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University Microfilms, a XEROX Company, Ann Arbor, Michigan DIFFRACTION by DOUBLY CURVED 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.
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