THE UNIVERSITY OF MICHIGAN AFCRC-TR-59-179 FINAL REPORT ON CONTRACT AF 19(604)-1949 by Ko M. Sieg.el Report No. 2591-5-F July 1959 Prepared for AIR FORCE CAMBRIDGE RESEARCH CENTER AIR RESEARCH AND DEVELOPMENT COMMAND BEDFORD, MASSACHUSETTS

THE UNIVERSITY OF MICHIGAN 2591-5-F This contract started on 1 October 1956 and ended with the publishing of this final report on 31 July 1959. The major effort was the writing of an instruction manual (Ref, 1). The purpose of this manual is to tell engineers, mathematicians, and physicists (with only an undergraduate course in electromagnetic theory and the usual undergraduate courses in applied mathematics) how to compute the radar cross sections of aircraft and missiles. Different methods presented are dependent upon. whether the ratio of the major dimensions of the scatterer are large or small with respect to the wavelength. This manual contains many examples and these examples should suffice for the many aircraft and missile shapes with which a member of the aircraft and/or electronics industry may be involved. With average judgement on methods to use, the inexperienced engineer should be able to reproduce the radar cross section of an aircraft within 6 db. When the missile dimensions are small with respect to the wavelength, the calculated cross section should agree within three db of the measured return. When the wavelength is small with respect to all radii of curvature of the missile again the cross section as calculated should agree very well with experiment. When the missile is long and thin and when the wavelength is large in respect to the radius of curvature in the front and rear but less than the missile's dimensions then much more experience and judgement are required to compute the cross section. In that case it is necessary to determine the reflection coefficient at the rear of the missile in order to determine the nose-on cross section. This is described n Reference 1 and utilized in Reference 2, Other major efforts on this contract include the following reports: 1, Dr. TeB. Ao Se.niors report (Ref, 3) which gave the exact solution for the imperfectly conducting wedge. 1

THE UNIVERSITY OF MICHIGAN 2591-5 -F 2. Dr, R. Goodrich's report (Ref. 4) which describes and applies Fock theory, He describes and uses this method to solve both two dimensional and three dimensional problems, 3. Dr. Fo Sleator applied the variational technique to obtain the scalar solution for a prolate spheroid at long wavelengths (Ref. 5). 4. Prof. R, Ritt and Prof, N. Kazarinoff analyzed the scalar scattering from a prolate spheroid at small wavelengths (Ref. 6), Information and knowledge gained on this contract allowed us to contribute all or part of the journal articles listed in References 7-18, Information and knowledge gained under this contract allowed us to perform more efficiently on many other United States Air Force contracts such as AF 30(602)-1808, AF 33(616)-5585, AF 33(600)-36793, AF 30(602)-1853, AF 19(604)-4993, AF 30(602)1982, AF 19(604)-5470, AF 19(604)-5553, and AF 33(600)-39476. At the termination of some contracts we::are aware of partial contributions which will be or have been completed under other contracts, Examples of such contributions are included in References 19-25, for this contract. One major effort is still not completed and should.play an important role in Contract AF 19(604)-4993. This effort was by Professor D. Darling (Refs, 26 and 27) and shows how to solve exactly (but presently mainly in potential theory) the Dirichlet and Neumann problems for a shape which is made up of surfaces which have the following property: Each surface is part or all of a surface obtained for a coordinate system for which Laplace's equation separates and one of the coordinates is set equal to a constant, Methods are known for using these solutions to obtain solutions to Maxwell's equations 2

THE UNIVERSITY OF MICHIGAN 2591-5-F when the wavelength is large with respect to the dimensions of the body. This method is important in that it applies to homogeneous surfaces for all values of dielectric constant and permeability. Thus ablating warhead's cross sections can now be found exactly in the long wavelength region. This will be an example of basic research performed under one contract leading to important applications on another contract. We: desire to take this opportunity to thank P. Blacksmith for the excellent cooperation we have received during the course of this effort. His efforts undoubtedly doubled our efficiency in performance under this contract. 3

THE UNIVERSITY OF MICHIGAN 2591-5-F REFERENCES 1. Crispin, J. W., Goodrich, R. F., and Siegel, KM., "A Theoretical Method for the Calculation of the Radar Cross Sections of Aircraft and Missiles", To be Published. 2. Hiatt, R. E., Siegel, K. M., and Weil, H., "CRAM (Counters to R.A.M. )", presented at the 2nd Annual R. A.M. Symposium, Rome Air Development Center, June 9-11, 1959. SECRET. 3, Senior, T. B. A., "Studies in Radar Cross Sections XXV - Diffraction by an Imperfectly Conducting Wedge", The University of Michigan Radiation Laboratory Report No. 25912-T, AFCRC-TN-57-591, AD 133746, October 1957. UNCLASSIFIED. 4. Goodrich, R. F., "Studies in Radar Cross Sections XXVI - Fock Theory"', The University of Michigan Radiation Laboratory Report 2591-3-T, AFCRC-TN-58-350, AD 160790, July 1958, UNCLASSIFIED. 5. Sleator, F. B., "Studies in Radar Cross Sections XXII - A Variational Solution to the Problem of Scalar Scattering by a Prolate Spheroid", The University of Michigan Radiation Laboratory Report No. 2591-1-T, AFCRC-TN-57-586, AD 133631, March 1957, UNCLASSIFIED. 6. Ritt, R. K, (with appendix by N. D. Kazarinoff), "Studies in Radar Cross Sections XXX - The Theory of Scalar Diffraction with Application to the Prolate Spheroid", The University of Michigan Radiation Laboratory Report No. 2591-4-T, AFCRC-TN58-531, AD 160791, August 1958. UNCLASSIFIED. 7. Goodrich, R. F., Kleinman, R. E., Maffett, A. L., Reitlinger, N. E., Schensted, C, E,, and Siegel, K. M., "Radiation and Scattering From Simple Shapes - I" Presented at Congres International Circuits et Antennes Hyperfr6quences, Paris, France, 21-26 October 1957. Published in L'Onde Electrique, Vol. I, No. 376 46-48(August 1958). 8. Goodrich, R. F., Kleinman, R. E., Maffett, A. L., Reitlinger, N. E., Shhensted, C. E., and Siegel, K. M,, "Radiation and Scattering From Simple Shapes - II" Presented at Congres International Circuits et Antennes Hyperfrequences, Paris, France, 21-26 October 1957. Published in L'Onde Electrique, Vol. I, No. 376 49-57(August 1958). 9. Siegel, K. M., "Increasing the Effective Dynamic Range of a Radar" American Institute of Electrical Engineers, Paper DP-58-519(1958). 10. Siegel, K. M., "Bistatic Radars and Forward Scattering", Aero Electronics 1958 National Conference Proceedings, 286-290(1958). 4

THE UNIVERSITY OF MICHIGAN 2591-5-F REFERENCES (Continued) 11. Siegel, K. M., "Far Field Scattering From Bodies of Revolution", Applied Scientific Research, Section B, Vol. 7, 298-328(1958). 12. Kazarinoff, N. D., and Ritt, R. K., "On the Theory of Scalar Diffraction and Its Application to the Prolate Spheroid", Annals of Physics, Vol. 6, No. 3, 277-299 (March 1959). 13. Senior, T. B. A., "The Currents on Strip Aerials", Electronic and Radio Engineer Vol. 36, 60-63 (1959). 14. Chernin, M. G., Shanks, H. E., Plummer, R. E., Goodrich, R.F., Kleinman, R.E., Maffett, A. L., Schensted, C.E., and Siegel, K.M., "Radiation From Slot Arrays on Cones", IRE Transactions on Antennas and Propagation, Vol. AP-7, No. 3 (July 1959). 15. Senior, T. B.A., "Diffraction by an Imperfectly Conducting Wedge", Communications on Pure and Applied Mathematics, Vol. XII, 337(1959). 16. Senior, T. B. A., "The Scattering of Electromagnetic Waves by a Corrugated Sheet", Canadian Journal of Physics, Vol. 37, 787-797(July 1959). 17. Siegel, K. M., "Comments on Far Field Scattering From Bodies of Revolution", Applied Scientific Research, Section B, Vol. 8 (1959). 18. Cottony, H. V., Elliott, R. S., Jordan, E. C., Rumsey, V.H., Siegel, K.M., Wait, J. R., and Woodyard, O. C., "U. S.A. National Committee Report URSI Subcommission 6.3 - Antennas and Waveguides and Annotated Bibliography", IRE Transactions on Antennas and Propagation, Vol. AP-7, No. 1, 87-98 (January 1959). 19. Kazarinoff, N. D., and Ritt, R. K., "Scalar Diffraction by an Elliptic Cylinder", The University of Michigan Radiation Laboratory Report No. 2871-2-T, June 1959, UNCLASSIFIED. 20. Kazarinoff, N. D., and Ritt, R. K., "Scalar Diffraction by an Elliptic Cylinder," Presented at URSI-Toronto Symposium 15-20 June 1959. To be Published in a Special Issue of IRE Transactions on Antennas and Propagation. 21. Brysk, H., "Electromagnetic Scattering by High Density Meteor Trails", The University of Michigan Radiation Laboratory Report No. 2871-1-T, June 1959. UNCLASSIFIED. 5

THE UNIVERSITY OF MICHIGAN 2591-5 F REFERENCES (Continued) 22. Brysk, H., "Electromagnetic Scattering by High Density Meteor Trails", Presented at URSI-Toronto Symposium 15-20 June 1959. To be Published in a Special Issue of IRE Transactions on Antennas and Propagation. 23, Goodrich, R. F., "Fock Theory", Presented at URSI-Toronto Symposium 15-20 June 1959. To be Published in a Special Issue of IRE Transactions on Antennas and Propagation. 24, Senior, T. B. A., and Siegel, K. M.,!The Asymptotic Expansion of Electromagnetic Scattering Functions at Long Wavelengths", The University of Michigan Radiation Laboratory Report No. 2871-3-T. (To be Published). UNCLASSIFIED. 25. Senior, T. B.A., and Siegel, K. M., "The Asymptotic Expansion of Electromagnetic Scattering Functions at Long Wavelengths", Presented at URSI-Toronto Symposium 15-20 June 1959. To be Published in a Special Issue of IRE Transactions on Antennas and Propagation. 26. Darling, D. A., "Brownian Motion and the Exterior Dirichlet Problem", The University of Michigan Radiation Laboratory Internal Memorandum No. 2591-501-M, 8 January 1957. UNCLASSIFIED. 27. Darling, D. A., "Brownian Motion and the Exterior Neumann Problem I", The University of Michigan Radiation Laboratory Internal Memorandum No. 2591-518-M, 5 June 1958. UNCLASSIFIED. 6

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