j_/2 1 (-cos 0) and 3P -\j2 ^(-cos 6 )/99* The general m m representation of P^y (-cos 0) is determined for large values of v in this appendix and then y and v are appropriately evaluated for the field in question. Similarly, 3P^y (-cos 0)/30 is found in general and then specifically evaluated as needed. The Legendre polynomials used by Stratton [25] are defined as ( - vm . m /2 ,_ (103) Pnm (X) - (yz*) F(-n, n+1; 1-m; ) . The usual definition, however, is given as (104) Pnm (X) - (£*) F(-n, n+1; 1-m; ^ ) 150 151 since it is better behaved when generalized to non-integer values of n and m, i.e., Legendre functions. As noted by Goodrich et a l . [3 9 ], this difference of definition involves a factor of (-l)m . In this appendix the results are obtained from the standard definition and then corrected. Note that only the electromagnetic case is affected since m = 0 for the acoustic case. A useful relationship from Erdelyi[40] (105) Pvy (~X) = Pvy where 0 < x < 1 > is combined with well-known asymptotic expansions, 1/2 /in£\ n U/ n\ F(V+y+l) I 2 (106) pv (cos e) - r(V+3/2) ; „ 3ln e «. (-1 ) (l/2+U)o(l/2-y)p • 2 — sin [ (v+£+l/2)0 + # + +IT ] £=0 £!(2 sin 0) (v+3/2)£ z 4 1 and TT (107) Q^Ccos 6 ) ~ 1 ' ~ r(v+3/2) \2 sin 6 “ (-1 ) (1 /2+p)^(1 /2-y) ^ /y II tt£ , ■ 2 p cos [ (V+£+l/2)0 + — + 7-ht+ — t? ], £=0 £!(2 sin0) (v+3/2)£ q £ 152 where O 11/2 (108) PvV c o s 0) . (iff-sis-ff) » (-l)\l/ 2+y) (1 /2-y) 2 ------*:------* sin [ (v+£+1/2)0 + [t - j - v h r + - ^ ]. Z=0 11(2 sin 0) (V+ 3/2)A A simplification is possible since r(v+a) a-b r (a-b)(a+b-1 ) , TTv+bT ~ [1 + 2v +-..J provided v ^ -a, -a -1, ..., -b, -b -1, ... as V ->■ °°. Evaluated for the immediate problem at hand this yields nocn r(v+u+D u-1/2 . (u -i/2 )(u + 3 /2 ) , (109) r(v+3/2) ~ v [1 + 2v +...]. The first terms of Eq. (108) are now expressed as 153 2___ \1/2 u-1/2 (y-1/2)(y+3/2) . (110) PvV c o 3 6) tt sin-I„ Cl0 1 V L ~ 2vo,, T . . . J (sin [(v+l/2)0 + - £ -v)tt] r(3/2+y)r(3/2-y) ______. A-fl H . u W l a- \ 2 v sin 0 r(l/2+y)r(l/2-y) [ (v4 3/2) 0 +(4 - 2 - v ) t t J + ...). With V = [V ]—1/2 and y = 0 the form of Eq. (110) applicable m to the acoustic problems is obtained as \l/2 (111) ? [V ]-l/2<-COS e> ~ TT sin 0 m ([V ]-l/2)1/2 U ' 8([V - 1/2> m sin[([v 1-1)0 -[v ]7r+57T/4] + ... ] (sin[ [vm ] (0—tt)+ -j ] 8 ([V 1-1/2) sin 0 + • • • ) m and with V = [Vm l~l/2 and y= -1 the electromagnetic form is found to be 1/2 -1 (112) P [V ]-l/2<-C0S 6> - TT sin 0 3/2[1 “ 8([V l-l/2)] m ([V ]-l/2) m m 3 sin [([v ]-l)0 -[v ]tt+7tt/41 sin[ [ v j (0-tt)+5tt/4] + 8([v ]-1/2) sin 0 + * * * m 154 which is corrected to agree with Eq. (103) as 1/2 ,-1 (113) P“^ /2(-cos 6) TT sin 0 3/2 [ 1- 8([v 1-1/2) m ([V 1-1/2) m m sin[[v ] (0-tt)+5tt/4] m sin[ ([v 1-1)0 -[V ]tt + 7tt/4] ______m______in______8 ([V ]-1/2) sin 0 m As shown in Appendix A s the first two terms in the expansion for [v 1 are of the same form for all cases and are given as m 1/3 [q ] e“j7T/3 + ... j y e-dTT/3(ic||-2/3 + _ _ = ka [1 + til ([q^] signifies the m root of the Airy function or its derivative whichever is appropriate to satisfy the boundary condition.) With this substitution the acoustic cases take the form (114) ? [V ]-l/2(-C0B e> ' \/i'tt ka sin 0 m" • a - i i l e - ^ /3 | i | f 2/3 + • cos [ [v ] (0-tt) + tt/4] in and the electromagnetic case becomes <115> Pii ]-l/2<-C0S e) ~ -\/W a sin e • ( 1 - ^ 1 e-i>/3 (k|]'2 / 3 + .. •sin [ [v ] (0-tt) + tt/4] m and in the corrected form is (U6) P^v }_x/2 (-cos 0) ~ \J^ka gin 0 (ka) m v •2/3 3[qiJ —j tt/3 |ka (1 - — f - e — |^f| + 156 These results are sufficient in evaluating the diffracted field to the accuracy desired. The electromagnetic field also contains a term of the general form 9P^(-cos 6)/90 . The expression for this term is determined first, followed by its asymptotic evaluation. In general, a straightforward differentiation of the asymptotic expansion for cos 0) does not yield the correct result; another method must be used. When the recurrence relation (117) (v+u) P|J_1 (X) = (v-y)x P/(X) " (l-X2)1/2^ +1(X) is substituted into the recurrence relation 2 dPv P the result is given by dpvy(X> -y y pv+l(x) (H9> -Ldx------, T- 2 p v,W(X) - -- ,, ---- 2,1/2 i-x (i-x ) A variable change of X = -cos 9 and the chain rule of differentia tion lead to the result 157 3P (-cos 0) ,, <120> 30------i i S r Pv (”cos 6) - pv (-cos e) which is expanded asymptotically by use of Eq. (108) to give 9P^(-cos 6) 2 l1/2vy+l/2 (121) 90 tt sin 0 • sin [(v+ 1/2)0 - (j + ^ + v)tt] . In the electromagnetic problem where v = [v^-1/2] and y = -1, the results are given by 8Pii )-i/2(-co3 e> 1/2 m 2 [, ‘V (122) 90 irka sin 0 1 4 I t ! ! • C O S [ [v ] (0-TT) + tt/4] m The corrected form is given by -1 9P [V ]-l/2(-COS 0) -2/3 m 1!2 \ — J tt/3 [ka (123) 1 - 90 TTka sin 0 4 6 ' 2 • cos [ [v ] (0-tt) + tt/4] m 158 The higher order terms in all of the above expansions do not take on such useful forms; they are not readily interpreted in terms of GTD. Therefore, it is doubtful that higher order expan sions will yield additional GTD information. APPENDIX E HUYGENS-KIRCHHOFF INTEGRAL FORMULATION OF THE DIFFRACTED FIELD AT BACKSCATTER The failure of the GTD to predict the backscattered field from surfaces of revolution results from the convergence of the diffracted rays to form an axial caustic; it is well known that ray optical solutions are not valid at caustics. This limitation is overcome by the use of Huygens sources, which are calculated by the GTD, in conjunction with the Kirchhoff integral; this integral is evaluated by the method of steepest descents. The resulting expression for the diffracted backscattered field remains finite and it checks exactly with results obtained by Senior [28] and Levy and Keller [12] using different methods. This technique is an extension of earlier work by Kouyoumjian [29]. Green's Theorem expresses the field at a point P in terms of the field and its normal derivative at Q on the closed surface S as S 159 160 "j kR where Gq is the free-space Green's function, e J /4ttR, R is the distance from Q to P, n is the outward normal to S, and u is the ' * .. .. _ field or a component of the vector fieiaVv*Tha g.eometry of the problem under consideration is shown in Figure 24. The diffracted field on the Huygens surface as expressed in the form of the GTD is 2 _ -ikr c “j V * ±0+a/ O 00 (D ) /a e e e m (125) ud (Q) ~ Z - j 2 ttv g( where D and v are the diffraction coefficient and attenuation m m factor, respectively, as defined in Chapter III, iJj is the phase shift introduced by caustics, and r, 0, and The Kirchhoff integral can be evaluated conveniently if a spherical surface of integration is chosen. Since the + and - angular dependence of Eq, (125) is associated with rays moving in the negative and positive 0- direction, respectively, the total diffracted field can be accounted for either by considering both types of rays over a tt radian range of 0 or by considering one FINAL POINT 0 =TT ~ /» To P at INITIAL „ POINT / e=-Tr-a/r 161 Figure 24. Huygens Surface for Spherical Scatterer. 162 type over a 2tt radian range of 8 c As a matter of convenience the latter approach is used here for a ray moving in the positive 0- direction with the shadow boundary of the incident rays, ice,, 0 = -tt - a/r, being considered the initial point for the 0 integration and the same point one encirclement later, i,ec, 0 = it - a/r, the final point as shown in Figure 24* (Additional -j27TV —1 encirclements are already included by the (1 + e m) term in u^(Q)c) On each passage through a caustic, icet, 0 = nTT, n = 0, ±1, c r - , a phase advance of tt/2 is experienced by a 1 ejnTT/2 ray so that — ■ ■ = — — --- c Since the caustic phase /sin 0 /sin (nTT+0) ^ G ^ shift at 0 = e is given by tt, it follows that ...... —— | /sin 0 | j TT eJ /sin 0 The surface normal and the radial vector are equal for a spherical surface so that 9/9n = 9/9tc Consequently the necessary derivatives are given by „ ,, -jv (TT+0+a/r) » (D )2«^ r jkle m 3u(Q) 3 u „ lV ^ e e i <126) I n ' 87 ' <-^k- 7> ® m“1 <1 + n m ) , W e r ~ -jku(Q) 163 and 3Gq (QsP) 9Go e~jkR 3r (127) (-Jk- R) -JkG0 (Q,P) |f 3n 3r 4ttR 3r From the cosine law with r « % and r « R as shown in Figure 24, the source-to-field distance is given by R~£+r-r cos 0. Hence on the spherical surface of radius r,(3R/3r) 0_ itt x»“Cons tsnt ~ 3(&+r)/3r -3(r cos 0)/3r= - cos 0 so that (128) 8Go - -jk cos 6 G0 (Q,P) 2 Substitution of these results along with ds (Q) = r sin 0 d0d TT TT-a/r (130) ud (P) ~ -f I G (Q,P)U(Q)(-jk)(l + cos 0) r 2 sin 0d0d4> e^ ik e-jk(«.+2r)^- ^ 4S e ------{ n 1 cp=0 wh e r e 164 2 -jVm (TT+a/r) 7T_a/r 00 (D ) e -jv +jkr cos 6 I = Z — —----- _.9 ---- / e m (1 + cos 0)/sin 6 d0. m=l /i . 0=-TT-a/r (1 + e ) The 2 tt 2 tt for scaler fields f g(d>)dd) = ^ jor _ vector „ fields,,,, .To determine I, * a representative r (p=0 integral of the form ff-n/r-jv e+jkr cos e m (131) I q = / e (1 + cos 0) /sin 0 d0 m0 0=-7T-a/r is considered; with the evaluation of this integral the backscattered GTD field is readily calculated. The form of Eq. (131) suggests the use of the method of steepest descents. In this technique 0 is considered to be a complex variable where the desired path of integration is along the real axis. (Alternatively, _i/2^ (-cos original m asymptotic expansion of the field can be represented as a complex function of a complex variable; however, the direct use of the GTD is desired here.) In the evaluation of Eq. (131) the path 165 of integration must be deformed to that it does not pass through the regions of the axial caustics, i.e., 0 = 0, ±rr, thus maintaining the condition ka sin 0 >> 1. The end points of integration, 0 = -u - a/r and 0 = 7T - a/r, must be carefully considered since their contributions appear to be of the order k \ while the desired saddle point terms are of the order k and k With these details in mind, I Q is written in the form of mo the steepest descents type of integral (132) I Q = / F(0) ekf(6) d0 c V ______where f (0) = -j( 0 - r cos 0) and F(0) = (1 + cos 0) /sin 0 and C is a path along the real 0-axis as shown in Figure 25. The existance of saddle points is confirmed by \ 3f(0 ) /V = 0 <133> s s = - 3 H £ + r sln es * TT / 3 which has the solution of sin 0g = - a/r (1 + q^/2 e ** -2/3 • (ka/2) +...). Since a/r is small and since the arcsine is a multivalued function, the saddle points within the range of integration are given as / ! V / Figure 25. Topological View of u(0) (Amplitude of Contours: ------Highest Level, — ------Intermediate High Level, ooo Reference Level, 166 u(0), ----- Intermediate Low Level, ..... Lowest Level). The contour C is deformed as shown in Figure 25 to cross these saddle points on the steepest descents path. This path is chosen so that the branch cuts introduced by /sin 0 are not crossed. The exponential function can be written as (134) f (0) = u(0) + jv(0) = r (\) 0, - v.0)+r sinh 0. sin 0 k r i l r i r - j (-r- (v 0 + V.0.) - r cosh 0. cos 0 Ik r r i i i r The topological plot of u(0), Figure 25, aids in observing the relative importance of the various contour segments to the total integral, I^q = 1^ + The evaluation of each of these integrals follows. 168 Within the geometrical optics shadow region and near the shadow boundary defined by 0 = sin ^ (-tt -a/r) the one or two mode form of the GTD is known to be deficient in representing the field. However, the ray optical methods are still valid if many modes are used or if the integral representations of Fock [7] are employed. Both of these forms indicate a rapid, linear phase variation of the diffracted field along the Huygens surface, S, indicating no saddle point at 0gl for these more accurate repre sentations of the field. (Nussenzveig [23] also has shown this phase variation exists.) Consequently, for the range of aspects in and adjacent to the shadow region, these alternative repre sentations of the field should be used in 1^. When this is done the contribution to 1^ for 0 ~ -TT is found to be negligible. This, in large part, is due to the fact that the obliquity factor, 1 + cos 0, is approximately zero in this range of aspects. Beyond this region the deformed contour of integration proceeds, with insignificant contribution to 1^, down into the deep valley where 1^ begins; the end point also contributes nothing due to its vanishingly small amplitude. Consequently, to the accuracy of this approximation, 1^ = 0. 169 The usual saddle point procedures are employed in evaluating I^. A saddle exists at kal‘2/3 — ^ I + ..o) ; this is 9S2 ■ - f + ^ *'F / 3 sufficiently far removed from the branch point at the origin because ka sin 0CO ~ ka — >> 1. The results are given by IT (135;(135-) 1 I2 ~ e^^^S2^e S2 ^I k f ..(es2)___ f (0S2) + 0 The contribution from this segment of the contour is readily seen to be negligible„ The exponential attenuation is infinitely great at the initial point of integration and monotonically increases to only e "^m77 at the end point „ This term is of the - fka/2')^^TT order of e ' * which is negligible; therefore, 1^ = 0. 170 The combined result is then jv a/r Im8 ■ *2 - ^ 2*'J W 4 . d + ^| e-^/3jk|)'2/3 + When Eq. (136) is substituted into Eq. (131) and this in turn into Eq. (130), the total diffracted field at backscatter is given by /10-,N d S a e ^ 7^ -jkR / 1 (137) u (P)v - — — ' L /2 \jl TTk . l_ (V 2 e !_ (1 + 5a e-j7i/3fks.'_2/3 + . . . ) , -j2v 7T U 4 2 m-1 <1 + e " ) s ! 1 where ( ) and |l/2 are used to denote the scaler (S) case and the vector (v) case. This is in complete agreement with Senior's results [28] which were obtained by first setting 0 = 0 and then taking the Watson Transformation, a procedure which eliminates the singularity at 0=0. The previous work may be generalized to treat the axial backscatter from a prolate spheroid. For a body of revolution, the attenuation and phase shift relative to the 0 = tt/2 plane 171 0-TT/2 V m is written as f(0) = - ^ / dt + jr cos 0o Then tt/2 V •M = - j,— — ^ -jr sin 0 = 0, where dt = dt(0) is known from the 90 Jkp d0 * g surface geometry,, The saddle point is found from this equation. In particular, the prolate spheroid solution of importance is -2/3 /k P. ?£» (i + e“j7T/3 + ... ) which is of the S2 form of the sphere solution where and p^ appropriately replace a. This indicates that the locus of saddle points on -2/3 K.U q \-2/31-2/3 (1 + — e 3 —§■ + ... ) is given for the sphere. These 2 | 2( two modifications lead to an expression for the generalized field of -jv TT , j3tt/4 -jkR __ “ m (138) u (P) ~ ----^ ---- v/2TTk Y. - j 2TTV m=l m s (1 + e TT -2/3 • / (dm (PB,Ptn)2 g tn a + \ 4 e j1T/3 4>«o * P t n g(4>)d(|> . 172 If Pg and ptn exhibit no dependence upon readily simplifies to a form similar to the spherical result. Levy and Keller [12], Eq. (11), show that the backscattered 1 / Q / / O O/O field from a prolate spheroid is k a (cosh t| sinh ri) 1/3 1/3 2 which reduces to k p p . Since (D ) has a dependence of tn g m k ^ ^ p the results of Eq. (138) exhibit the same k ^ ^ p p g , tn g behavior for a prolate spheroid, serving as a confirmation of the validity of the general form. REFERENCES CITED [1] Bouwkamp, C.J., "Diffraction Theory," Reports on Progress in Physics, Vol.XVII, 1954, pp.35-100. [2] Corriher, H.A., Jr., and Pyron, B.O., "A Bibliography of Articles on Radar Reflectivity and Related Subjects: 1957 - 1964," Proceedings of the IEEE, Vol.53, No.8, August, 1965, pp.1025-1064. [3] Keller, J.B., "The Geometric Optics Theory of Diffraction," presented at the McGill University Symposium on Microwave Optics, June, 1953; published in Symposium Proceedings, April, 1959, pp. 207-210. [4] Franz, W., and Depperman, K., "The Creeping Wave Theory of Diffraction," presented at the McGill University Symposium on Microwave Optics, June, 1953; published in Symposium Proceedings, April, 1959, pp.160-169. [5] Kazarinoff, N.D., and Ritt, R.K., "On the Theory of Scaler Diffraction and Its Application to the Prolate Spheroid," Annals of Physics, Vol.6, No.3, March, 1959, pp.277-299. [6] Kazarinoff, N.D., and Ritt, R.K., "Scaler Diffraction by an Elliptic Cylinder," IRE Transactions on Antennas and Propagation (Supplement), December, 1959, pp.S21-S27. [7] Fock, V.A., "The Distribution of Currents Induced by a Plane Wave on the Surface of a Conductor," Journal of Physics of the USSR, Vol.10, No.2, 1946, p.130 (in English); reprinted in Electro magnetic Diffraction and Propagation Problems, ed. V.A. Fock, New York: Pergamon Press, 1965, pp.10-22. [8] Keller, J.B., and Levy, B.R., "Decay Exponents and Diffraction Coefficients for Surface Waves on Surfaces of Nonconstant Curvature," IRE Transactions on Antennas and Propagation (Supplement), Vol.AP-7, Decembers, 1959, pp.S52-S61. [9] Franz, W., and Klante, K., "Diffraction by Surfaces of Variable Curvature," IRE Transactions on Antennas and Propagation (Supplement), Vol.AP-7, December, 1959, pp.S68-S70. 173 'au&auc . - ^ . ^ ._ [10] Hong, S., "Asymptotic Theory of Electromagnetic and Acoustic Diffraction by Smooth Convex Surfaces of Variable Curvature," Journal of Mathematical Physics, Vol.8, No.6, June, 1967, pp.1123- 1232. 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