Research on Problems of Diffraction, Scattering & Propagation of Waves: Facts & Ideas Derived from Analysis Part III

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 November 4, 2009 and was finished on January 11, 2010.

Last Update: January 30, 2010.

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


63. G.D. Malyuzhinets’s 1959 paper on Diffraction:

Edge rays

Field penetrates partially into the region of geometrical shadow

Huygens’ principle – integral equation, in general, can not be solved

Complexity of diffraction problems shown in his two 1958 papers [64] & [65]


Major contributor in Soviet Diffraction Theory

64. G.D. Maliuzhinets’ 1958 paper on the Sommerfeld Integral:

Presented by Academician M.A. Leontovich

Problem of Diffraction for an Ideal Rigid 2-dimensional Wedge

Sommerfeld found a solution for boundary condition S = 0 or ∂S/∂n = 0

Another approach to solution of problem of diffraction of a plane wave by wedge with boundary conditions ∂S/∂n + hS = 0 for Sommerfeld integral

Key paper: Asymptotic solution to a Diffraction problem

Acoustics Institute, Academy of Sciences, USSR


65. Maliuzhinets’ 1958 paper on Surface Waves and the Wedge:

Presented by Academician V.A. Fock

Key paper: Asymptotic solution to problem of Diffraction & two-dimensional Wedges with Surface Impedances

Institute of Acoustics, Academy of Sciences, USSR


66. A Review of the Malyuzhinets theory and Wedge Boundary Scattering:

Malyuzhinets technique: it was first presented in G.D. Malyuzhinets' doctoral dissertation (in Russian) while he was at the P.N. Lebedev Physics Institute, Moscow USSR) in 1950 and later in his 1958 paper [65].

The problem of determining the wave field scattered from the edge of a Wedge with arbitrary impedance conditions on either face.

Direct relationship between the Sommerfeld integral representation and the Laplace transform

Proof of the Inversion formula for the Sommerfeld integral & the crucial nullification theorem

Special functions

Solution details of the Malyuzhinets expressions for the impedance wedge's diffraction wave field

Edge value is expressed in terms of Malyuzhinets functions.

There is an issue of the Underlying assumption:

Two-dimensional fields & incident wave falls at a right angle to the edge. Scalar Helmholtz equation is used.

When the incident wave is oblique or skew to the edge, for the Electromagnetic case, Vector Maxwell's equations are required and for the Elastic case, the problem becomes Vectorial in nature, and there is no explicit solution.


No mention of Ufimtsev

67. Springer On-Line Reference on Huygens’ principle:

Space of odd dimensions

Not valid in 2-D space

Lacunary principle: One side of the propagating wavefront is zero.


68. R. Meneghini and J. Bay's 1982 paper on Maggi-Rubinowicz Theory of Edge Diffraction:

The Maggi-Rubinowicz (M-R) Method for Scalar and Vector fields

Theory of edge diffraction created by Thomas Young

A transformation of a surface integral to a line integral where the Line Integral corresponds to the physical edge of the scatterer

Observer approaching shadow boundary: a variable mesh integration is used.

Numerical comparisons between Physical Optics and M-R method: good agreement for the Diffraction & a Circular Aperture


69. H.M. Nussenzveig and W.J. Wiscombe’s 1987 paper on Tunneling and Diffraction:

NASA funded investigation on “Diffraction as Tunneling”

Short-wavelength tunneling & impenetrable sphere: reflection from smooth surface along with diffraction

Uniform asymptotic approximation: valid at large angles

Main problem is the uniform treatment of diffraction by the curved edge.

Large-angle diffraction is very accurately described by uniform approximation including edge amplitude

Surface waves (Evanescent waves) can arise from Tunneling.

New Physical picture of diffraction:
Can be applied to more general curved surfaces; and
Tunneling through an inertial barrier
Diffraction & tunneling are typical wave effect


Very novel approach: Non classical diffraction theory

70. V.E. Grikunov, G.P. Pelosi, and J.L Volakis’s 2001 Foreword on Millennium Day of Diffraction Seminars (2000):

First Seminar in 1968 (Soviet Union)

Leningrad Diffraction Theory School: V.A. Fock & V.I. Smirnov

Analytical focus to numerical modeling


Another Russian Diffraction Investigator: V.E. Grikunov

71. A.S. Kirpichnikova and V.B. Philippov’s 2001 paper on Diffraction caused by a Discontinuity:

Statement about importance of processes of Wave Propagation:

"...almost all engineering innovations are related to processes of wave propagation."

Understanding of Wave phenomena is critical to the fields of optics, radio engineering, electronic engineering, heating engineering, navigation, seismology, acousics, etc.

The solution to the Problem of Diffraction and Wave propagation is a necessity to the successful application of many novel technologies.

Problems of diffraction: surfaces with discontinuities (boundaries, media parameters)

Boundary discontinuities (Edges, Cracks): Jump of curvature of the boundary on the line, Jump of derivatives of curvature, etc.

Jump point, point of discontinuity of curvature.

This paper's diffracting body & jump point:

a cylindrical body with a boundary composed of a convex-cylindrical surface and a half-plane that are jointed together by a straight line; &

the jump point is in the penumbra region.

Cases of Dirichlet, Neumann, and impedance boundary conditions are considered.

Wave Field Determination Approach:

Incident plane wave & parallel to planar boundary; &

Wave field representation is partitioned by boundary segments (2) by jump point: Green's function for convex part of body and Parabolic equation method & Philippov's method for planar segment.

No numerical or experimental results are presented.


No mention of Ufimtsev

One can not judge the validity of this paper's solutions to the Problem of Diffraction by Boundary Discontinuity because no comparison is made between the paper's results against experimental results.

72. V.M. Babich, V.P. Smyshlyaev, D.B Dement’ev, and B.A. Samokish’s 1996 paper on Diffraction and Numerical Calculation:

Last remaining class of unsolved "canonical" problems: diffraction by arbitrary cross section cones

A new method of Numerical Calculation of Diffraction Coefficients based on the approach of V.P. Smyshlyaev


Another set of Russian Diffraction investigators

No mention of Ufimtsev

73. S.J. Chapman, J.R. Ockendon, and V.H. Saward’s 2000 paper on Edge Diffraction:

United Kingdom work on Edge diffraction of creeping rays

Multi-diffraction case:

Diffraction of surface creeping ray encountering sharp edge.

Tangent ray


No mention of Ufimtsev

74. T.B.A. Senior and P.L.E. Uslenghi’s 1971 paper on Keller's and Ufimtsev Theory:


Geometrical theory of diffraction: Keller

Ufimtsev: systematic improvement of Physical Optics surface currents by introducting non-uniform contribution and for field is subsequently obtained by integration on surface of scatterer

Both theories are heuristic.

Comparison is possible for a very limited number of geometrically simple bodies

Strip, 2-D

“We therefore conclude that Ufimtsev’s theory does not, in general, lead to the correct asymptotic expansion, even though it may yield good numerical estimates.”

“Finally, we point out that in recent work [12] Ufimtsev recognizes that his previous results [8] are incorrect beyond the leading term. He succeeds in finding the correct higher order terms only by employing function-theoretic results.”


75. S.W. Lee’s 1977 paper on the Comparison between Two theories of Edge Diffraction:

GTD – UAT (Uniform Asymptotic Theory) vs. PTD Comparison

Ufimtsev: J0 uniform surface current, constant over surface:

This is not exactly true (page 162 footnote)

J1, fringe current, difficult to estimate in general wedge diffraction problem


UAT & Ufimtsev’s theories for edge diffraction problem

UAT: explicit field solution Ufimtsev: physical optics integral must be asymptotically evaluated before the field solution is obtained

Simple 2-D example: agreement with UAT

“Of course, it remains to be verified that this agreement also holds for a more general case.”

Acknowledgement: K.M. Mitzner was providing critical comments to Dr. Lee (“that may have increased the objectivity of the comparison made in this paper.”)


American Diffraction investigator (University of Illinois)

76. P. Beckmann’s 1968 book on Electromagnetic Waves Depolarization:

"Diffraction – Ray Optics"

"Keller’s and Ufimtsev Theories" (page 105)

Scatterer: made up of canonical forms (page 106)

Perfect conductor

Generalize for scatter with finite conductivity has not yet been attempted

Diffraction coefficients

Special cases, rigorous solutions are known but general proof is still lacking (page 110)

"Rigorous Solutions" (page 118)

Issue of lack of rigorous solutions

Convergence of series is only in special limiting cases

Depolarization of the radiation scattered by sphere

Infinite rotational paraboloid: No edges or vertices

Keller – Ufimtsev theory would agree for this case: rigorous solutions would all be identical

Claim that Schensted (1955) found this to be true (page 128)

“this is an impressive confirmation of the Keller – Ufimtsev theory”


Since No fringe currents exist, Beckmann’s claim does not validate Ufimtsev’s theory (key aspect of PTD is in fact about Fringe [edge] currents & scattered/diffracted fields both exist)

77. C.E. Schensted’s 1955 paper on Electromagnetic and Acoustic Scattering:

Scattering by perfectly reflecting bodies with no "shadow" region

A Plane wave is incident on the body along the axis of a perfectly reflecting semi-infinite body of revolution.

Thus, No “shadow” regions is present for a 2-D Paraboloid body.

Exact Electromagnetic solution is given (in closed form): first term

S. Silver’s current-distribution method

“The physical optics method consists of replacing the fields inside the integrals by their geometric-optics approximation. In the electromagnetic case this corresponds to using the geometric-optics approximations for the current and charge distributions on the scattering surface.”


78. K.M. Siegel, J.W. Crispin, and C.E Schensted’s 1955’s paper on Electromagnetic and Acoustical Scattering:

University of Michigan

This paper is applying the Physical Optics (PO) approach to the problem of Back Scattering for a semi-infinite cone.

Radar cross-section determination (solution): approximations

Approximation approach is dependent on incident wave wavelength relative to scatterer's characteristic dimension.

Large wavelength: Rayleigh approximation is valid

Small wavelength: Physical-optics approximation computed by current distribution method.

Geometric optics method is not valid: Tip of cone is the major contributor to the radar cross section.

Exact nose-on radar cross section of a semi-infinite cone has been evaluated: polarized wave incident along the axis of the cone

Agreement of results between PO & exact Electromagnetic theory & experiment: within a factor of 2.

"Amount of experimental information on cone & other bodies, ... the paraboloid, is very small indeed."


No mention of Ufimtsev

79. L.A. Vainshtein’s very famous 1969 work/book (USSR 1966) on Open Resonators and Open Waveguides:

Open resonators: "systems that support modes with high Q's, provided that the oscillations in such systems are accompanied by radiation into free space"

"High-Q modes in open systems may be excited by virtue of one of three physical phenomena:
1) Reflection from the boundaries of the resonator;
2) the formation of caustic surfaces;
3) total reflection."
Open resonators:
1) the distance between the plane mirrors as compared with the wavelength is large;
2) "the oscillations are generated as a result of the reflection of a waveguide mode
at a frequency near the critical frequency of this mode"
"Diffraction at the Open end of Waveguide":

Plane-parallel open-ended waveguide;

Parallel semi-infinite half-planes (Infinitely thin) model;

Asymptotic behavior is not affected by the current density near the edge which is a different function for waves with different polarization. (page 21); &

Physical meaning:

"Waves propagating in a plane-parallel waveguide can be represented by the sum of two plane waves, i.e., by two beams of parallel rays";

With the Wave's frequency near its cut-off frequency, the rays make a small angle ε0 with the normal of the waveguide wall;

As a result of diffraction, the rays are "turned" through an angle twice ε0 where the returned rays form the reflected wave;

Thus, there is no conversion to modes of different orders (modes of radiative loss via the open end of the waveguide);

The reflection coefficient is also dependent on the diffractive divergence of a "pencil of rays" reflected by the waveguide walls.

"The Parabolic Equation and Diffraction at the Open End of a Waveguide":

Leontovich and Fock's Parabolic equation: reduces a 3-D to a 2-D Diffraction problem (Wave equation & boundary conditions);

Fresnel integrals appear in the mathematical formulation;

"Justification is provided by the rigorous treatment of diffraction by a half-plane & a semi-infinite plane-parallel waveguide";

Equivalency to the usual calculation of Fresnel diffraction in physical optics based on the Huygens-Fresnel principle and the assumption of small diffraction angles;

The approximate treament of diffraction at the open end makes it applicable to any pair of semi-infinite plates whose reflecting surfaces are parallel, separated by l and kl >> 1 (wave number k).

"Two-Dimensional Resonators formed by Plane Mirrors": is of great Significance;

Finite thickness of reflecting plates (mirrors);

Standing waves;

"The decrease of the current density to small values at the edges is characteristics for open resonators; due to this phenomenon the radiation losses are minimal."

"The Eigenfrequency Spectrum":

"The principal difference between open and closed resonators is that in the absence of losses in the walls (mirrors) the eigenfrequencies of closed resonant cavitites are real, whereas those of open resonators are complex, and this leads to a reduced mode density."

'Open resonator' applications in physics & engineering has existed for a long time such as in all musical instruments (organ pipes, etc.) & a number of acoustical devices (tuning fork, etc.). (page 51)

Ball Lighting (High frequency oscillations):
P.L. Kapitsa hypothesis
Thunderstorm, natural open resonators forms, localization of Electromagnetic field
Ball lighting is a result of focusing of Electromagnetic Waves

P.I. Kapitsa & V.A. Fock supported this work; V.P. Bykov & S.P. Kapitsa were involved in this work, too.

Open-Ended Waveguide Diffraction:

A very important phenomenon with novel applications such as "Open Resonators with new properties; &

Waveguide modes, propagating at frequencies only slightly exceeding their cut-off frequencies, upon reaching the end of the waveguide, are reflected back into the waveguide with a near unity (absolute value) reflection coefficient.

Extremely important concept:

Reflection from the open end of a waveguide = reflection from the edge of a waveguide.

80. Second Lieutenant USAF U.J. Macias’s 1986 M.S. Thesis:

Performed at Air Force Institute of Technology

Thesis Chairman: Andrew J. Terzuoli

A very important investigation performed at an U.S. Air Force facility comparing the effectiveness of various Radar cross section (RCS) prediction technologies

Computational methods & RCS experimental measurements

Abstract & DD Form 1473 19.

“It was found that although the Moment Method is the most accurate RCS prediction method, it takes too much CPU time for large plates (over 2.5 wavelengths). The Uniform Theory of Diffraction, on the other hand, is accurate for large plates and takes less CPU time than the Moment Method. The Geometric Theory of Diffraction is also accurate but fails near the edge of the plate. Finally, the Physical Theory of Diffraction and the Physical Optics approximation are relatively inaccurate.”

1953 Keller GTD

Fifties, PTD: Braunbek US & Ufimtsev

Uniform Theory of Diffraction: Kouyoumjian & Pathek

UAT: Uniform Asymptotic Theory: Lee & G. Deschamp

Moment Method: Numerical solution to an exact equation (late 1960s), computer speed dependent

Thesis Computational Methods Implementation: VAX 11/780 mainframe computer

The Avionics Laboratory Far-Field Radar Cross Section (RCS) Measurement Chamber was used for thesis experiment.

Radar Transmit Signal Frequency: 10 GHZ (3 cm. wavelength)


81. W.D. Burnside, C.L. Yu, and R.J. Marhefka's 1975 paper on GTD and the Moment Method (MM):

GTD applied to Antenna and Scattering Problems: a structure is large in terms of wavelength; and few geometries have diffraction solutions.

MM techniques to arbitrarily shaped objects: solution if overall size is small in terms of wavelength

Combination of the two approaches: arbitrary shaped and large structures

Use fewer current samples away from the point of diffraction: underlying assumption is that the Physical Optics (PO) current is normally quite reasonable in those regions.

Large number of current samples is used around a point of diffraction.

Analysis using examples:

Greatly decrease number of current samples by including a diffraction current term (based on GTD solutions)

2-Dimensional Diffraction Problems in the case of TE plane wave at a single wave number k

Scattering by a Simple Wedge (Perfectly Conducting, Infinite Wedge structure) example:

Using GTD solution for the total field, the total surface current is specified in terms of an expression that contains an unknown constant C. C is known from a canonical solution.

Field point is at a large distance from the point of incident with the wedge and not near a shadow boundary.

Claim that one can solve for C using MM simply by knowing the form of the diffraction current.

This approach can be extended to new 2-dimensional problems with diffracted field from a structural discontinuity such as a plate/cylinder junction.

The Total surface current density is partition into regions.

MM current around the edge are defined by simple basis functions: orthogonal pulse function (weighted) for pulse current samples.

Pulse currents occur around the Diffracting edge; GTD regions contain Matching points.


Simultaneous Linear equations

Diffracted current (complex-valued) is represented as a series. On a shadow boundary, diffracted current is the first term of the series

Number of current samples and Evaluation of Infinite Integrals: abbreviation of integration is dependent on orientation of plane wave to the edge.

Square cylinder example: pulse current samples and employing point matching

Circular cylinder example: partition into three regions:

The number of diffraction modes dictate the number of matching points in the GTD region that is used to obtain the necessary number of equations.

Higher order terms interact such that the solution is "somewhat" dependent on the location of matching points (Averaging scheme)

Axial waveguide example

Conclusions: claims results are in good agreement with exact solution for wedge and circular cylinder.

Few current samples needed around point of points of diffraction.

Shape of diffraction object dictates the number of current samples.

Use known GTD current form outside the current sample region.

One can apply this technique to concave and surface impedance structures. The technique may be extended to 3-D diffraction problems.


This GTD-MM technique has the same concept of methods that partition the surface into small regions or elements except this technique uses regions that can vary by size, i.e., variable size elements.

This GTD-MM technique for canonical forms and diffraction is very accurate. But how do you apply this technique to non-canonical objects?

This technique is dependent on the known form of the solution for the diffraction current.

The MM pulsed current samples are real-valued.

82. U. Jakobus and F. M. Landstorfer's 1995 paper on the Improvement of the PO-MoM Hybrid Method:

A current-based hybrid method is proposed by merging the Method of Moments (MoM) with the physical optics (PO) approximation, the Improved PO Method (IPO).

Approximate correction terms derivation is based on the uniform geometrical theory of diffraction (UTD).

Three-dimensional perfectly conducting scattering bodies of arbitrary shape & an incident wave of oblique incidence and arbitrary polarization

Perfectly conducting wedges & Ufimtsev's Physical Theory of Diffraction (PTD):

application of PTD "involves fictitious electric and magnetic line currents along the edges."

Modification of the standard PO current leads to surface current densities in the scattering body's MoM region & its PO region.

This leads to a uniform treatment of both regions in terms of representing "the surface current density as a linear superposition of basis functions defined over triangular patches with unknown coefficients."

The improved PO-current density is heuristically obtained by modifying the conventional PO current:

the correction terms for the edges contribution are based on the Sommerfeld's exact solution for the half-plane scattering problem; and

the additional correction terms for the boundary of the body's surface are based on perfectly conducting wedges.

Examples (comparison between IPO, MoM & PO results):

Perfectly conducting cube with side-length of twice of the incident wavelength: close agreement of IPO with MoM

Convex corner reflector: good agreement of IPO with MoM

Half-wavelength dipole antenna: good agreement of IPO with MoM


The authors of this paper make such a definitive statement about the fundamental concept of the PTD of the validity of electric and magnetic line currents along the edges of a scattering body.

83. P.Y. Ufimtsev's 1997 comments on Jakobus and Landstorfer's 1995 paper:

Ufimtsev states that Jakobus and Landstorfer's IPO method is in fact his PTD concept.

Jakobus and Landstorfer's statement about PTD's model of edge currents is false: "the PTD does not contain any fictitious edge currents." "It involves real elementary edge waves"

The IPO currents (wedge edges & half-plane edges) "are created simply by the mentioned elementary edge waves."

The analog between the PTD's nonuniform current and the IPO's currents (wedge edges & half-plane edges) correction terms is stated.

The scattered field that is created by nonuniform current is found by the use of 1-D integrals along the edges instead of 2-D surface integration over the scattering surface.

IPO is based on the PTD current concept and represents its hybrid with the moment of methods (MoM).


It is strange that Ufimtsev makes conflicting statements about the nature of elementary edge waves (EEW). Per his 2007 book [2] on page 3, "the elementary edge wave is a wave radiated by surface sources, induced in the vicinity of an infinitesimal element of the edge." But in paragraph two, he claims that the IPO/PTD wedge edges & half-plane edges currents are created by EEW.

Additional references to Russian works

84. U. Jakobus and F. M. Landstorfer's reply to Ufimtsev's 1997 comments on their 1995 paper:

"Ufimtsev is totally right that in the PTD formulation, the nonuniform (fringe) current component is based on elementary edge waves (EEW) … and not on fictitious electric and magnetic line currents along the edges."

For practical applications, edge equivalent currents based on PTD are used: Method of Equivalent Edge Currents.

"the PTD integration approach is very seldom useful for practical calculations."

Referring to "Practical implementations" of PTD: based on equivalent electric and magnetic line currents along the edges: "these line currents are indeed fictitious since the currents generally depend on the direction of observation that is not true for physical currents."

Equivalent line currents

Attempt to avoid the use 1-D integrals in finding the scattered field

Linear superposition of basis functions

Continuous current flow across the MoM & asymptotic regions can exist.

Electric field integral equation-Magnetic field integral equation (EFIE-MFIE) hybrid technique

For the IPO method, "additional 1-D line integrals along edges are avoided."

The IPO proposal is different than the hybrid methods that Ufimtsev mentioned in his 1997 comments [83]:

the IPO method allows for two coupling mechanisms between the MoM and the asymptotic region; and

Allows the source to be an antenna located in the near-field region


German investigators are questioning the work of PTD in terms of its practical implementation and, more importantly they are able to countering Ufimtsev comments about their Improved PO-MoM hybrid technique and paper.

85. G. Tiberi, S. Rosace, A. Monorchio, G. Manara, and R. Mittra's 2003 paper on EM Scattering from Large Faceted Conducting Bodies:

Efficient and rigorous Solution for EM Scattering Problems from Faceted bodies:
Analytical derived Characteristic Basis Function (CBF)
Matrix equation Approach
CBF & boundary conditions on scatter
Galerkin method 
Electrical large problems
Method of Moments (MoM) and Rao-Wilton-Glisson (RWG) Basis functions

Proposed Solution analytically derived CBF in a computationally efficient manner

Primary and Secondary CBF models the induced surface currents density.

Suitable correction coefficients where the Galerkin method is used to solved for these coefficients

Open and Closed 2-D bodies

Dimensions of matrix equal the number of CBF.

Inversion is necessary to obtain the Induced current density.

Scattering by a perfectly conducting strip and 2-D corner reflector: results (CBF vs. MoM) suggest validity of approach.

CBF Method is numerically rigorous (the characteristic basis functions are analytically derived).

Technique can be used at frequencies lower than the regime of the asymptotic method and can be extended to 3-D faceted bodies


Important Computational alternative to Ufimtsev's PTD

Method is still working with Current-based approach.

Still have issue with determining correction coefficients (complex-valued).

86. S. M. Rao, D. R. Wilton, and A. W. Glisson's 1982 paper on EM Scattering by Arbitrary Shaped Surfaces:

Simple and efficient Numerical procedure for treating problems of scattering by arbitrarily shaped object

Objects modeled using planar triangular surface patches

Electric field integral equation (EFIE) & Moment Method (MoM)

Problems with Wire-grid modeling

Surface patch modeling and Integral equations

Triangular patches and arbitrary shapes

Surface geometry and current

Numerical solution of problem of scattering by either Open or Closed Arbitrarily shaped conductors

Triangular patches modeling and EFIE formulation

Crucial: Special basis functions defined on triangular patches which are free of fictitious line or point charges

Special set of basis functions and MoM: linear system of equations for surface current

"The calculation of accurate surface currents is much more stringent test of a numerical approach than ultimate calculation of far-field quantities"

Work with Lockheed Missiles and Space Company


No mention of Ufimtsev

87. V.S. Prakash and R. Mittra's 2003 paper on Characteristic Basis Function Method:

Method of Moments (MoM) Application to EM scattering problems: Discretization of a body's geometry and Matrix equations

A Novel approach for efficient solution of these equations:

Characteristic Basis Function (CBF): Reduce matrix size because the conventional domain discretization of wavelength/20 parameter has been increased.

Proposed discretization range: Wavelength / 20 to Wavelength / 10

Non-iterative and Decrease Matrix size

CBF are high-level expansion functions.

The set of basis functions are characteristic of the particular chosen domain (e.g., patch).

Solve for current distributions.

Direct inversion of matrix equation of reduced size

The proposed method does not result in a deterioration of the condition number of matrix.


88. R. Mittra and K. Du's 2008 paper on Characteristic Basis Function Method Applications:

Numerically modeling of Complex Electromagnetic (EM) Systems

Computational Electromagnetics (CEM)

Method of Moments (MoM)

Need CEM Technique for Large Scale EM Systems for Microwave Circuits and Problems of Open radiation and Scattering

Characteristic Basis Method of Moments (CBMoM): a direct method (non-iterative)

CBMoM can be adapted to a Highly parallel computer application

Solve Scattering problems involving perfectly electrically conducting (PEC), Dielectric and dielectric-coated PEC bodies


No mention of Ufimtsev

Gianni Tiberi and Agostino Monorchio are past students of Dr. Mittra.

89. D. Erricolo, S.M. Canta, H.T. Hayvac1, and M. Albani’s 2008 paper on Diffraction:

American and Italian diffraction investigators

ITD – UTD: Incremental theory of diffraction is an extension of Uniform theory of diffraction

Current-based methods – PTD: methods to localize fringe field contribution & Mitzner’s ILDC

Field-based methods – ITD: incremental field contributions from the field of local canonical problem

II. Essentials of the ITD:

Using Simple contour integration, incremental contributions due to points along the contour of obstacle

Locality principle: incremental contribution interactions are not taken in consideration.
Canonical Perfect Electric Conductor (PEC) Straight wedge
PEC Circular disc
A hole in PEC plane
Single Diffraction ITD

Simple recipe for the application of ITD

“The single diffraction ITD shares a limitation that is common among ray-based methods in that it cannot properly describe physical situations that involve interactions among edges.”


90. I.E. Dzyaloshinskii 2001 Interview (Online):

UCI Physics Professor

Received Ph.D. from P.L. Kapitsa’s Institute of Physical Problems in Moscow in 1957

1953 Moscow State University

Student of Lev Landau

Three Part Interview reveals extremely valuable information:

Part I:

Clearance is required for FIAN (P.N. Lebedev Physics Institute) or Kapitsa’s Institute

He arrived at the Institute of Physical Problems in fall of 1954.

He talks about P.L. Kapitsa but makes no mention of L.A. Vainshtein (Note: 1956 photo of Landau group)

Part II:

“We were trained never to use uncontrollable approximations. Landau would kill anyone who was using them.”

“The only relevant model, you cannot solve. You have to wait for a new generation of machines.”

Part III:

P.L. Kapitsa’s deputy

Russian series, Classics of Science, Kapitsa official editor He was virtual editor.

Theory of quantum crystals

Landau’s two-fluid hydrodynamics


91. L.A. Vainshtein’s 1959 Part I paper on Antenna Current:

Institute of Physical Problems

Waves of Current & Transmitting Antenna


92. L.A. Vainshtein’s 1959 Part II paper on Antenna Current:

Institute of Physical Problems Waves of Current & Transmitting Antenna

Current in Passive Oscillator

Summary: “I express my gratitude to V.A. Fok for valuable comments.”


93. L.A. Vainshtein’s 1961 Part III paper on Antenna Current:

Institute of Physical Problems

Current Waves


94. L.A. Vainshtein’s 1961 Part IV paper on Antenna Current:

Institute of Physical Problems

Current Waves

Numerical calculations by use of an “Ural” electronic computer


“It must be noted that complete understanding of the precision of these formulas can be reached only by comparing the results obtained by these formulas with more precise results. These more precise results can be obtained, for example, by the method presented in an earlier work [5].”


95. P.L. Kapitsa, V.A. Fok, and L.A. Vainshtein’s 1960 paper on a Static Boundary Problem:

Finite Length Hollow Cylinder


“The relations which have been obtained make it possible to solve electrostatic problems for cylinders by means of high-speed computing machines. The appropriate numerical results will be published by us in the near future.”


Joint paper by three brilliant Soviet scientists

Paper demonstrates the Soviet's use of High-speed computing.

96. P.L. Kapitsa, V.A. Fok, and L.A. Vainshtein’s 1960 paper on Symmetric Electric Oscillation:

Ideally Conducting Finite Length Hollow Cylinder


“The relations which are obtained allow us to solve a problem of this kind by means of high-speed computing machines.”


Joint paper by three brilliant Soviet scientists

Paper demonstrates the Soviet's use of High-speed computing.

97. B.D. Tartakovskii’s 1961 paper on First Soviet Diffraction Symposium:

Combined Symposium on Theory of Wave Diffraction, Odessa 9/26/60 – 10/1/60

Greater than 400 Scientists (~75 doctors of science) attended

Some 100 papers presented

Academician V.A. Fock opened the symposium: asymptote as a “new ‘quality’”, inherent in the diffraction phenomenon

G.A. Grinberg: method of shadow currents & its application to problems of diffraction

“Some important problems of electrostatics & electrodynamics reduce to the solution of integral equation which connects the value of the current (or charge) density on the surface of a hollow cylinder with the values of the vector (or scalar) potential on the same surface.”

L.A. Vainshtein’s researches report for 1957-1960: ka >> 1 & ka ~ 1 (calculation applications of computer)

G.D. Malyuzhinets: diffraction phenomenon, a process of diffusion of wave amplitude along the propagation wave front

P. Ya. Ufimtsev; E.N. Maizel’s & Ufimtsev: Numerical calculations agree well with experiment.

V.M. Babich, A.I. Sivov, B.Z. Katsenelenbaum, and A.G. Sveshnikov

Electromagnetic problems calculations on an electronic computer

Effective methods using fast acting computing machines for calculation of wave propagation in arbitrary waveguides

Large amount of work presented at this Symposium: great scientific & applied value

Symposium: proof of Soviet diffraction theory is proceeding successfully.


98. B.D. Tartakovskii’s 1963 paper on Second Soviet Diffraction Symposium:

Second All-Union Symposium on Wave Diffraction, June 4 – June 9, 1962, Gor’kii

314 persons attended

Professor G.D. Malyuzhinets open symposium

Moscow & Leningrad

Second Plenary Session:

P.Ya. Ufimtsev:

“approximate solution to diffraction problems for convex ideally conducting bodies with surface discontinuities”

“self-evident physical considerations as to the nature of scattered-field formation”

Multiple diffraction

V.A. Fock, G.D. Malyuzhinets, & L.A. Vainshtein paper “Lateral Diffusion of Shortwaves at a Convex Cylinder”

V.M Babich and A.F. Filippov presented their work.

“Numerical methods in solution of wave propagation and diffraction problems – widespread introduction of electronic computers

L.A. Vainshtein: Periodic gratings, “Toward an Electrodynamic Theory of Gratings”

Symposium: large amount of work carried out in the Soviet Union:

Problems of diffraction, wave propagation in inhomogeneous media, wave emission


99. Y.A. Kravtsov, C.M. Rytov, and V.I. Tatarskii’s 1975 paper on Statistical problems in Diffraction Theory:

Statistical theory of Diffraction & Wave propagation

Linear & classical problems of statistical theory of waves

Statistics of antenna: difficulties in specifying the fluctuating currents along antenna flare in justified manner

Diffraction of Partially Coherent Fields

Synthetic aperture method: phase & amplitude of radar signals

Diffraction by Bodies Having Random Shapes or Positions

Multiple Scattering

Airplane: random character of the behavior of the scattering pattern (page 126)


100. V.G. Veselago’s 1968 paper (Published in Russian in 1964) on "Left-Handed" Substances (Negative Index of Refraction):

Soviet Scientist at P.N. Lebedev Physics Institute

Discussion of material properties

Index of refraction of a Left-handed medium relative to vacuum is negative.

ε-μ diagram

Isotropic & anisotropic substances (ε and μ tensors)

Gyrotropic substances (ε and μ tensors)

Plasma in magnetic field: tensor ε and scalar μ

Magnetic material: scalar ε and tensor μ

Gyrotropic substances: both ε and μ are tensors, pure ferromagnetic metal & semiconductors

Applied material research

Figure 8. (page 514), a very important diagram:

Left-handed isotrophic substance, spherically irradiated, isotropic compression, Dirac monopoles.


Landmark paper: provides the basis for future research on development of devices of Stealth, Invisibility, and SuperLens.