NOTE 1: You have arrived at the __INVESTIGATION: ANALYSIS SECTION__ of this website.

This section is Under-Edit or Analysis Addition if necessary: Construction began on December 7, 2010 and was finished on April 28, 2011.

NOTE 2: This section contains the relevant facts and associated ideas that have been derived from each reference that is used in this investigation. They are presented in Summary form using the following SECTION GUIDE:

**Item Number. Reference Description:** Derived Fact (s) from the Reference Document **[Reference No.]** Derived Ideas.

NOTE 3: GTD: Geometrical Theory of Diffraction; PTD: Physical Theory of Diffraction

**A23. Y.A. Kravtso and P.Ya. Ufimtsev's 1989 paper on Virtual field Actualization:**Fields of Diffracted Waves : Multiple representations of this field

Physical interpretable: individual components directly measurable with the aid of some physical device

Field representation - superposition of "virtual rays"

L.A. Vainshtein's 1982 & 1986 papers in Russian

"Spectrum" of Virtual Waves"

Problem of actualization (detection of virtual waves)

"Actualization of Virtual Waves of the Ray Type": Nonuniform medium

"Virtual Field Measurement with a Curvilinear Extended Antenna":

Point probe: amplitude & phase

Wave uncertainty principle

Theory of quantum mechanical measurements

"Examples":

1. "Plane Wave Actualization with a Linear Antenna";

2. "Huygen's Source Actualization":

"Scattering of the initial wave field by various obstacles can also be considered
as detecting a certain part of virtual waves "hidden" in the continual integrals.";

3. "Detecting Virtual Fields Related to Diffraction Rays"; and
4. "Detection of a Virtual Field in the Wedge Diffraction Problem": Knife-edge Wedge.

"Discussion":

Procedure of Actualization is equivalent to an inversion of the initial field representation.

Important to Measure Virtual fields of ray type - diffraction factors are not yet calculated:

Experiment studies - "developing a geometric theory of diffraction on bodies of a complicated shape"

Measurement of virtual fields of the caustic type: theory of catastrophes

"Acknowledgements":

"The authors wish to express their gratitude to L.A. Vainshtein for his critical comments on this paper.
Ufimtsev:

Born in Altai region, USSR in 1931;

Graduated from Odessa State University (Physical & Mathematical Department), Odessa, in 1954;

Received Candidate of science degree in radio engineering (Central Research Institute of Ministry of the Radio Industry, Moscow) in 1959; and

Received Doctor of science degree in theoretical and mathematical physics (Leningrad State University, Leningrad) in 1970.

**[A23]**Altai region, USSR: located in Southwestern Siberia, Russia.

Ufimtsev - Undergraduate Physics and Mathematics Education from a Ukrainian university & Ph.D. equivalent from Leningrad State University

It is important to realize that L.A. Vainshtein was still involved with Ufimtsev (Acknowledgement of Dr. Vainshtein's contribution to this 1989 paper). Note Dr. Vainshtein died in 1989.

**A24. T.B.A. Senior's June 1990 report on Approximate Boundary Conditions:**M.A. Leontovich, 1948: Impedance boundary condition, properties of scatterer

Scattering Problem Simplification: More general boundary conditions proposed by Senior & Volakis (1989)

Using method by S.M. Rytov (1940) to examine effects of surface curvature & material variations:

Applied to the formulation developed by Leontovich

Inhomogeneous dielectric body (lossy):

complex permittivity ε and permeability μ

N complex refractive index of dielectric where |N| is large

N = v/q where v is a function of position & q is a small parameter

Using Geometrical Optics as a guide: complex exponentials with constants

**A** &

**B****A** &

**B** can be expressed as a power series in q

Orthogonal Curvilinear Coordinates

__Lossy Body's Surface__ is a

__Coordinate Surface__ in an orthogonal curvilinear coordinate system.

Zeroth Order Solution (power series terms independent of q):

standard impedance boundary condition

Analogous to eikonal equation of Geometrical optics

First Order Solution (involving q terms):

Discovery of error in Leontovich 1948 formulation

Only normal variation of impedance has any effect

Anisotropy is a consequence of the curvature of surface S

Second Order Solution (terms involving q square):

Complex derivation

Examination of Results:

"the form of the boundary condition is independent of any variation in the material properties of the surface"

"Any lateral variation of the properties is taken care of by the gradient operation, and a variation in the normal direction affects only the second order terms in the expression for the effective surface impedance" -> material is homogeneous

Final Point (sequence of boundary conditions):

Zeroth order condition involves only a single scalar parameter, independent of any material inhomogeneity and the shape of the surface

First order condition involves a parameter that may be a function of the variation of the material properties in the normal direction, as well as the curvature of the surface

Second order condition involves two parameters and all material variations are in the normal direction & embedded in one of these parameters

Conclusions for Approximate Boundary Condition Derivations:

"Treatment is based on an asymptotic expansion of the interior fields in powers of 1/N where N is the complex refractive index of the dielectric"

Zeroth order condition (in 1/N): Leontovich impedance condition, same for any surface (planar or curved) is independent of any variation of the material properties

Higher order condition (in 1/N): surface curvature affects the boundary condtion, -> a generalized boundary condition developed for a planar surface may not be applicable to a non-planar surface

S.M. Rytov 1940 Paper published in the Russian (Soviet) Journal of Experimental and Theoretical Physics: Translation by V. Kerdemelidis and K.M. Mitzner

Appendix: Homogeneous Circular Cylinder approximate boundary condition

**[A24]**__A very important work on Approximate boundary condition modeling for an inhomogeneous dielectric body with a curved surface__ that was published in a University of Michigan report by Professor Senior, a distinguished Electromagnetics expert in 1990.

But this subject dielectric body has No Edges.

Unfortunately, no geometrical visualization of the problem is presented.

It is very interesting that Dr. Senior possessed an English translation by V. Kerdemelidis and K.M. Mitzner of a Russian (Soviet) 1940 paper by S.M. Rytov.

V. Kerdemelidis received a Ph.D. at California Institute of Technology in 1966 (dissertation: "A Study of Cross-polarization Effects in Paraboloidal Antennas").

K.M. Mitzner, B-2 Stealth Bomber designer is a 1964 Ph.D. graduate of California Institute of Technology.

**A25. E. Marx's 1990 paper on Computed fields and Dielectric Wedge's Edge:**Scattering of plane monochromatic wave by a homogeneous dielectric wedge, sharp edge

Perfectly conducing wedge, Electromagnetic field sharp edge: test case

Solve integral equation: point-matching method

Problem with definition of normal (n) pointing out from the wedge at the sharp edge

Results for Dielectric Wedge and Conclusion

Difficulty in finding analytical solution

Singularity of the boundary functions

Rigorous solution to fields near edge of a dielectric wedge missing

Method of Solution for dielectric obstacles with sharp edge:

"Caution", computed near field (sharp edge dependent) & far fields and radar cross section (do not depend strongly on sharp edge)

**[A25]**E. Marx is an employee of the U.S. government (National Institute of Standards & Technology).

The analytical solution to the problem of scattering by an infinite dielectric wedge is not known to exist (difficulty with matching the mixed boundary conditions).

**A26. J. Van Bladel's 1991 book on Singular electromagnetic fields and sources:**"Singularities at an Edge" (Chapter 4):

Wedges, found in practical structures, mixed material, and assumed infinite sharpness of edge

Mathematical nature of these singularities:

Study of the field near charged Strip -> extrapolated to other structures with sharp edge

Edge energy density requirement

Field, currents and charges

Field singularities are only potentially present.

Predominance of Singular field: singular behavior (first few points close to edge)

Edges encountered in actual devices are not perfectly sharp

"Uniqueness and edge condition": Green's theorem

Singularities & dielectric wedge

**[A26]**Importance of Perfectly conducting strip: edge model

**A27. S. Wang's 1995 paper on Diffraction:**(Huygens & Fresnel) & Kirchhoff: integral formulas

Sommerfeld: boundary diffraction theory, half-infinite conduction plane

Experimental - Theory inconsistencies:

i) Physical meaning of the i (imaginary no.) in diffraction integral formula : pi/2 phase jump but this is not observed in practice

focal shift (interference superposition point)

ii) axial bright point in Fresnel diffraction region

Proposed theory:

"…diffraction is the result of interference between 2 waves: one is the wavelet with pi phase jump produced by boundary diffusion, the other is the envelope of the wavelets from the original source directly."; &

E = E

_{d} + E

_{b}exp(-i * pi) where E

_{d} is optical field from original source & E

_{b} is the field from the diffraction boundary.

Funding: National Hi-Tech Inertia/Confinement Fusion Committee

**[A27]**Amazing proposal by a Chinese scientist to correct Theory of Diffraction after 3 hundred years

Phase transformation: Half wave (wavelength/2):

Bright and Super-diffraction limited beams in the far field

**A28. S.H. Lui's interview with Vladimir Arnol'd in 1995:**Professor Arnol'd obtained a Ph.D. from Moscow State University in 1961.

A student of A. N. Kolmogorov at Mechmat (Moscow State University Mechanics and Mathematics Faculty).

On page 433 in the paragraph discussing Pontriagin, a disclosure is made about the KGB's role about the publication of a mathematics paper:

"... in his autobiography published in the Russian Mathematical Surveys. When he submitted this paper to the Editorial Board, the KGB representative suggested that the article should not be published as it was because of its extreme openness. I would prefer to see the original text published--what you now find is rather softened."

**[A28]**Very important fact: Independent confirmation of the powerful role of the KGB (a member of the Soviet Editorial Board) in the control of information that is distributed via the open literature (scientific paper).

**A29. L. Knockaert, F. Olyslager, and D. De Zutter's 1997 paper on Diaphanous Wedge and Diffraction:**Perfectly conducting wedge vs. dielectric wedge (incomplete analytical theory)

References: MacDonald; Bowman, Senior & Uslenghi; Van Bladel; Bouwkamp; Meixner and Marx

Diaphanous (isorefracting) body: same wavenumbers inside & outside body

Integral equations: fields

Static Solution: the Mellin transform

Dynamic solution: Kantorovich-Lebedev

Applications: Line Source Excitation and Plane wave excitation

Results:

Figure 3 contours of constant field amplitude results for right-angle wedge and line source excitation:

"infer the jumps of the derived fields across the boundary of the diaphanous wedge

Finiteness of the energy integral (wedge condition)

"Conclusion":

Diaphanous wedge solved;

General wedge problem, matching conditions for the two different Kantorovich-Lebdev transforms "almost impossible to meet, at least analytically"

**[A29]**Belgium scientists

Numerical work, No experimental results presented.

**A30. J.H. Meloling and R.J. Marhefka's December 1997 paper on Diffraction and Curved Edge:**i) Uniform geometrical theory of diffraction (UTD) and Curved edge (symmetric)

Caustics of diffracted field

High-frequency Electromagnetic field (highly localized)

CC (Caustic corrected)-UTD assumption (page 1847)

Asymptotically expand the diffraction at high-frequency:

Appendix: Two Uniform Asymptotic Expansions

Region of validity

ii) Short Monopole (source) mounted on an elliptic disk

Far zone radiated field

Directive gain Dg of antenna

Diffracted field expressions become singular

**[A30]**American investigators/scientists: J.H. Meloling, U.S. Navy supported; R.J. Marhefka, Ohio State University

Antenna design work

New Curvature dependent diffraction coefficient

Very important to be able to model a radiating antenna mounted on a finite perfectly conducting plate (Curved edged):

Directive gain and fast way to compute radiation pattern

**A31. T.L. Zinenko, A.I. Nosich, and Y. Okuno's 1998 paper on plane wave scattering by resistive and dielectric strip periodic grating:**Problem of Wave Scattering by a flat-strip periodic grating with zero thickness:

It is one of the canonical problems in scattering theory.

1st Reference is to H. Lamb's 1898 paper on metallic grating & electric waves

Regularization: analytical inversion of the static part of the full-wave integral equation (IE):

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

Absorption vs. PEC (Perfectly electric conductor) boundary conditions

Key problem: scattering of a plane wave from a resistive-strip grating

This paper: Analytical-regularization-base algorithm for method of solution

"Resistive Strip Grating":

Figure 1 geometry of problem

2-D Helmholtz equation

Boundary conditions do not hold at strip edges

Condition of finite stored energy

Radiation condition: "no sources at infinity"

Duel Series equations (DSE)

"Numerical Results":

"the fact that the resistive boundary condition leads us immediately to a Fredholm second-kind IE for the current; hence, no analytical regularization is needed. The unknown current density function here has no singularity at the edges of a resistive strip."

Figure 2: relative Truncation error

Figure 3: Wood anomalies are observed

Figure 4: scattered and absorbed power fractions decremented with the strip width

Figure 5: existence of a broad maximum of which amplitude and position depend on the relative width of the strip and the angle of incidence

"Dielectric Strip Grating":

"different edge behavior from the resistive-edge problem"

DSE regularized matrix equations

"Numerical Results":

Figure 6, 7, 8, and 9

Wood anomalies

"thin 'wooden' grating of flat strips can serve as a narrow-band isolating screen"

"Modified Lamb's Formulas":

Regularized matrix equations --> analytical solutions by iteration (only for Fredholm 2nd-kind IE)

"Conclusions":

This paper describes "...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."

Accuracy is limited only by the precision of the computer used

This work revealed the following:

1) "existence of an optimum real-valued electric resistivity delivering a maximum absorption by a constant-resistivity strip grating,"

2) "a similarity of low-frequency behavior of the E- and H-wave curves for a dielectric strip grating, and"

3) "a quasi-total screening of the E-wave by a thin dielectric-strip grating near the first Wood's anomaly."

DSE Analysis Appendix

**[A31]**Wood anomalies (a result of natural singularities) for resistive and dielectric diffracting gratings are generated by this paper's numerical code.

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

**A32. S. Anoknov's 1999 paper on Diffraction:**Chinese Wang's 1995 publication: key

Inaccuracy of Kirchhoff's description of diffraction

Rigorous theory of diffraction

Half-wave sudden change in phase at boundary (edge of ideally conducting half-plane) of the Geometrical Shadow

Kirchhoff theory: aperture diffraction, valid only for plane waves

"Condition on a rim" (a rim does not radiate)

Not any reasonable explanation of the physical reality of geometrical and boundary waves

**[A32]**Ukrainian paper questioning the current theory of diffraction.

**A33. Symposium paper in 2000 by Ufimtsev's UCLA associates and himself on Singular Edge Current:**Singularity behavior of the current near sharp edges for a TM incident field

Singular current behavior: solely dependent on geometry

Also, source specific, greatly depending on the amplitude and direction of incident field

2-D Wedge: Perfectly Conducting (PEC) scatterer and infinite ground plane

Image Theory

References D.R. Wilton et al 1977 work and Bowman et al 1987 work

Current tends to zero due to source dependency of Edge condition.

(kr)^v where v = pi/alpha -1 where r is the distance from the edge and alpha is the wedge angle

Simulation results using Method of Moments (MoM)

No experimental results presented.

**[A33]**B. Khayatian, Y. Rahmat-Samii, Ufimtsev: UCLA Dept. of Electrical Engineering

**A34. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii's 2000 paper on Singular Edge behavior:**Repeat of 2000 symposium paper

Singular behavior of a parallel edge current

Image theory, PEC infinite ground plane

"Conclusion":

Last 2 sentences??:

"In other words, without clear knowledge of the characteristics of the incident field, it would not be advisable to impose the singular behavior of the current solely from the wedge geometry. Consequently, it is safer not to impose any singular edge current behavior in advance; instead let the numerical computations to properly construct the correct edge behavior."

**[A34]**P. Ufimtsev: UCLA & Northrop-Grumman, Defense Contractor (B2 Stealth Bomber)

Singular edge behavior

Last sentence of this paper does not make sense. He is discounting the previous PTD work.

**A35. M. Levy's 2000 book on Parabolic equation method for electromagnetic wave propagation:**Parabolic equation (PE) is an approximation of the wave equation (paraxial formulation)

Leontovich and Fock (1940s); & Malyuzhinets (1940s)

"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."

Advent of digital computers: PE approximation, finding numerical solution rather than closed-form expressions

"Parabolic equation framework" (Chapter 2):

Perfectly conducting Half-plane screen diffraction, neglecting edge current effect:

Paraxial formulation neglects backscatter due to edge effects: differs from the Sommerfeld expression

"Vector PE" (Chapter 14):

F117 Aircraft (14.4.5):

Forward bistatic RCS (radar cross section) of F117, Figure 14.16 & 14.17

**[A35]****A36. P. Cecchini, F. Bardati, and R. Ravanelli's 2001 paper on Edge Singularity Extraction:**Electromagnetic-field near composed wedges

2-D Boundary-element method (BEM) extension

Application: Microstrip transmission line with a strip of finite thickness

J. Meixner: field near edge can be locally expressed in a series; 1st term - singular behavior; & 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

Method requires knowledge of the field singularity order.

Factorization into a regular part & singular part: regular part is used for solution.

Each edge is treated independently of the others.

Restriction: well-defined tangent along the edge (locally straight edge)

Numerical Results: No theoretical solution is available for the direct comparison.

**[A36]**Italian Investigators

An important paper:

Composite geometry with more than one edge; &
Singular behavior of Electromagnetic field near Composed Wedges with Edge Singularity.

**A37. G.A. Kalinchenko, A.M. Lerer, and A.A. Yachmenov's paper from the 2002 Conference on Mathematical Simulation of Impedance Diffraction Gratings:**MMET*02 Proceedings

Eigenwaves of Periodic Impedance diffracting grating: impedance dielectric strips

Wave diffraction modeling & simulation

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.

"Results and Conclusions":

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).

**[A37]**This paper's results presentation 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.

**A38. The 70th Birthday (2000) of V.M. Babich:**Professor Vasilii M. Babich is an undisputed leader of the St. Petersburg School on the Mathematical Theory of the Diffraction and Propagation of Waves

St. Peterburg diffraction historical contributions from V. I. Smirnov, S.L. Sobolev, and V.A. Fock

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)

Propagation of waves in curved structures

Diffraction on cones and elastic wedges

**[A38]**Professor Babich is a winner of USSR State Prize for 1982 and of V.A. Fock's Prize of Russian Academy of Science (1998).

No mention of Ufimtsev.

**A39. Ufimtsev's 2006 paper on Improved Physical Theory of Diffraction (PTD):**Under conditions of Grazing incidence, PTD is not valid: singularity (infinite field values in forward direction).

New definition for the Uniform component of the Surface current (& a new Non Uniform component of the surface current that generates regular edge waves in all directions):

Current induced on the Half-plane tangential to the illuminated face of the scattering edge (& to the edge itself)???

References Fock's high-frequency localization principle

Elementary edge waves

Figure 2 Tangential Wedge ???

Wedge diffraction problem

Rubinowicz diffraction cone (A. Rubinowicz, German publication, 1924) - Keller Cone

Complex derivation of interested quantities

Sommerfeld integrals, Half-plane diffraction problem

**[A39]**Another interesting paper of Ufimtsev's effort to correct his theory of diffraction, now in 2006.

**A40. Ufimtsev's 2006 paper on Acoustic Diffraction and forward scattering:**Continuation of same idea (Improved PTD) but applied to Acoustics systems

Acoustic edge waves

Improved theory: forward scattering, grazing incident, grazing singularity

New definition for Uniform component of the Surface field (& a new Non Uniform component of the surface field that generates regular edge waves in all directions)

Tangential Wedge, figure 2 ????

Presents derivation with No results: analytical or Experimental

**[A40]**
**A41. S. Anokhov's 2007 paper on Fresnel integrals and Diffraction:**Fresnel integrals application to diffraction models

Integral transformation of complex integral integrals F(U)

Analytic simulation of Fresnel integrals

"Mystery of Young's Boundary Wave":

Young's boundary wave

Knife-edge diffraction of a paraxial light beam:

Figure 5. "Picture of a near field at diffraction of a laser beam by a sharp edge.";

D wave (diffraction + dislocation): 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).

**[A41]**
**A42. S. Anokhov's 2007 paper on Diffraction by a transparent half-plane:**Plane wave diffraction by a perfectly transparent half-plane, whose edge operates on the wave as a phase step

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

Initial wave modulation (before its contact with transparency)

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

Basis for concept of edges in PTD (reference to Ufimtsev's 1991 paper)

Study of the Physical Mechanism of Diffraction needed: investigate the distinct role of Amplitude and Phase disturbances in the formation of a diffracted field

**[A42]**A very novel numerical experiment created by Anokhov to study the physical mechanism of diffraction and the backward wave from an edge.

**A43. V.M. Babich, M.A. Lyalinov, and V.E. Grikurov's 2008 book on The Sommerfeld-Malyuzhinets Method in Diffraction Theory:**Sommerfeld-Malyuzhinets Method for solutions to problems of diffraction in wedge-shaped or angular domains with various boundary conditions (either "perfect" (Dirichlet or Newmann), or impedance)

Malyuzhinets' system of functional equations: can not be solved explicitly.

Results from the Sommerfeld-Malyuzhinets method can be used by the Geometrical Theory of Diffraction: justified by the (localization principle).

"Meixner conditions which ensure the boundedness of the wave-field energy near the vertex"

"Radiation conditions to eliminate other (except those for the incident wave) sources of radiation at infinity"

Sommerfeld integral (a contour integral) --> Fresnel integral (diffraction by infinite half-plane)

Far field asymptotics (kr >> 1): saddle-point method

Elimination of singularities

Sommerfeld-Malyuzhinets Technique:

Sommerfeld integral form for an arbitrary angle domain for various boundary conditions; &
Malyuzhinets functional (difference) equations.

"The Helmholtz Equation and Boundary Condition" (2.1)

"Vicinity of the Edge and Meixner Conditions" (2.2):

"The solution to the Helmholtz equation with the classical boundary conditions cannot be smooth (except for some special cases) in a neighbourhood of the wedge's edge (vertex)."

"Plane-Wave Incidence and the Geometrical-Optic Part of the Solution" (2.3)

"Radiation Conditions and Completion of the Formulation of the Problem" (2.4)

"Uniqueness of the Solution to the Impedance-Wedge Diffraction Problem" (2.5)

"Uniqueness of the Solution to the Perfect-Wedge Diffraction Problem" (2.6)

"Uniqueness Theorem and the Limiting-Absorption Principle" (2.7)

"Concluding Remarks":

"Key point: reduction of a diffraction problem to a system of functional (function-difference) equations."

This book's problems have been "problems with one propagation speed"

"Multi-speed problems" (diffraction by a transparent or elastic wedge): these problems have no known exact solutions

Computation of the Malyuzhinets Function (A): MATLAB code implementation, Unintelligible code

**[A43]**This book is a key source for the study of Diffraction and Scattering. It is written by key Russian Diffraction investigators that are member of the St. Peterburg School of Diffraction.

**A44. A.V. Guglielmi's 2010 paper on the 70th Anniversary of the Leontovich boundary condition:**A "Letter to the Editors" in regards to a 2009 paper by V.I. Alshits & V.N. Lyubimov: "Generalization of the Leontovich approximation for electromagnetic fields on a dielectric-metal interface")

Subject: M.A. Leontovich's 70-year old boundary condition at the surface of a well conducting body

Alshits & V.N. Lyubimov's paper asserts "that this boundary condition is more accurate than Leontovich himself believed."

Guglielmi's claim is that S.M. Rytov's 1940 paper ("written in the suggestion of Leontovich") is proof that Leontovich was fully aware of exactly how accurate his proposed boundary condition was.

Rytov's (& Leontovich) work pertains to the Asymptotical theory of the skin-effect.

Leontovich impedance boundary condition:

Formulated in late 1930s;

Holds true approximately on surfaces of well conducting bodies; and

Is expressed as

**E**t = ζ

**H**t x

**n**where

**E**t &

**H**t are tangent components of the electric & magnetic fields, respectively;

**n** is the inward normal to the body surface, and ζ is the surface impedance.

From the formulas presented in the Rytov's 1940 paper, one can see that corrections to the above formula for certain electromagnetic wave propagation models "start with the cubic term rather than with the quadratic term with respect to small parameter ζ."

"Corrections cubic in ζ break the local character of the boundary condition.

The boundary condition will acquire terms on the order ζ to the power of two if the surface curvature is taken into account for case where the body surface is not a plane:

Guglielmi asserts that this fact was known to Leontovich.

**[A44]**A.V. Guglielmi is associated with the O Yu Schmidt Institute of the Earth, Russian Academy of Sciences.

Remember that S.M. Rytov's paper was published in Russian in 1940. We know that there is some relationship between M.A. Leontovich and Rytov but what is it. Leontovich is the key developer of the idea of the asymptotical theory of the skin-effect but Rytov writes this important paper by himself supposedly. And, why is Gugielmi concerned with this issue now?

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