# Addendum to Research on Problems of Diffraction, Scattering, and Propagation of Waves: Diffractive Edge and Singularity

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This section is Under-Edit if necessary: Construction began on December 3, 2010 and was finished on April 28, 2011.

ADDENDUM to RESEARCH on PROBLEMS of DIFFRACTION, SCATTERING, and PROPAGATION OF WAVES: DIFFRACTIVE EDGE and SINGULARITY

```by Darrell A. Nolta
April 28, 2011
```

TABLE OF CONTENTS

INTRODUCTION

FACTS DRAWN from DOCUMENTS and ASSOCIATED IDEAS

DISCUSSION

••I) P.Ya. Ufimtsev and the Physical Theory of Diffraction (PTD)

••II) The Problem of Diffraction (Scattering) and Shadow

••III) Physics and Mathematics of Singularity (Edge & EM field Singularity)

••IV) Numerical Modeling of Scattering & Diffraction Problems

••V) Important Scattering/Diffraction Objects with an Edge

••Summary of Wave Processes Paper and Book References

CONCLUSIONS

ADDENDUM LIST of REFERENCES

INTRODUCTION

The purpose of this Addendum to the "Research on Problems of Diffraction, Scattering, and Propagation of Waves" paper is to present the results of a study of the Physics and Mathematics of Edge Diffraction and Singularity. Prior to reading this Addendum, one should read the initial paper to understand the purpose of this Addendum.

The diffraction, scattering, and wave propagation are one of the most important phenomena of the observable and unobservable physical world. Almost an infinite number of important problems dealing with the interaction of electromagnetic or acoustic waves with 4-D (space and time) objects of macroscopic or microscopic scale exist in the Universe. Most if not all of these objects are composed of edges and other types of surface (& material property) discontinuities where the surface normals become discontinuous.

In the macroscopic world, we have important examples dealing with military systems such as stealth weapons (e.g., aircraft, submarine), and seismic propagation of energy through the earth and its possible aftermath of a propagation of a tsunami across an ocean (Japan's 2011 Killer Earthquake).

Consider interesting examples of Russian and Chinese prototype stealth aircraft. On January 29, 2010, the Russian government demonstrated its "fifth-generation" Stealth fighter aircraft, the Sukhoi PAK FA Advanced Tactical frontline fighter T-50. The T-50 Russian stealth prototype aircraft performed its flight tests in Russia's Far East.

Consult the Sukhoi Company website for the T-50 Russian stealth aircraft (fighter) information & photographs.

In addition, China is now engaged in this apparent arms race to develop stealth weapons. We learned on January 7, 2011 that the Chinese military is developing a prototype (experimental version) of a stealth fighter (J-20).

Check out a story on the J-20 stealth fighter from the 1/12/11 L.A. Times.

Check out the additional information and photograph for the J-20 stealth fighter.

What is the nature of the Stealth technology and applied radar absorbing materials in the T-50 and the J-20?

What is amazing after so much scientific study of diffraction (scattering) is that the fundamental basis of this phenomena is not known. Counter to what Albert Einstein believed (1933), the nature of scattering and diffraction is not the realization of the simplest conceivable mathematical ideas.

Rigorous mathematical solutions to the specific Problems of Diffraction, Scattering, and Wave Propagation for multi-edge natural or man-made objects are not known. In most cases the physical models of these problems are not rigorous, either.

After centuries of study of diffraction (scattering), we know that the geometry and the electromagnetic (acoustic) properties of the diffracting (scattering) object and the geometry of the observation space are involved in dictating the complexity of the solution. In the relatively simple cases, the wave equation is simplified; the problem and coordinate system allows for separation of variables; and/or 2-D diffraction integral containing mathematical irregularities is altered. Or a high frequency asymptotic approach is used. In the end, the solution is an approximation.

But in the cases such as non-separable 2-D or 3-D problems (boundary & media discontinuities) where these mathematical simplifications can not be used, the 2-D or 3-D vectorial diffraction integrals that may contain mathematical irregularities can not be solved in an exact (closed form) solution.

That is why the scientific literature is written in term of theories and solutions of approximations to these problems. This is why an approximate solution from a new approach (model) is compared to the approximate solution of an old approach (model). Sometimes the new result is compared with experimental data but this data contains experimental error, too.

A direct result of this representation problem is that the natural phenomena of waves and their interaction with a physical object's surfaces and edges are extremely difficult to accurately model and simulate in a computational machine.

So we are faced with asking fundamental questions about the conceptual nature of diffraction (scattering). Further, if possible, it is important that the "Original literature" be studied so that one can avoid the simplification and the distillation of the difficult concepts. The first principles of wave interaction with matter must be revisited at the microscopic and macroscopic level.

But what are the correct first principles or concepts involved in a spherical, cylindrical, or plane wave's interaction with a scattering or diffracting 3-D surface or edge (moving or stationary). And what is a diffractive edge in 2-D & 3-D space, i.e., how is an edge physically and mathematical defined? An Edge can be straight, curved, discontinuous, or a composite of these shapes.

The results of this study are based upon the Analysis of a set of Documents (papers & books) that are delineated in the List of References that is found below. Consult the Facts Drawn from Documents and Associated Ideas analysis section for the specifics of the original analysis of an individual document.

In the Discussion section below, we will consider the following topics:

I) P.Ya. Ufimtsev and the Physical Theory of Diffraction (PTD);

II) The Problem of Diffraction (Scattering) and Shadow;

III) Physics and Mathematics of Singularity [Edge & Electromagnetic (EM) Field Singularity];

IV) Numerical Modeling of Scattering & Diffraction Problems; and

V) Important Scattering/Diffraction Objects with an Edge.

At the end of the Discussion section, a Summary of Wave Processes Paper and Book References is given.

Through these topics, we will attempt to bring to light some key facts, ideas, and questions about diffraction (scattering) and its level of understanding by the scientific community in the Discussion and Conclusion section.

FACTS DRAWN from DOCUMENTS and ASSOCIATED IDEAS (Link below)

FACTS AND IDEAS DERIVED FROM ANALYSIS of DOCUMENTS PART I: No. A1 - A7.

FACTS AND IDEAS DERIVED FROM ANALYSIS of DOCUMENTS PART II: No. A8 - A22.

FACTS AND IDEAS DERIVED FROM ANALYSIS of DOCUMENTS PART II: No. A23 - A44.

DISCUSSION

I) P.Ya. Ufimtsev and the Physical Theory of Diffraction (PTD)

We begin this discussion by updating our knowledge about P.Ya Ufimtsev and the Physical Theory of Diffraction in regards to the open questions about Dr. Ufimtsev and his role in U.S. stealth aircraft development (e.g., F-117 Nighthawk, B2 Stealth Bomber, F-22 Raptor, etc.).

In the Ufimtsev's 1968 paper [A11] (Parabolic equation application to dipoles), we learned about the contribution of Professor L.A. Vainshtein to Ufimtsev's work ("The author thanks L.A. Vainshtein for formulating the problem and for his interest."). Also, we learn in the Yu. A. Kravtsov and Ufimtsev's 1989 paper [A23] (Virtual fields) the contribution to Ufimtsev's work by Dr. Vainshtein (" The authors wish to express their gratitude to L.A. Vainshtein for his critical comments on this paper.").

It is still a mystery about the details of the working relationship between Ufimtsev and Vainshtein in terms of the development of Soviet contributions to the theories of diffraction and scattering and their applications. It is important to realize that Dr. Vainshtein died in Sept 1989 in the USSR and Ufimtsev left the Soviet Union in 1990 (about 1 year before the collapse of the country).

Given the supposed impact on the military spending on stealth technology (PTD applications) by the U.S, it is a fair question to ask about the release of the Soviet scientific works through the "Open" literature. The interview of the famous mathematician Vladimir Arnol'd by S.H. Lui [A28, page 433] we learned about the role of the KGB on Editorial Boards in terms of control of information content of papers to be published (or not to be published).

This Soviet control of publication of "Open" literature via KGB censorship of scientific papers is real. If we consult Richard Lourie's book SARKAROV A Biography (2002, page 88), we learned about the Stalin decree (1946), Glavlit, and censorship of scientific articles. This censorship continued up to the fall of the Soviet Union (1991) and probably continues today. So it is a fair question to ask about the fidelity and intent of the contents of key Soviet papers of L.A. Vainshtein, Ufimtsev (e.g., 1957 & 1958 papers & 1962 monograph), and many other Soviet sources.

Finally, after the publication of key papers, monograph & books, Dr. Ufimtsev appears to be searching for new ideas/approaches while still crusading for PTD.

In the 1968 paper [A11], Ufimtsev applied the Parabolic Equation (PE) method to find an approximate solution to the Problem of diffraction by a thin cylindrical wire (dipole) and by a ribbon. He states that the properties of current waves in thin vibrators have become a recent study in the Theory of Dipoles. This Ufimtsev paper is important because it indicates the following ideas:

i) Ufimtsev's study of another diffraction approximation method: Parabolic equation method (PE); &

ii) his work with Current Waves.

In the 1989 paper [A23], Ufimtsev (& Kravtsov) studied the concept of "Virtual Rays" (Virtual Fields & Virtual Waves) and is trying to develop a new version of the Geometrical Theory of Diffraction. What is wrong with PTD?

In the Ufimtsev (& B. Khayatian and Y. Rahmat-Samii) 2000 papers, [A33, A34], we learned about another issue concerning the use of PTD: the application of the wedge model and the imposition of the Singular Edge behavior.

And in the 2006 Ufimtsev paper [A39], we learned about the idea of the Improved PTD to solve a very serious flaw of PTD (issue of the Grazing Singularity). He decides to create a new definition of the Uniform surface component and a new Non-uniform Surface current component. Also, in his acoustic version of PTD, he needs an Improved version of PTD (2006, [A40]), too.

II) The Problem of Diffraction (Scattering) and Shadow

At this point, let us look at some important works on the Problem of Diffraction (Scattering) and Shadow that include the authorship by H.M MacDonald [A1], A. Sommerfeld [A6], J.B. Keller [A7], L.A. Vainshtein [A8 & A9], T.B.A. Senior et al ([A12] & ), M. Born et al , A.E. Heins , S. Wang [A27], S. Anokhov [A32, A41, & A42], A.P. Kiselev et al [A38], and V.M. Babich et al [A43].

The problem of the general edge diffraction (scattering) is found in many natural objects and man-made applications (physical constructs). There are a number of geometrical models such as the half-plane, wedge with straight edge, & curved wedge with curved edge, and other canonical forms that are used to study this problem of diffraction (scattering).

From the H.M. MacDonald's 1915 paper [A1] we learn about the approach of deriving an approximate analytical result for the problem of diffraction of sound waves by a wedge. This approach is based on the commonality of two harmonic functions series (one for diffraction and the other for potential) to determine the appropriate constants of the diffraction series.

In 1915, did the technology exist to measure a wedge's acoustic diffraction field? How do we know that the MacDonald's approach is indeed valid in general?

Optics, the 1954 book [A6] by Professor Sommerfeld, is a foundational (thus, extremely important) scientific work in the fields of Electrodynamics of Light and the Theory of Diffraction. It contains many unique theoretical ideas, mathematical formulations, and experimental results including important works of Russian scientists. Dr. Arnold Sommerfeld (1868 - 1951) is one of the founding fathers of modern theoretical physics; thus, he is one of the most famous physicists of the 20th century. This book was published three years after he died in 1951; the preface of his book was written by him about two years before his death.

Scientific investigators in the field of the Theory of Diffraction reference through their papers and books directly or indirectly the works of Professor Sommerfeld (e.g., Optics).

For example, consider two extremely important works such as the 1957 and 1962 papers on the Geometrical Theory of Diffraction by J.B. Keller (New York University) where the concepts of the Law of Edge Diffraction, the "Cone of Diffracted Rays," and the Diffraction Coefficient are discussed.

It is important to realize that Optics does not provide a complete mathematical derivation that supports the existence (structure) of the "Cone of Diffracted Rays" nor does it address the issue of the energy distribution in such a light cone. One could consult Sections 44 C (Rubinowicz & E. Maey) and 36 E for a very incomplete discussion on the "circular half-cone" of radiation and "light fans," respectively.

Sommerfeld does not specify in detail (physical & mathematical description) the phenomenon of the Edge Wave (simply stated as "a kind of reflection"). Note that there is no such thing as a "sharp" Edge according to him.

Also, no solution to the Problem of Diffraction by a Wedge is presented in Optics.

The "luminous edge" of a diffracting structure is not real according to Sommerfeld.

Further, one must understand the key concept of the Rigorous Definition of a Problem of Diffraction from a physical perspective, a mathematical perspective, and their mutual limitations so that one can derive the correct diffraction solution, if possible.

Singularities in Diffraction fields (i.e., infinities of field values) do not exist physically. They are problems of constructs of mathematical models of Diffraction.

This very important fact about field singularities implies that the mathematical or physical models of the Diffraction phenomena do not actually describe the true phenomenon of Diffraction in Nature. Further, the generally accepted Theories of Diffraction may not be correct, too.

Finally, there are common ideas between Optics (Optical Electrodynamics) and Hamiltonian dynamics (mechanics).

Thus, what new model (s) of Diffraction can be found using these ideas?

J.B. Keller's 1957 paper [A7] is part of a collection of key papers on The Geometrical Theory of Diffraction (GTD). Dr. Keller was a key Diffraction investigator in U.S. who worked at the Institute of Mathematical Sciences, New York University. This paper/research was funded by the U.S. Air Force, Air Force Cambridge Research Center.

This paper does not prove physically or mathematically why the "Cone of Diffracted Rays" exists as an edge diffraction phenomenon nor does it provide a clear mathematical description of the "Cone of Diffracted Rays": Keller refers to the Fermat's principle (stationarity of optical path length) as a justification for this light cone. But how can an infinitely many diffracted rays come from one incident ray after this ray interacts with an edge?

An Edge is a caustic with an infinite number of diffracted rays being emitted at a given point of diffraction (principle of energy conservation?).

Keller claims that Sommerfeld's 1954 book Optics provides key proof to support his GTD work and the necessary associated Diffraction Coefficients. Thus, Sommerfeld's work serves as a key to determining the fidelity of the ideas of Keller.

L.A. Vainshtein, a famous Soviet theoretical physicist, wrote two extremely important papers (1963 [A8] & 1964 [A9]) on the subject of the role of Diffraction in Open Resonators and Open Waveguides. The wave propagation in an Open Waveguide exhibits a very unique behavior where

Diffraction by Open Ended Waveguide Edge = Modal Reflection (Waveguide mode conversion):

Not normal reflection --> Diffraction (deviation from refraction and reflection laws).

Dr. Vainshtein does not provide any physical explanation into why this unique reflection or turning (diffraction) of a "modal" wave (superposition of two traveling plane waves) occurs.

T.B.A. Senior (& P.L.E. Uslenghi) 1971 and 1972 papers contribute greatly to the discussion about the scientific existence of the "Cone of Diffracted Rays". In the 1972 paper [A12], they display a photograph that appears to show an "intersection of the edge-diffraction cone with the screen." They claim it provides experimental evidence of the validity of GTD (i.e., the law of edge diffraction). The experimental apparatus included a laser (source of incident visible light, lambda = 632.8 nm, 8-mW output power), razor blade (edge), screen, and photographic device.

The razor blade edge produces a cone of diffracted rays (in all directions away from the diffraction point). They claim that their 1971 paper [A12] provides the diffraction matrix for a wedge that supports the existence of a nonzero diffracted field.

From a conservation of energy point-of-view, how is the finite energy (intensity) of the incident laser beam distributed to produced a cone of an infinite number of diffracted rays whose intensity (energy) is finite and visible (obviously, the intersection of the diffracted light with the screen exposed the film and is visible to the eye).

The 1980 book of Max Born (1954 Nobelist in Physics) and E. Wolf [A21] contains a very important chapter on the "Rigorous Diffraction Theory."

The first rigorous solution was developed by Sommerfeld (1896) for a 2-D plane wave, infinitely thin perfectly conducting half plane. The solution is expressed exactly and simply in terms of the Fresnel integrals

Doctors Born and Wolf appear to be concern with and knowledgeable about the work of experimental verification of the derived formulas of Rigorous Diffraction Theory and Edged Obstacles. Max Born and E. Wolf are working with current and radiated waves.

The 1982 paper by A.E. Heins [A22] addresses the Sommerfeld's half-plane diffraction problem with the increase complexity of mixed boundary conditions.

The mathematical solution is based on solving a system of Wiener-Hopf integral equations. But no physical insight is provided to the nature of the problem or the solution.

S. Wang, a Chinese scientist, published a very important paper in 1995 [A27] that exposes the inconsistencies of the classical Theory of Diffraction [(Huygens and Fresnel), Kirchhoff & Sommerfeld]. And then he makes an amazing proposal to correct this theory of Diffraction after three hundred years:

Phase transformation: Half wave (wavelength/2), pi phase jump, not pi/2.

S. Anoknov, a Ukrainian scientist, publishes three important papers (1999, January and December 2007); in them he questions the validity of the current Theory of Diffraction.

He refers to the 1995 paper of S. Wang in the 1999 paper [A32]. He present the "Condition on a rim" (a rim does not radiate), a contradiction to Young's concept; and he states that "…there is not any reasonable explanation of the physical reality of geometrical and boundary waves, thus putting questions about the validity of the solution."

In the January 2007 paper [A41], he investigated the Young's boundary wave and his D wave by looking at the case of the Knife-edge diffraction of a paraxial light beam:

D wave (diffraction + dislocation): a Singular cylindrical wave;

"D field has a zero field on the boundary of a geometrical shadow" (in contrast to the boundary wave);

Young's boundary wave is similar to the D wave but they are not the same:

At the edge diffraction of a light beam, a real D wave arises (a Young's boundary wave almost coincides structurally); &

But at the geometrical boundary of shadow, they are structurally different in terms of continuity of wave front amplitude (D wave continuous, Young wave discontinuous). . And in his September 2007 paper [A42], Dr. Anoknov created a novel numerical experiment to study the physical mechanism of diffraction and the backward wave from an edge: the plane wave diffraction by a perfectly transparent half-plane, whose edge operates on the wave as a phase step. He found the following results:

d-wave (diffraction + dislocation) = edge dislocation wave = D-wave: singular wave;

Initial wave modulation (before its contact with transparency); and

Backward wave: can be detected at a long range from producing edge.

Paper's conclusion, a Study of the Physical Mechanism of Diffraction is needed to investigate the distinct role of Amplitude and Phase disturbances in the formation of a diffracted field.

Thus, Dr. Anoknov's work is challenging the current Theories of Diffraction and our fundamental understanding of the diffractive phenomena.

In the 2003 paper by A.P. Kiselev and V.P. Smyshlyaev [A38] that celebrates the 70th birthday of V.M Babich, we learn about the research of Dr. Babich at the St. Peterburg School of Diffraction in Russia. They involve Ray Theory in its real and complex versions:

High-frequency wave field, working in different Geometries (Riemannian in case of isotropic media; Finslerian in case of anisotropic media); and

Propagation of waves in curved structures; and Diffraction on/by cones and elastic wedges.

In the 2008 book by V.M. Babich (& M.A. Lyalinov and V.E. Grikurov) [A43], we learn about the Sommerfeld-Malyuzhinets Techniques in solving diffraction problems involving angular or wedge-like domains and various boundary conditions [either Perfect (Dirichlet or Newmann), or impedance].

This book's diffraction problems have been "problems with one propagation speed." In the case of "Multi-speed problems" (diffraction by a transparent or elastic wedge), these problems have no known exact solutions.

III) Physics and Mathematics of Singularity (Edge & EM field Singularity)

Next, let us focus on the main issue of this Addendum, i.e., the physical and mathematical representation of Edge Diffraction. The implementation of the current theory of diffraction (scattering) creates solutions (mostly approximate) that are based on rigorous Integral equations (IE) that contain singularities and methods to extract those singularities.

The following key ideas are involved in this subsection of discussion:

2-D or 3-D Edge: Maxwell equations in differential form & boundary conditions;

Finiteness of field values & energy radiating from diffracting obstacle;

Uniqueness of Diffraction (& Scattering) EM field;

Wave equation (derived from Maxwell Laws in differential equation form) of 2nd order derivatives in space and time: Finiteness of Field and Energy (Continuity of derivatives);

Physical & Mathematical representation of Edge Diffraction in the Creation of Rigorous Integral Equation (IE) with Singularities;

Singularities: a function of position of Observation point; and

IE Approximation with Extraction/Removal of Singularity using 2-D Complex plane techniques.

For the works on Singularity, we have the P.A.M Dirac's 1931 paper [A2], the Einstein et al 1935 paper [A3], the J. Elliott's 1951 paper [A5], and the B.L. Moiseiwitsch's 1977 book [A15].

Dr. Dirac (1933 Nobelist in Physics) investigated the idea of Quantised Singularities in an EM Field in quantum mechanical systems in his 1931 paper [A2]. He addresses the problem of the advancement of Theoretical Physics and the increasing abstraction of Mathematics by stating:

"…perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities…"

Dirac wrote about the existence of singularities in electromagnetic field in regards to wave functions (particle's motion) and to the end points of nodal lines that are the same for all wave functions.

In the famous 1935 paper by Professors A. Einstein (1921 Nobelist in Physics) and N. Rosen (a problem in the General Theory of Relativity) [A3], they addressed the problems of field representations containing singularities. In their work (page 73), they state the following concepts:

"For these reasons writers have occasionally noted the possibility that material particles might be considered as singularities of the field." and

"Every field theory, in our opinion, must therefore adhere to the fundamental principle that singularities of the field are to be excluded."

So their approach was to modify the equations in a natural way so that they are free of denominators (thus, the Singularities are removed). This natural exclusion of singularities occurs because a new physical realization of the problem is made.

J. Elliott's 1951 paper [A5] on Singular Integral Equations of Cauchy Type provides valuable solutions of Cauchy-type Integral Equations that involve a complex plane contour and is expressed as a Cauchy principal value (normalized by pi). Those solutions are used in the solving diffraction problems.

B.L. Moiseiwitsch's 1977 book [A15] on Integral Equations provides an in depth study of the Theory of Integral Equations (e.g., Riemann & Lebesgue integration, Singular integral equations, etc.)

For the works on Edge and EM Singularity, we have the C.J. Bouwkamp's 1946 paper [A4], the D.S. Jones' 1964 book [A10], the J. Meixner's 1972 paper [A13], the D.R. Wilton et al 1977 paper [A17], the E. Marx's 1990 paper [A25], the J.Van Bladel's 1991 book [A26], and the P. Cecchini et al 2001 paper [A36].

In the C.J. Bouwkamp's 1946 paper [A4], the problem of the sharp edge and diffraction is treated in regards to the issue of the Singularity behavior in a mathematical description versus energy finiteness of a field. Wave functions with finite derivatives up to a certain order may or may not be obtainable in a complete space (Edge singularity & Uniqueness theorem applicability). And if there is a Maxwell's field equations solution with a singularity, it must be configured so that no energy is actually generated in the apparent sources at those places.

In the D.S. Jones' 1964 book [A10], the "Diffraction by Obstacles with Edges" chapter addresses the importance of the Uniqueness Theorem (a unique solution of a diffraction problem must be found even when many solutions are possible given the presence of an edge of the diffracting obstacle). And the "Edge Condition" must be taken in consideration since no energy is radiated from the edge.

In the J. Meixner's 1972 paper [A13], the solution to the problems of diffraction at sharp edges of a diffracting obstacle is presented. In these cases, the EM field vector may become infinite (singular). Order of this singularity is subject to the so-called edge condition.

Edge condition: "states that the electromagnetic energy density must be integrable over any finite domain even if this domain contains singularities of the electromagnetic field. In other words, the electromagnetic energy in any finite domain must be finite."

In the D.R. Wilton and S. Govind's 1977 paper [A17], they address the problem of singular current in a conducting edge in problems of EM scattering and radiation. They report their investigation in the use of the Edge Condition in the moment method solution in the cases of singular edge behavior. Failure to incorporate the correct edge behavior can result in erroneous currents and anomalous behavior of the solution near edges.

In the E. Marx's 1990 paper [A25], the problem of Computed fields and Dielectric Wedge's Edge is addressed. There are problems with the definition of normal (n) pointing out from the wedge at the sharp edge and Singularity of the boundary functions. For his Method of Solution for dielectric obstacles with sharp edge, he states a "Caution": computed near field is sharp edge dependent; and the computed far fields and radar cross section do not depend strongly on sharp edge.

The J.Van Bladel's 1991 book [A26] addresses the subject of singular EM fields and Sources. In the chapter "Singularities at an Edge", the following key ideas are presented: Wedges, found in practical structures, mixed material and assumed infinite sharpness of an edge; the Mathematical nature of edge singularities; the study of the field near charged Strip as a model whose results may be extrapolated to other structures with sharp edge; and Edge energy density requirement. Also, edges encountered in actual devices are not perfectly sharp.

In the P. Cecchini (& F. Bardati and R. Ravanelli) 2001 paper [A36], the issue of Composed Wedges and Edge Singularity Extraction is discussed. The following points are made:

J. Meixner: field near edge can be locally expressed in a series; 1st term - singular behavior; and static solution (quasi-static limit in region whose dimension, compared with wavelength, are small);

Field singularities models for numerical computations: speed up convergence & decease memory utilization; and

The Edge Singularity Extraction Method requires knowledge of the field singularity order for each edge.

IV) Numerical Modeling of Scattering & Diffraction Problems

This subsection's topic of Numerical Modeling of Antenna and associated objects' interaction with an incident or transmitting electromagnetic wave (scattering & diffraction) is extremely important given the advent of very fast computational machines of today. Once the technical issues (questions) that are raised in subsection two and three are answered, computational modeling and simulation can be used to develop incredible (&powerful) new technology for applications in communications, imaging, and weapon (& counter weapon) applications.

We will consider the following interesting papers (& book) for this topic that have been written by P. Parhami et al (1977, [A18]), P. Parhami et al (1980, [A19]), T.B.A. Senior (1990, [A24]), J.H. Meloling et al (1997, [A30]), M. Levy (2000, [A35]), and A.V. Guglielmi (2010, [A44]).

In this important 1977 paper by P. Parhami (& Y. Rahmat-Samii and R. Mittra) [A18], a computational technique is presented for modeling (&simulating) a problem of scattering by a monopole antenna mounted on a finite conducting body (rectangular plate). This problem belongs to the problem class of arbitrary incident field, unknown induced current (plate & antenna), and scattered field determination. This technique is focused on speeding up the evaluation of the current on the structure by developing a numerical Green's function (using finite-difference method) and E-integral equation for the antenna. The issue of "Incorporation of edge condition" is addressed. The authors claim that "The procedure developed in this paper is capable of handling the case of a thin, vertical, wire antennas arbitrarily located on a rectangular ground plate."

In the 1980 paper by P. Parhami (&Y. Rahmat-Samii and R. Mittra) [A19], the issue of efficient evaluation of Sommerfeld Integrals is discussed. These integrals appear in problems dealing with radiating current elements existing over lossy ground (imperfect ground). This efficient evaluation of Sommerfeld Integrals is based on an approach using a computational (numerical) evaluation of the integration performed on the steepest descent path. Asymptotic and exact expressions are valid only when No Singularities are intercepted for the steepest descent path deformation. This evaluation technique can be in the computational analysis and synthesis of Electromagnetic Pulse (EMP) simulators and other weapon systems.

In the 1990 paper by T.B.A. Senior [A24], he addresses a very important problem of scattering by an obstacle whose boundary conditions must be approximated so that an analytical and numerical solution can be found. This important work is using an inhomogeneous dielectric body with a curved surface as its subject obstacle. But this obstacle has no edges. Dr. Senior is using the method developed by S.M Rytov (1940) and formulated by M.A. Leontovich (1940). He states that beyond the zeroth order approximation, the surface geometry affects the boundary conditions. Unfortunately, no geometrical visualization of the problem is present, nor any numerical results are presented.

In the 1997 paper by J.H. Meloling and R.J. Marhefka [A30], the problem of Diffraction by a Curved Edge is considered. An improvement to the Uniform Theory of Diffraction (UTD) is created so the high frequency field near a caustic is calculated correctly (a curved edge's caustic's near field is in error for a UTD-based calculation). A new curvature dependent diffraction coefficient has been created. This paper has an important ramification to antenna design: work toward satisfying a need to be able to accurately and quickly model a radiating antenna that is mounted on a finite perfectly conducting plate (determine Directive gain and a fast way to compute radiation patterns).

In the 2000 book by M. Levy [A35], the modern application of the Parabolic Equation (PE) method for electromagnetic wave propagation is discussed. The PE method that was developed by Leontovich and Fock (1940s) and Malyuzhinets (1940s) is an approximation of the wave equation for the cases of paraxial wave propagation. Levy states that the "Russian workers pioneered the idea of simplifying the wave equation for certain types of radiowave propagation problems and solved a number of these problems of special functions." The advent of digital computers have brought back the possible applications of the PE approximation technique: determine the numerical solution rather than closed-form expressions. Levy presents the Vector PE method for the modeling of the Forward bistatic RCS (radar cross section) of the F-117.

In the 2010 paper by A.V. Guglielmi [A44], the approximate boundary conditions are discussed in regards to the Leontovich impedance boundary condition (under the guise of the 70th Anniversary of the Leontovich boundary condition). M.A. Leontovich was the key developer of the idea of the asymptotical theory of the skin-effect but S.M. Rytov wrote a Russian paper in 1940 about it. The computational ability to model and simulated boundary conditions of non-perfectly conducting (finite conductivity, dielectric, or mixed material) curved edged bodies is extremely valuable for developing many advanced applications of antennae and weapons of directed energy & stealth.

The ideas and results found in this subsection's described papers (& book) have extremely important ramifications for modeling & simulating the following physical structures:

1) Carbon nanotube antenna array (Transmit/Receive) and its associated supporting object for possible applications such as:

i) Normal transmit/Receive Antenna for Communications, and

ii) Directed energy weapons; and

2) Biological-base antenna array & its associated supporting object: life-form (Oriental Hornet model).

Note there are many Antenna-base structural possibilities: metal-metal, metal-dielectric, and organometallic-dielectric.

V) Important Scattering/Diffraction Objects with an Edge

Finally, for our last topic, we will look at significant real world diffracting (scattering) objects with an Edge (or Edges). The problem of analytical determining the actual near and far electromagnetic (EM) fields (diffracting & edge) of these diffracting (scattering) objects has not been solved. Only approximate results have been determined at this time.

Interesting papers for this topic have been written by A.D. Rawlins (1977, [A16), C. Joo et al (1980, [A20]), L. Knockaert et al (1997, [A29]), T.L. Zinenko et al (1998, [A31]), and G.A. Kalinchenko et al (2002, [A37]).

One must note that the dielectric wedge is very different than a perfectly conducting wedge, i.e., it possesses a transmitting & reflecting capacity instead of exclusively reflecting capacity. Also, it is important to be able to analytically describe the behavior of the electromagnetic field near the vertex (tip) of dielectric wedge.

In the 1977 paper by A.D. Rawlins [A16], approximate expressions for the solution of the problem of EM Diffraction by an arbitrarily angled Dielectric Wedge are presented. For a line source incident field, the 3-D polarization problem is reduced to a 2-D scalar problem since the polarization vector (E or H vector) is parallel to the edge.

The solution is in the form of Fredholm integral equation: solution to this equation is made by the use of the standard perturbation technique (Rayleigh-Gans-Born approximation). The perturbation parameter is dependent on the refractive index of the dielectric wedge. The Vertex field obeys the "Edge condition" (by D.S. Jones, 1964). And the Radiation condition is taken into consideration. A Green's function singularity (actually a double pole) exists.

For the case of the diffraction of an E-polarized line source by a right angled dielectric wedge, far field results via graphical plots are provided in the paper. Rawlins claims that the technique can be extended beyond the 1 <= n (index of refraction) <= (2)^1/2. But he says that "it is difficult to prove this rigorously."

In the 1980 paper by C. Joo (&, J. Ra, and S. Shin) [A20], the results on the derivation of the asymptotic (approximate valued) diffracted fields in the case of a Right-angled Dielectric Wedge and incident plane wave are presented.

The Dual integral equation is first formulated for this two wave numbers and regions EM problem.

Physical Optics (PO) approximation is made and then a correction is performed where the PO field consists of reflected plane waves; refracted plane waves; and an edge-diffracted field (cylindrical waves). Corrected edge-diffracted field is made (Sommerfeld integral along the steepest descent path).

A new Dual integral equation is arrived at: Virtual source in free space and in unbounded dielectric; a arbitrary multipole line source is now assumed to exist at the origin; and a Sommerfeld integral is formed and a dual series equation is created.

The total field is the sum of the reflected & refracted plane waves and the corrected edge-diffracted waves.

The dual series is solved numerically by multipole expansion coefficients truncation.

Unfortunately, the paper's graphical results figure is difficult to understand.

In the 1997 paper by L. Knockaert (& F. Olyslager and D. De Zutter) [A29], the problem of scattering by a Diaphanous Wedge is considered. This Wedge type is characterized by identical wave numbers inside and outside of the wedge (εinμin = (εoutμout), i.e., an isorefractive body.

The results are obtained from an integral equation for the wedge fields on the wedge. This equation is solved by the Mellin transform for the static case and the Kantorovich-Lebedev transform for the dynamic case. Results are obtained for the application of the Line Source Excitation and the Plane wave excitation.

The authors claim that the Problem of Scattering by a Diaphanous wedge has been solved. Note that only numerical results are presented but no experimental results are presented.

Further, in regards to the general wedge problem, matching conditions for the two different Kantorovich-Lebdev transforms are "almost impossible to meet, at least analytically."

In the 1998 paper by T.I. Zinenko (& A.I. Nosich and Y. Okuno) [A31], the problem of the plane wave scattering by resistive and the dielectric strip periodic gratings is addressed. This problem belongs to the class of Problems of the EM Wave Scattering by a flat-strip periodic grating with zero thickness.

The paper discusses the idea of Regularization: analytical inversion of the static part of the full-wave integral equation (IE) and the

Original first-kind IE --> Fredholm second kind one with a smooth kernel.

The Analytical-regularization-base algorithm is applied for this paper's method of solution.

Duel Series equations (DSE) are created for the Resistive Strip grating.

DSE regularized matrix equations are created for the Dielectric Strip Grating

Numerical (computational) results are presented for the cases of Resistive Strip Grating and the Dielectric Strip Grating. Accuracy is limited only by the precision of the computer used.

But no comparison is shown between the paper's numerical results and experimental results.

Concluding the paper, the authors claim they have developed "a simple but numerically exact algorithm for computing the transmission, reflection, and absorption characteristics of an E- or H-polarized plane wave incident on a resistive or dielectric flat-strip periodic grating."

The 2002 paper by G.A. Kalinchenko (& A.M. Lerer and A.A. Yachmenov) [A37] presents an important work in Russia about the Mathematical Simulation of Impedance Diffraction Gratings. This work (Wave diffraction modeling & simulation) deals with the calculation of Eigenwaves of Periodic Impedance diffracting grating, impedance dielectric strips.

The authors' approach is based on the approximate solution of a Rigorous Integral Equation (IE) (developed in a 2001 paper by Kalinchenko et al):

Approximate boundary conditions (ABC) -> Approximate Integral Equation; & Impedance Boundary conditions (IBC): applied in this work;

Half-analytical solution to paper's problem IE [described in L.A. Vainshtein's 1966 Russian book: The Theory of Diffraction and Method Factorization];

IE for 2-D structure -> 1-D IE;

Current, J(x) has a singularity on the border of metal strip, avoid by change of variables;

IE Singularity Existence and Extraction --> IE of 2nd Kind with Smooth kernel;

The simplified IE is solved numerically by the Collocation & Galerkin Methods and Formula of quadrangles is used.

The author's results and conclusion is that their method has good convergence; Wave diffraction by high number of strips (up to 100) can be investigated.

"Dispersion characteristics, the windows of transparency and phase synchronism conditions for first and second harmonics (Fig 1, 2)" are supposedly presented:

Table and Figure 1: presentation of results? and

Figure 2: |S11|, reflection coefficient results vs. wavelength & N (number of strips).

This paper's presentation of results is not clear: difficult to understand, maybe, Russian to English translation difficulties. And no experimental results are presented for comparison purposes.

But this paper is very important because of the Need to computationally model and simulate a Diffracting Periodic Grating, a very important canonical scattering structure. It combines the ideas of diffraction grating, wave reflection, impedance interface, approximate boundary conditions, edge boundary singularity and its removal from diffraction integral and the work of L.A. Vainshtein.

Summary of Wave Processes Paper and Book References: Lead Author (Co-Author)

SOVIET SCIENTISTS:

L.A Vainshtein: A8, A9
```P.Ya. Ufimtsev (USSR): A11
P.Ya. Ufimtsev (U.S): A34 (B. Khayatian and Y. Rahmat-Samii)
P.Ya. Ufimtsev (U.S): A39, A40
```
Yu.A. Kravtsov: A23 (P.Ya. Ufimtsev)

T.L. Zinenko (Ukraine, Japan): A31 (A.I. Nosich and Y. Okuno)

S. Anokhov (Ukraine): A32, A41, A42

G.A. Kalinchenko (Russia): A37 (A.M. Lerer and A.A. Yachmenov)

A.P. Kiselev (Russia): A38 (V.P. Smyshlyaev)

A.V. Guglielmi (Russia): A44

AMERICAN SCIENTISTS:

A. Einstein: A3 (N. Rosen)

J. Elliott: A5

J.B. Keller: A7

T.B.A. Senior: A12 (P.L.E. Uslenghi), A14 (P.L.E. Uslenghi), A24

D.R. Wilton: A17 (S. Govind)

P. Parhami: A18 (Y. Rahmat-Samii and R. Mittra), A19 (Y. Rahmat-Samii and R. Mittra)

A.E. Heins: A22

E. Marx: A25

J.H. Meloling: A30 (R.J. Marhefka)

B. Khayatian: A33 (Y. Rahmat-Samii and P.Ya. Ufimtsev)

EUROPEAN SCIENTISTS:

H.M. MacDonald (England): A1

P.A.M. Dirac (England): A2

C.J. Bouwkamp (Netherlands): A4

J. Meixner (Germany): A13

A.D. Rawlins (Scotland): A16

L. Knockaert (Belgium): A29 (F. Olyslager and D. De Zutter)

P. Cecchini (Italy): A36 (F. Bardati, and R. Ravanelli)

OTHER:

C. Joo (South Korea): A20 (J. Ra, and S. Shin)

S. Wang (Chinese Scientist): A27

S.H. Lui (Hong Kong): A28

MATHEMATICS and ELECTROMAGNETICS/OPTICS Books:

A. Sommerfeld (Germany): A6

D.S. Jones (England): A10

B.L. Moiseiwitsch (United Kingdom, Belfast): A15

M. Born (Germany: A21 (E. Wolf)

J. Van Bladel (Belgium): A26

M. Levy (United Kingdom): A35

V.M. Babich (Russia): A43 (M.A. Lyalinov and V.E. Grikurov)

CONCLUSIONS

This Addendum has presented the results of a study of the Physics and Mathematics of Edge Diffraction and Singularity and of the computational modeling of this phenomenon. The motivation for this Addendum should be self-evident by now if the reader, you, has read the "Research on Problems of Diffraction, Scattering, and Propagation of Waves" paper before reading this Addendum.

One of the basic problems of the wave theory is the physical, analytical, and computational description of the diffraction (& scattering) by edged obstacles.

As Russian mathematician A.S. Kirpichnikova has stated in her 2001 paper : "It is hard to mention an application area that is not related to problems of diffraction and propagation of waves."

It is amazing that after 300 years the problems of Diffraction (& Scattering) due to Wave interaction with a physical body still has not been solved in an exact analytical form (closed form). Many brilliant scientists, mathematicians, and engineers (e.g., H.M. MacDonald, P.A.M Dirac, Albert Einstein, Max Born, A. Sommerfeld, V.A. Fock, M.A. Leontovich, G.D. Malyuzhinets, L.A. Vainshtein, V.M. Babich, J.B. Keller, T.B.A. Senior, R. Mittra, etc.) have invested many years to develop the underlying Physics and Mathematics of Wave processes.

But why is the Problems of Diffraction and Scattering so difficult to solve. A. Sommerfeld in Optics, his 1954 book [A6], stated "Any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction is called diffraction." Note that Scattering is a more generalized phenomenon of wave-body interaction. As a result of Diffraction (&Scattering), the phenomenon of Shadow can exist in many complex shapes.

Key to these problems is how a wave actually interacts with a diffracting body with or without an edge (s), the geometry of the body and its surrounding space, and the material properties of the body. We simply don't understanding the underlying processes and the mathematics used to solve these problems fails us. So we use physical and mathematical approximations and tricks to solve these problems.

Let us summarize some of the key discoveries we have found for each Discussion topic:

I) P.Ya. Ufimtsev and the Physical Theory of Diffraction (PTD)

The validity of PTD is still an open question (e.g., PTD issues & fixes in 2000s paper). The reason for the Dr. Ufimtsev's 1957 & 1958 diffraction papers (4) and the1962 PTD monograph publication in the "Open" Soviet literature is still unknown. And L.A. Vainshtein played an important role in Ufimtsev's research and publication of work on wave processes.

We now know that Ufimtsev has worked on four Diffraction methodologies: PTD, Parabolic Equation Method, "Virtual Ray" (Virtual Field), and Improved PTD.

II) The Problem of Diffraction (Scattering) and Shadow

H.M. MacDonald (1915 paper) developed an approach of deriving an approximate analytical result for the problem of diffraction of sound waves by a wedge. This approach is based on the commonality of two harmonic functions series (one for diffraction and the other for potential) to determine the appropriate constants of the diffraction series. In 1915, did the technology exist to measure a wedge's acoustic diffraction field? How do we know that the MacDonald's approach is indeed valid in general?

A. Sommerfeld's Optics (1954 book) does not provide a complete mathematical derivation that supports the existence (structure) of the "Cone of Diffracted Rays" nor does it address the issue of the energy distribution in such a light cone.

The Edge Wave is "a kind of reflection" and the "luminous edge" of a diffracting structure is not real according to Sommerfeld.

It is important to understand the Rigorous Definition of a Problem of Diffraction from a physical perspective, a mathematical perspective, and their mutual limitations so that one can derive the correct diffraction solution, if possible.

Singularities in Diffraction fields (i.e., infinities of field values) do not exist physically. They are problems of constructs of mathematical models of Diffraction.

This very important fact about field singularities implies that the mathematical or physical models of the Diffraction phenomena do not actually describe the true phenomenon of Diffraction in Nature. Further, the generally accepted Theories of Diffraction may not be correct, too.

Keller (1957 paper) claims that Sommerfeld's 1954 book Optics provides key proof to support his GTD work and the necessary associated Diffraction Coefficients.

Dr. Vainshtein (1963 & 1964 papers) does not provide any physical explanation into why the unique reflection or turning (diffraction) of a "modal" wave (superposition of two traveling plane waves) occurs in an Open Resonator or Open Waveguide.

T.B.A. Senior (& P.L.E. Uslenghi) published a photograph (1972) that appears to show an "intersection of the edge-diffraction cone with the screen." They claim it provides experimental evidence of the validity of GTD (i.e., the law of edge diffraction).

Chinese scientist S. Wang (1995 paper) exposes the inconsistencies of the classical Theory of Diffraction [(Huygens and Fresnel), Kirchhoff & Sommerfeld]. And then he makes an amazing proposal to correct this theory of Diffraction after three hundred years: Phase transformation: Half wave (wavelength/2), pi phase jump, not pi/2.

S. Anoknov, a Ukrainian scientist, questions the validity of the current Theory of Diffraction.

Dr. Anoknov (1999 paper) references S. Wang 1995 paper. He present the "Condition on a rim" (a rim does not radiate), a contradiction to Young's concept; and he states that "…there is not any reasonable explanation of the physical reality of geometrical and boundary waves, thus putting questions about the validity of the solution."

He (January 2007 paper) investigated the Young's boundary wave and the D wave by looking at the case of the Knife-edge diffraction of a paraxial light beam. He found that the real D wave (diffraction + dislocation), a Singular cylindrical wave is not the same as the Young's boundary wave.

Dr. Anoknov (September 2007 paper) created a novel numerical experiment to study the physical mechanism of diffraction and the backward wave from an edge: the plane wave diffraction by a perfectly transparent half-plane, whose edge operates on the wave as a phase step. He found the following results: d-wave (diffraction + dislocation) = edge dislocation wave = D-wave: singular wave; Initial wave modulation (before its contact with transparency); and Backward wave: can be detected at a long range from producing edge.

The paper's conclusion, a Study of the Physical Mechanism of Diffraction is needed to investigate the distinct role of Amplitude and Phase disturbances in the formation of a diffracted field.

Thus, Dr. Anoknov's work is challenging the current Theories of Diffraction and our fundamental understanding of the diffractive phenomena.

III) Physics and Mathematics of Singularity [Edge & Electromagnetic (EM) Field Singularity]

Dr. Dirac (1931 paper) investigated the idea of Quantised Singularities in an EM Field in quantum mechanical systems. He addressed the problem of the advancement of Theoretical Physics and the increasing abstraction of Mathematics by stating: "…perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities…" Dirac wrote about the existence of singularities in electromagnetic field in regards to wave functions (particle's motion) and to the end points of nodal lines that are the same for all wave functions.

Professors A. Einstein and N. Rosen (1935 paper) addressed the problems of field representations containing singularities. On page 73, they state the following concepts:

"For these reasons writers have occasionally noted the possibility that material particles might be considered as singularities of the field."

And "Every field theory, in our opinion, must therefore adhere to the fundamental principle that singularities of the field are to be excluded."

So their approach was to modify the equations in a natural way so that they are free of denominators (thus, the Singularities are removed). This natural exclusion of singularities occurs because a new physical realization of the problem is made.

C.J. Bouwkamp (1946 paper) addresses the problem of the sharp edge and diffraction in regards to the issue of the Singularity behavior in a mathematical description versus energy finiteness of a field. Wave functions with finite derivatives up to a certain order may or may not be obtainable in a complete space (Edge singularity & Uniqueness theorem applicability). And if there is a Maxwell's field equations solution with a singularity, it must be configured so that no energy is actually generated in the apparent sources at those places.

D.S. Jones (1964 book) addresses the importance of the Uniqueness Theorem (a unique solution of a diffraction problem must be found even when many solutions are possible given the presence of an edge of the diffracting obstacle). And the "Edge Condition" must be taken in consideration since no energy is radiated from the edge.

J. Meixner (1972 paper) presents the solution to the problems of diffraction at sharp edges of a diffracting obstacle. In these cases, the EM field vector may become infinite (singular). Order of this singularity is subject to the so-called edge condition.

Edge condition: "states that the electromagnetic energy density must be integrable over any finite domain even if this domain contains singularities of the electromagnetic field. In other words, the electromagnetic energy in any finite domain must be finite."

D.R. Wilton and S. Govind (1977 paper) address the problem of singular current in a conducting edge in problems of EM scattering and radiation. They report their investigation in the use of the Edge Condition in the moment method solution in the cases of singular edge behavior. Failure to incorporate the correct edge behavior can result in erroneous currents and anomalous behavior of the solution near edges.

E. Marx (1990 paper) addresses the problem of Computed fields and Dielectric Wedge's Edge. There are problems with the definition of normal (n) pointing out from the wedge at the sharp edge and Singularity of the boundary functions. For his Method of Solution for dielectric obstacles with sharp edge, he has a "Caution": computed near field is sharp edge dependent; and the computed far fields and radar cross section do not depend strongly on sharp edge.

J.Van Bladel (1991 book) addresses the subject of singular EM fields and Sources. The key ideas are as follows: Wedges, found in practical structures, mixed material, and assumed infinite sharpness of an edge; the Mathematical nature of edge singularities; the study of the field near charged Strip as a model whose results may be extrapolated to other structures with sharp edge; and Edge energy density requirement. Edges encountered in actual devices are not perfectly sharp.

P. Cecchini (& F. Bardati and R. Ravanelli) (2001 paper) discusses the issue of Composed Wedges and Edge Singularity Extraction. The following points are made:

J. Meixner: field near edge can be locally expressed in a series; 1st term - singular behavior; and static solution (quasi-static limit in region whose dimension, compared with wavelength, are small);

Field singularities models for numerical computations: speed up convergence & decease memory utilization; and

The Edge Singularity Extraction Method requires knowledge of the field singularity order for each edge.

IV) Numerical Modeling of Scattering & Diffraction Problems

P. Parhami (& Y. Rahmat-Samii and R. Mittra) (1977 paper) presents a computational technique for modeling (&simulating) a problem of scattering by a monopole antenna mounted on a finite conducting body (rectangular plate). This problem belongs to the problem class of arbitrary incident field, unknown induced current (plate & antenna), and scattered field determination. This technique is focused on speeding up the evaluation of the current on the structure by developing a numerical Green's function (using finite-difference method) and E-integral equation for the antenna. The issue of "Incorporation of edge condition" is addressed. The authors claim that "The procedure developed in this paper is capable of handling the case of a thin, vertical, wire antennas arbitrarily located on a rectangular ground plate."

P. Parhami (&Y. Rahmat-Samii and R. Mittra) (1980 paper) discuss the issue of efficient evaluation of Sommerfeld Integrals. These integrals appear in problems dealing with radiating current elements existing over lossy ground (imperfect ground). This efficient evaluation of Sommerfeld Integrals is based on an approach using a computational (numerical) evaluation of the integration performed on the steepest descent path. Asymptotic and exact expressions are valid only when No Singularities are intercepted for the steepest descent path deformation. This evaluation technique can be in the computational analysis and synthesis of Electromagnetic Pulse (EMP) simulators and other weapon systems.

T.B.A. Senior (1990 paper) addresses the problem of scattering by an obstacle whose boundary conditions must be approximated so that an analytical and numerical solution can be found. This important work is using an inhomogeneous dielectric body with a curved surface as its subject obstacle. But this obstacle has no edges. Dr. Senior is using the method developed by S.M Rytov (1940) and formulated by M.A. Leontovich (1940). He states that beyond the zeroth order approximation, the surface geometry affects the boundary conditions.

J.H. Meloling and R.J. Marhefka (1997 paper) consider the problem of Diffraction by a Curved Edge. An improvement to the Uniform Theory of Diffraction (UTD) is created so the high frequency field near a caustic is calculated correctly. A new curvature dependent diffraction coefficient has been created. This paper has an important ramification to antenna design: work toward satisfying a need to be able to accurately and quickly model a radiating antenna that is mounted on a finite perfectly conducting plate (determine Directive gain and a fast way to compute radiation patterns).

M. Levy (2000 book) discusses the modern application of the Parabolic Equation (PE) method for electromagnetic wave propagation is discussed. The advent of digital computers have brought back the possible applications of the PE approximation technique: determine the numerical solution rather than closed-form expressions. Levy presents the Vector PE method for the modeling of the Forward bistatic RCS (radar cross section) of the F-117.

A.V. Guglielmi (2010 paper) discusses the approximate boundary conditions in regards to the Leontovich impedance boundary condition. M.A. Leontovich was the key developer of the idea of the asymptotical theory of the skin-effect but S.M. Rytov wrote a Russian paper in 1940 about it. Again, the computational ability to model and simulated boundary conditions of non-perfectly conducting (finite conductivity, dielectric, or mixed material) curved edged bodies is extremely valuable for developing advanced applications of antennae and weapons of directed energy and stealth.

V) Important Scattering/Diffraction Objects with an Edge

A.D. Rawlins (1977 paper) presents approximate expressions for the solution of the problem of EM Diffraction by an arbitrarily angled Dielectric Wedge. For a line source incident field, the 3-D polarization problem is reduced to a 2-D scalar problem since the polarization vector (E or H vector) is parallel to the edge. The solution is in the form of Fredholm integral equation: solution to this equation is made by the use of the standard perturbation technique (Rayleigh-Gans-Born approximation). Rawlins claims that the technique can be extended beyond the 1 <= n (index of refraction) <= (2)^1/2. But he says that "it is difficult to prove this rigorously."

C. Joo (&, J. Ra, and S. Shin) (1980 paper) presents their results on the derivation of the asymptotic (approximate valued) diffracted fields in the case of a Right-angled Dielectric Wedge and incident plane wave. A Dual integral equation (after Physical Optics approximation is made and then a correction is performed) is formulated for this two wave numbers and regions EM problem. The total field is the sum of the reflected & refracted plane waves and the corrected edge-diffracted waves. The dual series is solved numerically by multipole expansion coefficients truncation. Unfortunately, the paper's graphical results figure is difficult to understand.

L. Knockaert (& F. Olyslager and D. De Zutter) (1997 paper) consider the problem of scattering by a Diaphanous, i.e., an isorefractive body. The results are obtained from an integral equation for the wedge fields on the wedge. This equation is solved by the Mellin transform for the static case and the Kantorovich-Lebedev transform for the dynamic case. The authors claim that the Problem of Scattering by a Diaphanous wedge has been solved. Further, in regards to the general wedge problem, matching conditions for the two different Kantorovich-Lebdev transforms are "almost impossible to meet, at least analytically."

T.I. Zinenko (& A.I. Nosich and Y. Okuno) (1998 paper) addresses the problem of the EM plane wave scattering by resistive and the dielectric flat-strip periodic gratings with zero thickness. Concluding the paper, the authors claim they have developed "a simple but numerically exact algorithm for computing the transmission, reflection, and absorption characteristics of an E- or H-polarized plane wave incident on a resistive or dielectric flat-strip periodic grating."

G.A. Kalinchenko (& A.M. Lerer and A.A. Yachmenov) (2002 paper) presents an important work in Russia about the Mathematical Simulation of Impedance Diffraction Gratings. This work (Wave diffraction modeling & simulation) deals with the calculation of Eigenwaves of Periodic Impedance diffracting grating, impedance dielectric strips. This paper is very important because of the Need to computationally model and simulate a Diffracting Periodic Grating, a very important canonical scattering structure. It combines the ideas of diffraction grating, wave reflection, impedance interface, approximate boundary conditions, edge boundary singularity and its removal from diffraction integral, and the work of L.A. Vainshtein.

As a result of this study on the Physics and Mathematics of Edge Diffraction and Singularity, interesting (& important) questions come to mind.

The work of A. Sommerfield, S. Wang, and S. Anokhov (& others) questions the validity of the current Theory (ies) of Diffraction and Scattering. Thus, a drastic revision of the fundamental concepts or new ideas of Diffraction and Scattering needs to be made or discovered, respectively. This revision to produce a new Theory of Diffraction and Scattering must be able to provide solutions to the Problem of Edge Diffraction (& other wave phenomena) that are accurate (& its solutions are physical and mathematical rigorous) and not approximations.

And new Mathematical abstractions and formulations need to be discovered so that the Mathematics of Asymptotic Solution (i.e., high-frequency, far field, and paraxial approximations) and Singularity Extraction are no longer used in developing solutions to the Problems of Diffraction and Scattering.

If this question (s) can not be answer, the advancement of the field of Computational Electromagnetics into novel biomedical, communications (& radar), and directed energy and stealth weapons applications will be thwart. Also, this situation will be true for Acoustics and other elastic wave applications.

So let us close by asking (considering) some additional questions as follows:

What is the physical & mathematical definition of a 2-D edge (discontinuity) and of a 3-D edge (discontinuity) from a Classical and Quantum Mechanical perspective?

What is the unit normal vector n and the unit tangent vector t to a 3-D Edge?

What is the correct physical and mathematical description of a plane, cylindrical, or spherical wave front interacting with an arbitrary 2-D (3-D) edge (discontinuity) (or infinitely thin edge, finite thin edge) of a perfectly conducting, finite conducting, or dielectric body?

Do real edges actually radiate?

What is the correct physical model for a 3-D Edge and for a non-stationary 3-D Edge?

Are Edge Singularities real (Do they have physical existence)? Or are they just mathematical artifacts?

In the 3-D Diffraction and Mixed boundary conditions world - do real Singularities exist?

And, finally, what physical and computational (numeric) experiments would you create to answer these questions?

ADDENDUM LIST OF REFERENCES

Note: UCI SL, University of California Irvine Science Library.

[A1] H.M. MacDonald, "A Class of Diffraction Problems", Proceedings of the London Mathematical Society, Vol. s2-14, Issue 1, pp. 410-427, 1915.

[A2] P.A.M. Dirac, "Quantised Singularities in the Electromagnetic Field", Proceedings of the Royal Society London A, Vol. 133, pp. 60-72, 1931.

[A3] A. Einstein and N. Rosen, "The Particle Problem in the General Theory of Relativity", Physical Review, Vol. 48, pp. 73-77, July 1, 1935.

[A4] C.J. Bouwkamp, "A note on Singularities occurring at sharp edges in Electromagnetic Diffraction Theory", Physica, Vol. 12, No. 7, pp. 467-474, 1946.

[A5] J. Elliott, "On Some Singular Integral Equations of the Cauchy Type", The Annals of Mathematics, Vol. 54, No. 2, pp. 349-370, September 1951.

[A6] A. Sommerfeld, Optics, Academic Press Inc., New York, N.Y., 1954.

[A7] J.B. Keller, "Diffraction by an Aperture", Journal of Applied Physics, Vol. 28, No. 1, pp. 426-444, April 1957.

[A8] L.A. Vainshtein, "Open Resonators for Lasers", Soviet Physics-Technical Physics, Vol. 17, No. 3, pp. 709-719, September 1963.

[A9] L.A. Vainshtein, "Diffraction in Open Resonators and Open Waveguides with Plane Mirrors", Soviet Physics-Technical Physics, Vol. 9, No. 2, pp. 157-165, August 1964.

[A10] D.S. Jones, The Theory of Electromagnetism, New York: Pergamon Press, 1964, UCI SL: QC670 .J63.

[A11] P.Ya. Ufimtsev, "Current Waves in a Thin Wire and on a Ribbon", USSR Computational Mathematics and Mathematical Physics, Vol. 8, Issue 6, pp. 250-260, 1968.

[A12] T.B.A. Senior and P. L. E. Uslenghi, "High-frequency backscattering from a finite cone", Radio Science, Vol. 6, No. 3, pp. 393-406, March 1971.

[A13] J. Meixner, "The Behavior of Electromagnetic Fields at Edges, IEEE Transactions on Antennas and Propagation, Vol. AP-20, No. 4, pp. 442-446, July 1972.

[A14] T.B.A. Senior and P. L. E. Uslenghi, "Experimental Detection of the Edge-Diffraction Cone", Proceedings of the IEEE, Vol. 60, No. 11, pp. 1448, November 1972.

[A15] B.L. Moiseiwitsch, Integral equations, Longman Inc., New York, 1977.

[A16] A.D. Rawlins, "Diffraction by a Dielectric Wedge", Journal of the Institute of Mathematics and Its Applications, Vol. 19, pp. 231-279, 1977.

[A17] D.R. Wilton, and S. Govind, "Incorporation of Edge Conditions in Moment Method Solutions", IEEE Transactions on Antennas and Propagation, Vol. AP-25, No. 6, pp. 845-850, 1977.

[A18] P. Parhami, Y. Rahmat-Samii, and R. Mittra, "Technique for calculating the radiation and scattering characteristics of antennas mounted on a finite ground plane", Proceedings of the Institution of Electrical Engineers, Vol. 124, No. 11, pp. 1009-1016, November 1977.

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[A20] C. Joo, J. Ra, and S. Shin, "Edge Diffraction by Right-Angled Dielectric Wedge", Electronics Letters, Vol. 16, No. 24, pp. 934-935, November 20, 1980.

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 Addendum to Research on Problems of Wave Processes: Diffractive Edge and Singularity, Facts and Ideas Derived from Analysis Part III