# Fundamentals of the Physical Theory of Diffraction

 Copyright Year: 2014 Author(s): Pyotr Ya. Ufimtsev Book Type: Wiley-IEEE Press Content Type : Books Topics: Fields, Waves & Electromagnetics

## Abstract

The book is a complete, comprehensive description of the modern Physical Theory of Diffraction (PTD) based upon the concept of elementary edge waves. The theory is demonstrated with examples of the diffraction of acoustic and electromagnetic waves at perfectly reflecting objects.
Readers develop the skills to apply PTD to solve various scattering problems. The derived analytic expressions clearly illustrate the physical structure of the scattered field. They additionally describe all of the reflected and diffracted rays and beams, as well as the fields in the vicinity of caustics and foci. Shadow radiation, a fundamental component of PTD, is introduced and proven to contain half the total scattered power. The equivalence relationships between acoustic and electromagnetic diffracted waves are established and emphasized. Throughout the book, the author enables readers to master both the theory and its practical applications.
- Plotted numeric results supplement the theo y and facilitate the visualization of individual contributions of distinct parts of the scattering objects to the total diffracted field
- Detailed comments help readers understand and implement all the critical steps of the analytic and numeric calculations
- Problem sets in each chapter give readers an opportunity to analyse and investigate the diffraction phenomena

• ### Front Matter

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The prelims comprise:
Half Title
Title
Contents
Preface
Foreword to the First Edition
Preface to the First Edition
Acknowledgments
Introduction View full abstract»

• ### Basic Notions in Acoustic and Electromagnetic Diffraction Problems

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In two-dimensional problems, the physical theory of diffraction (PTD) is valid for both electromagnetic and acoustic waves. This chapter presents the theoretical fundamentals for acoustic and electromagnetic waves. To complete formulation of the diffraction problem and ensure the uniqueness of its solution, the wave equation and boundary conditions are supplemented by the Sommerfeld radiation condition for the scattered field. The high-frequency physical optics (PO) approach is widely used in acoustic and electromagnetic diffraction problems. Among the properties of the shadow radiation, the most significant are the shadow contour theorem and the total power of shadow radiation, which are verified for acoustic and electromagnetic waves, in the chapter. Taking into account the diffraction and nonuniform components of surface fields, PTD overcomes the PO shortcomings and provides more accurate asymptotic results for high-frequency scattered fields.

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• ### Wedge Diffraction: Exact Solution and Asymptotics

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As the wedge diffraction problem is the basis for the construction of physical theory of diffraction (PTD), its solution is considered in this chapter in detail. First its solution is derived in the form of infinite series and then converted it to Sommerfeld integrals convenient for asymptotic analysis namely, Sommerfeld asymptotics and Pauli asymptotics. The chapter provides the expressions that relate to the excitation of the field by a cylindrical wave, with a specific source term around the wedge in a specific region. These expressions can be modified for excitation by a plane wave. The chapter shows the derivation of asymptotic expressions under the condition that the incident wave does not undergo double and higher-order multiple reflections at faces of the wedge; this is an extension of the Pauli technique. It also demonstrates the use of magic-zero procedure for fast convergent integrals and uniform asymptotics.

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• ### Wedge Diffraction: The Physical Optics Field

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In this chapter, the calculation of physical optics (PO) part of the scattered field is shown, that is, the field generated by the uniform component of the induced surface scattering sources. This calculation can be used to examine the field radiated by the nonuniform sources as the difference between the exact and physical optics fields. The chapter demonstrates the conversion of PO integrals to the canonical form. It also demonstrates the derivation of fast convergent integrals and asymptotic expressions for the PO diffracted field. The related results of a numerical comparison of the PO approximation against the exact solution for the wedge diffraction problem have been presented by Hacivelioglu et al.

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• ### Wedge Diffraction: Radiation by Fringe Components of Surface Sources

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The integral and asymptotic expressions can be constructed for a field radiated by the nonuniform/fringe component of the surface sources which are induced at the wedge by the incident wave. The contribution to the diffracted field by the nonuniform/fringe component is the difference between the exact total field and its physical optics (PO) part; this contribution is investigated in the chapter. The integrals derived for the fringe field are taken along the steepest descent path and are very convenient for numerical and analytical analysis. The chapter demonstrates the reduction of the three-dimensional diffraction problem for the oblique incidence to a two-dimensional problem for the normal incidence. The solution for the oblique incidence can be found automatically using simple replacements in the solution for the normal incidence.

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• ### First-Order Diffraction at Strips and Polygonal Cylinders

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General asymptotic expressions have been derived for first-order edge-diffracted waves generated by both uniform and nonuniform components of the surface sources. In this chapter this general theory is applied to high-frequency diffraction at strips and cylinders with triangular cross-sections. These specific diffraction problems have been studied comprehensively and reported in the literature. In particular, the uniform asymptotic expressions for the directivity pattern and for the surface field at the strips have been derived by Ufimtsev. High-frequency diffraction at polygonal cylinders was investigated by Morse and Borovikov. The chapter considers the problems to demonstrate the first applications of physical theory of diffraction (PTD). The physical optics (PO) part of a scattered field generated by the uniform components of the surface sources is determined by the integrals. The chapter presents the numerical analysis of a scattered field by utilizing different approaches.

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• ### Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution

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This chapter develops the first-order physical theory of diffraction (PTD) for acoustic waves scattered at bodies of revolution with sharp edges. It deals with axially symmetric scattering. This situation occurs when an incident plane wave propagates in the direction along the symmetry axis of a body of revolution. It also determines the field scattered at certain bodies of revolution. The chapter studies symmetrical scattering at bodies of revolution whose illuminated side is an arbitrary smooth convex surface with nonzero Gaussian curvature. In addition, it presents three types of calculation for the cone. The first type consists of calculation for conformal paraboloids, which differ by their length; the second type relates to the transformation of paraboloids into the disk, and the third type reveals the influence of the shadowed base of paraboloids on backscattering. Finally, the chapter investigates the influence on backscattering of the shadowed part of the spherical segment.

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• ### Elementary Acoustic and Electromagnetic Edge Waves

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The relationships exists between acoustic and electromagnetic elementary edge waves (EEWs) propagating in directions that belong to the diffraction cone. The goal is to derive high-frequency asymptotics for EEWs. The appropriate tangency point must be the origin of the diffracted ray coming to the observation point on the tangential wedge. The chapter discusses the analytical properties of EEWs. It presents the results of numerical calculations of the elementary edge-diffracted waves radiated by the non-uniform scattering sources. The central idea of physical theory of diffraction (PTD) is the separation of surface scattering sources into uniform and non-uniform components in such a way that they would be the most appropriate for calculation of the scattered field. The asymptotic theory is well suited for the calculation of bistatic scattering in the case, when both planar faces of the edge are illuminated by the incident wave.

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• ### Ray and Caustic Asymptotics for Edge Diffracted Waves

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The asymptotics for edge-diffracted waves can be presented in another form that reveals their ray structure. Ray asymptotics were postulated in the geometrical theory of diffraction (GTD). GTD can be interpreted as the ray asymptotic form of physical theory of diffraction (PTD) for total diffracted field. In contrast to PTD, GTD is not applicable in regions where the field does not have a ray structure and where the actual diffraction phenomena happen. The ray asymptotics are not valid at caustics, where they predict an infinitely large field intensity. Caustic asymptotics are presented for both acoustic and electromagnetic waves. These asymptotics have the same structure and differ only in the coefficients. This chapter discusses the relationships between PTD and GTD. PTD is a source-based technique and GTD belongs to ray-type theory.

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• ### Multiple Diffraction of Edge Waves: Grazing Incidence and Slope Diffraction

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This chapter investigates two special cases. The first case is a grazing incidence of edge waves on acoustically hard planar plates. In the asymptotic theory, the incident wave is approximated by an equivalent plane wave. The second case that also needs special treatment occurs when the scattering edge is located in the zero of the incident wave. This is the case for slope diffraction. The chapter considers the important one to be the slope diffraction of the first order, when the first derivative of the incident wave is not equal to zero. Such a situation occurs, for example, in reflector antennas, when one tries to decrease side lobes, and in the process of multiple diffraction between several scatterers or between different parts of the same scatterer. The chapter founds ray asymptotics using the stationary-phase technique.

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• ### Diffraction Interaction of Neighboring Edges on a Ruled Surface

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This chapter considers a diffraction interaction of two edges with a common face. If this face is bent, the edge wave already undergoes diffraction on its way along the face to another edge. The ray asymptotics of the diffracted field can be found using the stationary-phase technique. At the shadow boundary the normal derivative of the total field is continuous and equal to one-half of the normal derivative of the incident wave. The uniform high-frequency asymptotics with arbitrary high precision have been developed there for the scattered field and for the surface currents on the strip. The chapter employs the first-order high frequency approximations for the scattered field. It presents the secondary wave predicted by the exact asymptotic theory.

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• ### Focusing of Multiple Acoustic Edge Waves Diffracted at a Convex Body of Revolution with a Flat Base

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This chapter focuses on multiple acoustic edge waves diffracted at a convex body of revolution with a flat base. It analyses the field generated by the total scattering sources. The first-order (primary) edge waves excited directly by the incident wave are determined by the integral expression applied to the circular edge. The total scattered field on the focal line also includes the reflected rays in front of the object (z < 0) and the shadow radiation behind the object (z > 0). This approximation for the scattered field actually represents an incomplete asymptotic expansion, because it includes only the first term in the individual asymptotic expansion for each multiple edge wave. The higher-order edge waves arise due to the slope diffraction of waves running along the flat base of the scattering object.

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• ### Focusing of Multiple Edge Waves Diffracted at a Disk

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This chapter presents disk diffraction problem, where it is necessary to take into account the edge waves propagating along both faces of a disk. This problem is complicated by the fact that the wave traveling along one face of the disk generates (due to diffraction at the edge) higher-order waves not only on the same face but also on the other face. The chapter discusses about the multiple soft diffraction, the multiple hard diffraction, and the multiple diffraction of electromagnetic waves. Finally, it investigates the diffraction of a plane wave at a perfectly conducting disk.

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• ### Backscattering at a Finite-Length Cylinder

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A solid circular cylinder with flat bases is illuminated by the incident plane wave. The scattered field is evaluated for the backscattering direction in the far zone. Physical optics (PO) far fields backscattered by acoustically hard and soft objects differ from each other only in sign. Hence, it is sufficient to exhibit the PO calculations only for the case of scattering at a hard cylinder. This chapter calculates the far field scattered by the left base/disk of the hard cylinder. Most of the maximums in the soft fringe field are located in the vicinity of the angular positions of the minimums of the PO field. The opposite situation is observed for the hard fringe field; its maximums are positioned near the maximums of the PO field. The chapter describes the original Physical theory of Diffraction (PTD) of electromagnetic waves scattered from a finite perfectly conducting cylinder.

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• ### Bistatic Scattering at a Finite-Length Cylinder

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The shadow radiation is the constituent part of the physical optics (PO) field. It was noted there that this field concentrates in the vicinity of the shadow region. This chapter verifies this property by numerical investigation of the shadow radiation generated by the finite-length cylinder. The most appropriate procedure for doing this work would be the direct application of the shadow contour theorem. It presents numerical investigation of the scattered field. The chapter considers its physical structure and presents simple high-frequency asymptotics for the directivity pattern. It deals with the E-Polarization and the H-Polarization. Finally, the chapter discusses about the PO shooting-through rays and their cancellation by Fringe rays, and the refined asymptotics for the specular beam reflected from the lateral surface.

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• ### Conclusion

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• ### References

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• ### Appendix to Chapter 4: MATLAB Codes for Two-Dimensional Fringe Waves and Figures (F. Hacivelioglu and L. Sevgi)

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• ### Appendix to Chapter 6: MATLAB Codes for Axial Backscattering at Bodies of Revolution (F. Hacivelioglu and L. Sevgi)

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• ### Appendix to Section 7.7: MATLAB Codes for Diffraction Coefficients of Acoustic Elementary Fringe Waves (F. Hacivelioglu and L. Sevgi)

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• ### Appendix to Section 7.8.3: MATLAB Codes for Diffraction Coefficients of Electromagnetic Elementary Fringe Waves (F. Hacivelioglu and L. Sevgi)

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• ### Appendix to Section 7.9.2: Field d Radiated by Modified Uniform Currents Induced on Elementary Strips (P. Ya. Ufimtsev)

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