Maliuzhinets Function Research Articles

Physical Optics Scattering of Waves by a Half-Plane With Different Face Impedances

The theory of physical optics is extended for the diffraction problem of waves by a half-plane with different face impedances. The kernel of the modified theory of physical optics integral is expressed in terms of the Maliuzhinets function. The resultant integral is evaluated asymptotically for large values of the wavenumber. The results are examined numerically.

A Uniform Geometrical Theory of Diffraction for predicting fields of sources near or on thin planar positive/negative material discontinuities

Relatively simple and accurate closed form Uniform Geometrical Theory of Diffraction (UTD) solutions are obtained for describing the radiated and surface wave fields, respectively, which are excited by sources near or on thin, planar, canonical two‐dimensional (2‐D) double positive/double negative (DPS/DNG) material discontinuities. Unlike most previous works, which analyze the plane wave scattering by such DPS structures via the Wiener‐Hopf (W‐H) or Maliuzhinets methods, the present development can also treat problems of the radiation by and coupling between antennas near or on finite material coatings on large metallic platforms. The latter is made possible mainly through the introduction of important higher‐order UTD slope diffraction terms which are developed here in addition to first‐order UTD. The present solutions are simpler to use because, in part, they do not contain the complicated split functions of the W‐H solutions nor the complex Maliuzhinets functions. Unlike the latter methods based on approximate boundary conditions, the present solutions, which are developed via a heuristic spectral synthesis approach, recover the proper local plane wave Fresnel reflection and transmission coefficients and surface wave constants of the DPS/DNG material. They also include the presence of backward surface waves in DNG media. Besides being asymptotic solutions of the wave equation, the present UTD diffracted fields satisfy reciprocity, the radiation condition, boundary conditions on the conductor, and the Karp‐Karal lemma which dictates that the first‐order UTD space waves vanish on a material interface.

Exact Evaluation of the Maliuzhinets Half-Plane Function

The Maliuzhinets half-plane function psipi(z) has been used for many applications, but its closed-form has not been known. In this Communication, we exactly evaluate a known integral representation of psipi(z) to formulate an exact closed-form of the function, which is expressed in terms of the dilogarithm function. The exact formulation is numerically verified by comparing it with the results obtained by numerical integration of a integral representation of general Maliuzhinets function. A closed-form of psipi/2(z) is also obtained since it can be expressed in terms of psipi(z)

A simple approximation of the Maliuzhinets function for describing wedge diffraction

A new approximation of the Maliuzhinets function /spl psi//sub /spl Phi//(/spl alpha/) by a single cosine function is proposed. This is the simplest approximation of the function available in the literature. The use of the cosine approximation permits representing fields scattered by nonmetallic wedges in terms of elementary functions only. The accuracy of the approximation, as applied to the case of an impedance wedge under normal illumination, is estimated both analytically and numerically.

Diffraction by an Anisotropic Impedance Half-Plane at Skew Incidence

ABSTRACT The problem of scattering from an anisotropic impedance half-plane at skew incidence is considered. Application of Maliuzhinets’ technique in combination with an approximation leads to a solution for the scattered field expressed in terms of Maliuzhinets functions. Even though the approximation is not based on the assumption that the obliquity of the incident wave is small, the exact solution is identically recovered in the cases of perpendicular incidence and skew incidence on an isotropic half-plane.

Approximation of the Maliuzhinets function

The Maliuzhinets function appears in the important problem of diffraction by an impedance wedge. A more convenient normalized Maliuzhinets function is introduced and new methods of approximating this function are proposed. The Osipov's algorithm is adapted to this normalized function, which provides more accuracy, an extension of the validity domain of the approximation, and more efficiency in computation times. Alternate new methods which permit significant saving computation times are then developed.

Calculation of Maliuzhinets function in complex region

Numerical computation of the Maliuzhinets (1958) function is a difficult and important problem encountered in electromagnetic scattering by a wedge with impedance faces. We employ the tanh transformation to successfully solve this problem. The computation of our result with the precision values shows that the computing precision of our method is high.

Reflections, diffractions, and surface waves for an interior impedance wedge of arbitrary angle

The asymptotic-impedance wedge solution for plane-wave illumination at normal incidence is examined for interior wedge diffraction. An efficient method for calculating the diffraction coefficient for arbitrary wedge angle is presented, as previous calculations were very difficult except for three specific wedge angles for the uniform geometrical theory of diffraction (UTD) expansion. The asymptotic solution isolates the incident, singly reflected, multiply reflected, diffracted, surface wave, and associated surface wave transition fields. Multiply reflected fields of any order are considered. The multiply reflected fields from the exact solution arise as ratios of auxiliary Maliuzhinets functions; however, by using properties of the Maliuzhinets functions, this representation can be reduced to products of reflection coefficients which are much more efficient for calculation. A surface-wave transition field is added to the surface wave boundaries. Computations are presented for interior wedge diffractions although the formulation is equally valid for both exterior and interior wedges with uniform but different impedances on each face for both soft and hard polarizations. In addition, the accuracy of the high-frequency asymptotic expansion is examined for small diffraction distances by direct comparison of the exact and asymptotic solutions. >

Diffraction by a metallic wedge covered with a dielectric material

In the literature, the usual formulation which assumes a constant impedance boundary condition does not permit one to express both the value of the geometrical optics field and the actual excited surface wave, since the impedance associated with the material for the reflection and for the surface wave are generally different. To obtain these elements in the field expression, we replace the notion of constant impedance by a differential operator. The general solution presented here preserves enough degrees of freedom to satisfy the continuity of fields at the internal junction of the two materials which cover each face of the wedge. The numerical calculations are based upon a function, related to the Maliuzhinets function, in a form which is easily computable.

Analytic expressions for a function occurring in diffraction theory

Simple expressions are derived for the Maliuzhinets (integral) function which arises in connection with diffraction by an impedance wedge. The given expressions allow the computation of the function with remarkable accuracy throughout the entire complex plane for any exterior wedge angle.

Polynomial approximation of Maliuzhinets' function

Approximate formulas for the Maliuzhinets functions which give the solutions for the wedge diffraction with face impedance, are derived using the Chebyshev polynomials for various wedge angles. The coefficients of the polynomials are listed in a table. Although the formulas are derived for real variables, it is found that they can be used for the restricted range of complex variables. Since the validity of the formulas depends on both real and imaginary part when applied to complex variables, the equi-error contour of the formulas is given for the special case where the exact solution is available. An alternative but empirical formula for Maliuzhinets' functions is also obtained, which can be applied to arbitrary wedge angles.

Simple expressions for a function occurring in diffraction theory

The Maliuzhinets (integral) function arises in connection with diffraction by a half-plane. Two simple expressions are derived which, when used in conjunction with known identities, serve to approximate the function to a high degree of accuracy throughout the entire complex plane.