Scattering of electromagnetic waves by a metallic object partially immersed in a semi-infinite dielectric medium

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 2, FEBRUARY 2001 223

Scattering of Electromagnetic Waves by a MetallicObject Partially Immersed in a Semi-Infinite

Dielectric MediumR. T. Ling, Student, IEEE,and P. Y. Ufimtsev, Fellow, IEEE

Abstract—The concept of the generalized scattering amplitudeis applied to the electromagnetic (EM) scattering of an arbitrarilyshaped, perfectly conducting object partially immersed in a semi-infinite dielectric medium. The dielectric medium can have eitherelectric loss or magnetic loss or both. In a two-dimensional (2-D)formulation, after the outgoing cylindrical wave is factored outfrom the scattered wave, the remaining wave envelope componentin the scattered wave is defined as the generalized scattering am-plitude. The transformed Helmholtz equation in terms of the gen-eralized scattering amplitude can be solved numerically using a fi-nite-difference method over the entire scattering domain includingboth the semi-infinite free-space (vacuum or air) and the semi-in-finite dielectric medium. Example problems of scattering by in-finitely long, perfectly conducting cylinders of circular and trape-zoidal ship-shaped cross sections are solved to demonstrate the the-oretical formulation and numerical method. The radial profiles ofthe generalized scattering amplitude and the total field over theentire scattering region are also presented, and their propertiesare discussed. The far-field bistatic cross section and induced cur-rent density on the obstacle’s surface are also presented. These re-sults show that the method can be used to yield complete and accu-rate solutions to 2-D EM scattering problems involving arbitrarilyshaped metallic objects partially immersed in a penetrable semi-in-finite dielectric medium.

Index Terms—Optical theorem, radar cross section, scatteringamplitude, scattering diffraction.

I. INTRODUCTION

I N CASES where the scattering object is not isolated in asingle medium, but instead is immersed in two interfacial

media, the theoretical and numerical analysis of the scatteringproblem becomes much more difficult. One has to deal with notonly the usual target surface boundary condition and the far-field radiation condition, but also the media interface boundarycondition. The additional media interface condition accounts forthe reflection and refraction of waves at the media interface.This media interface extends from the scattering object to in-finity, making the analysis and computation difficult. The en-forcement of the media interface boundary condition is criticalin the solution of scattering problems involving two differenthalf-space media. Theoretically, it can only be simplified whenone of the two media is perfectly conducting, allowing imagingtheory to be applied.

Manuscript received July 21, 1998; revised July 28, 2000.R. T. Ling was with the Northrop Grumman Corporation. He is currently with

the Technische Universität Hamburg-Harburg, Hamburg D-21073, Germany.P. Y. Ufimtsev is with the Electrical Engineering Department, University of

California at Los Angeles, Los Angeles, CA 90095-1594 USA. He was withNorthrop Grumman Corporation, Pico Rivera, CA 90660-3783 USA.

Publisher Item Identifier S 0018-926X(01)01271-6.

Scattering of electromagnetic (EM) waves by buried andpartly buried bodies of revolution has been treated by Changand Mei [1]. Xu and Butler [2] have computed current inducedon partially buried cylinders at the interface between two media.Recently, Cuiet al. [3] studied EM scattering by multiple di-electric and conducting objects buried under multilayeredmedia and Cui and Chew [4] discussed an efficient method forthe evaluation of Sommerfeld integrals involved in scatteringsof buried objects in an integral equation formulation. In thispaper, a differential equation approach is proposed for thestudy of scattering of EM waves by a metallic object partiallyimmersed in a semi-infinite dielectric medium.

When an EM wave is incident upon an object immersed in twointerfacial media, with parts of the object lying in one mediumand the rest in the second medium; in addition to being scatteredby the obstacle, the incident wave is also reflected and trans-mitted at the media interface. The primary reflected wave andthe transmitted wave can subsequently be scattered by the ob-stacle; causing secondary scattered waves, reflected waves, andtransmitted waves. The complicated interactions among the inci-dent, reflected, transmitted, and scattered waves need to be takeninto account in order to fully describe the scattering process. Forthis purpose, we will represent the total field throughout the firstmedium where the incident wave originates as a linear superpo-sition of the incident wave, reflected wave, and a first-mediumscattered wave. Throughout the second medium, the total fieldwill be represented as a linear superposition of the transmittedwaveandasecond-mediumscatteredwave.Thescatteredwave ineither medium will be conveniently written as an outgoing cylin-dricalwave [in two-dimensional (2-D)problems]modulatedbyageneralized scattering amplitude. The concept of the generalizedscattering amplitude has been introduced in solving acoustic andEM scatterings by objects in a single medium [5]–[7]. Recentlyit has also been applied to quantum potential scattering governedby general central potentials [8]. Its extended concept has beenapplied to integral equation approach to scattering problems [9].

II. THEORETICAL FORMULATION

The problem under consideration is the scattering of a planeEM wave, propagating in free-space (vacuum or air) and inci-dent normally upon an infinitely long, perfectly conducting, andarbitrarily shaped cylinder. The cylinder is partially immersed ina semi-infinite dielectric medium. The schematics and the co-ordinate system are depicted in Fig. 1. In Region 1 of Fig. 1,

0018–926X/01$10.00 © 2001 IEEE

224 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 2, FEBRUARY 2001

Fig. 1. Schematics.

the incident EM wave of wavelengthand frequency prop-agates in free-space toward the dielectric medium (Region 2)in the direction of wavenumber vector which lieson the – plane and makes an angle with the axis. Theabsolute vacuum permittivity and permeability are denoted by

and . The relative vacuum permittivity and permeabilityare unity, , . The incident plane wave is repre-sented by , where is the vacuumwavenumber, is the angular frequency, andis theradius vector. The time-harmonic dependence is as-sumed for all dependent variables and subsequently suppressed.Fig. 1 depicts the electric field polarization case where the inci-dent electric field is parallel to theaxis. In Region 2 of Fig. 1,the semi-infinite dielectric medium is characterized by relativepermittivity and relative permeability , both of which ingeneral are complex. All dependent variables are assumed to beindependent of in the present 2-D formulation.

A. Governing Equations and Boundary Conditions

The theoretical formulation will be described in detail firstfor -field polarization case, then the necessary changes for the

-field polarization case will be given. In the-field polar-ization case, the component of the electric field satisfies theHelmholtz equation. In region 1 (free-space), we have

(1)

Similarly in region 2 (dielectric medium), we have

(2)

where is the wavenumber in the dielectric re-gion. In general, is complex.

The boundary conditions that need to be enforced upon thesolution of the Helmholtz equation, (1) and (2), include the usualboundary condition on the surface of a perfectly conductingscattering body and the radiation condition in the far field. Forthe -field polarization case, thecomponent of the total elec-tric field vanishes on the perfectly conducting obstacle’s sur-face, i.e.,

and on the scatterer's surface(3)

In the far-field, the radiation condition applies to the scatteredwave

(4a)

and

(4b)

where and are the scattered wave components of thetotal wave in the first and second medium, respectively. In addi-tion to these two boundary conditions, a new interface boundarycondition is to be imposed on the media interface that extendsfrom the scatterer’s surface to the far field on the– plane

. The tangential components of the total electric andmagnetic fields are required to be continuous across the inter-face, i.e.,

(5)

and

(6)

on the – plane.For the 2-D -field polarization case, Maxwell’s equations

provide

and (7)

and (6) can be rewritten as

(8)

Therefore, the media interface conditions, (5) and (8), implythat the total electric field strength and its gradient (as scaledby the permeability) in the direction normal to the interface arecontinuous across the media interface.

One notes that there is another interface boundary conditionthat requires the normal component of the magnetic flux vector

to be continuous across the interface

(9)

From the second of the Maxwell’s equations in (7), this implies

(10)

Therefore, the gradient of the total-field tangential to the in-terface is continuous across the interface. This condition is sat-isfied automatically if (5) is satisfied for all points on th–plane.

B. Formulation in Terms of the Generalized ScatteringAmplitude

We now apply the concept of the generalized scattering am-plitude to the scattering problem under consideration. The pre-vious theoretical development and formulation [5]–[7] for anisolated obstacle in free-space will be modified for an obstaclepartially immersed in the dielectric medium while the basic con-cept of the generalized scattering amplitude will be retained.

LING AND UFIMTSEV: SCATTERING EM WAVES BY METALLIC OBJECT 225

The presence of the semi-infinite dielectric medium complicatesthe scattering process by causing the incident plane wave to un-dergo reflection and refraction (or transmission) processes at themedia interface. The scattering (or diffraction) process now in-volves not only the original incident wave but also the reflectedwave and the refracted wave. To explicitly account for the addi-tional waves and to incorporate the concept of the generalizedscattering amplitude, the total field is expressed in the followingforms. In free-space (Region 1), , we write

(11)

and in the dielectric medium (Region 2), , we write

(12)

In (11), the first term on the right-hand side is the incident planewave with unit amplitude and with the incident angle rel-ative to the positive axis. The second term represents the re-flected plane wave with amplitude and propagating in thedirection relative to the positive -axis. The third termis the scattered wave in free-space and it is composed of an out-going cylindrical wave modulated by the generalized scatteringamplitude . In (12), the first term on the right-hand side repre-sents the refracted (or transmitted) wave with amplitudeandwith refracted (or transmitted) angle relative to the negative

axis. The second term on the right-hand side of (12) is the scat-tered wave in the dielectric medium, and it is composed of anoutgoing cylindrical wave modulated by the generalized scat-tering amplitude . The generalized scattering amplitudesand are explicitly dependent on bothand , in contrast to theordinary far-field scattering amplitude which depends ononly.Note also that is complex for lossy dielectric medium. Thetransmitted wave is thus, in general, an inhomogeneous planewave. The angular orientations are shown in Fig. 1.

The explicit expressions for the amplitudes of reflected andrefracted waves in terms of incident angle and wavenumbers inboth media can be found in [10]

(13)

(14)

In deriving (13) and (14), the relationship between the incidentangle and the transmitted angle

(15)

has been used.The purpose of writing the total fields in the forms given in

(11) and (12) is to avoid dealing with the oscillations in the totaland scattered fields in the physical space. The oscillation in the

total field is due to the interference among various wave compo-nents in (11) and (12). By explicitly factoring out the outgoingcylindrical wave from the scattered waves, we also avoid the os-cillatory component of the scattered field in the radial direction.The merit of the concept of the generalized scattering amplitudehas been demonstrated [5]–[8].

The plane wave components in (11) and (12) satisfy theHelmholtz equation automatically. The substitution of (11)and (12) into (1) and (2) results in the following transformedHelmholtz equation for the generalized scattering amplitudes

(16)

(17)

for the interior of physical space bounded by the scatter’s sur-face, media interface, and the far-field computational circle onwhich the radiation condition is applied.

Substituting the expressions for the scattered field from (11)and (12) into the far-field radiation condition, (4), one obtains

for both and (18)

The second media interface condition, (8), can be equivalentlyexpressed in the cylindrical coordinates as

(19)

at the interface and . Substitution of the total fieldexpressions, (11) and (12), into this media interface conditionresults in

(20)

for both and . In (20)

and

are the incident plane wave, reflected plane wave, outgoingcylindrical wave in free-space, transmitted plane wave, and out-going cylindrical wave in the dielectric medium, respectively.

In enforcing the media interface boundary condition, itis found that, for lossy dielectric medium with complex


wavenumber , the outgoing cylindrical wavein the dielectric medium becomes vanishingly small forlarge due to the exponentially decreasing factorin . After finite-differencediscretization of (20), the matrix elements containing the

factor become exceedingly small for far-fieldinterface grid points, and they make the matrix nearly singular.This numerical difficulty can be alleviated by absorbing theexponentially decreasing factor into the unknowngeneralized scattering amplitude in the lossy dielectricmedium.

For this purpose, we defined a new unknownand an undamped outgoing cylindrical wave

so that the scattered wave termin (12) becomes . The governing equation (17) thenbecomes

(17)

The media interface condition (20) is also modified with thereplacement of and by and .

The radiation condition (18) for the dielectric medium be-comes

(18)

With for nonlossy dielectric, (17)′, (18)′, and (20) reduceto the corresponding (17), (18), and (20). For lossy dielectricin region 2, the transmitted wave in (12) represents a system ofinhomogeneous plane waves [10]. The physical interpretation ofthe scattering process in such a case is quite different from thatfor the nonlossy dielectric. However, Snell and Fresnel’s lawsare still mathematically correct and the theoretical formulationpresented above remains formally valid.

C. Radar Cross Sections and Optical Theorem

The measurable quantity in the far field which is usually ofthe most interest in a scattering problem is expressed in termsof the radar cross section as defined by

(21)

where is the scattered wave in thethmedium. For the unit amplitude incident plane wave in (11), itfollows that

or (22)

In this paper, the first medium is taken to be free-space, so that.

A useful check of the numerical results is the application ofthe optical theorembased on the conservation of power flux.In the case of the scattering process under consideration here,if the dielectric medium is nonlossy, then the optical theoremstates that the decrease in the power flux in the directions of thereflected wave and the refracted wave due to the presence of theobstacle is exactly compensated by scatterings in all possibledirections. For a nonlossy dielectric medium, when, , ,

, and are purely real, it can be shown that the opticaltheorem assumes the following form:

(23)

This is a generalized form of the one-medium case [5]–[7]. Itfollows from

(24)

where is the complex Poynting vector andis an outward normal surface element vector on the exte-

rior surface of a large computational circular cylinder whoseaxis coincides with the-axis. The left-hand side of (24) rep-resents the net power flux passing through the closed surface.For nonlossy medium, this integral vanishes. By using (11) and(12) for the expression of total field and the correspondingexpression for in (24) and letting the radius of the large com-putational circular cylinder go to infinity, one obtains the opticaltheorem in (23). The method of stationary phase is used in eval-uating the cross-term integrals involved. The optical theoremderivation procedure presented above generally follows that forthree-dimensional (3-D) quantum potential scatterings [11]. It isto be noted that the exact form of the optical theorem dependson the specific form of the scattered wave defined. If, in (11) and(12), the generalized scattering amplitude is defined to containan extra phase factor, such that , then theright-hand of (23) would involve only the imaginary part of thenew generalized scattering amplitude.

D. H-Field Polarization

For the -field polarization case, with the Maxwell’s equa-tions providing

and (7)

the general theoretical formulation described above is appli-cable with the replacement of by and by in(1), (2), (4), (11), and (12) and with changes in the expressionsfor reflection and transmission waves amplitudes, the body sur-face boundary condition, and the interface boundary condition.


For the -field polarization case, the amplitudes of the reflectedand refracted waves are given by [10]

(13)

(14)

The obstacle surface boundary condition in the-field polar-ization case is the requirement that the normal gradient ofon the scatterer’s surface vanishes, i.e.,

and (3)

where is the outward normal unit vector on the scatterer’s sur-face. The media interface condition in the-field polarizationcase is given by

(19)

The optical theorem for the -field polarization case assumesthe same form as that in (23), but with the replacement of

by .

III. N UMERICAL METHODS

The boundary value problem formulated above can be solvedby a finite-difference method. The detailed numerical methodsfor scattering problems of an obstacle in a single medium havebeen discussed in detail in previous article [5]–[8]. These arti-cles and others [12], [13] describe the conversion of partial dif-ferential equations into simultaneous algebraic difference equa-tions, stretched coordinates transformation, grid generation [12]for arbitrary obstacle shapes, and time-dependent type iterativeprocedure [13] for the solution of large sparse matrix equations.In the following, we discuss the particular numerical details in-volved in the scatterings of objects partially immersed in twomedia separated by a planar interface.

As in the problem of scattering by an airfoil-shaped object,sets of computational space body-fitted coordinated system

can be generated numerically. The variablesand aregeneral angular and radial coordinates, much like a set of gen-eralized and polar coordinates. Most grid generation codesuse Cartesian coordinates for the description of body-fittedgrids, i.e., defining and as functions of and . The trans-formed Helmholtz equations in cylindrical coordinates (16)and (17) are first transformed into the Cartesian coordinates [6]then transformed into the computational body-fitted coordinatesystem . The resultant equations actually solved are ofthe form

(25)

where coefficients , , , etc., involve components of thecovariant metric tensors of the body-fitted coordinate system.

The grid points with and various values fall righton the obstacle’s surface. The grid points are densely popu-lated near the obstacle’s surface to better resolve the expectedrapid variation of the generalized scattering amplitude in thenear field. The distribution of grid points become sparser radi-ally outward and eventually fall on a large circle centered at theobstacle with the interface line as a diameter. For present prob-lems, in order to accommodate the enforcement of the interfaceboundary condition, the interface line (i.e., the-axis) is usedas a boundary in grid generation of two sets of grids based onthe obstacle’s geometry inside each medium. This is to ensurethat the two sets of grids share common grid points at the mediainterface. In this manner, a set of C-shaped grid network is gen-erated for each medium region. Common grid points are gen-erated along the interface line. The two sets of C-shaped gridscombine into a series of O-shaped grids for the computation ofmetrical coefficients and numerical solution of (25).

Similar to the enforcement of internal boundary condition atthe potential well boundary between the interior and exterior re-gions in quantum potential scattering [8], the grid points on themedia interface are assumed to have double unknowns (bothand ) so that the field continuity relation (5) and gradient con-tinuity relation (8), (19), or (20) can be satisfied at the interfaceusing one-sided finite-difference expressions for the derivatives.In the numerically generated coordinate system , the sur-face boundary condition for the -field polarization case, (3)′,is transformed into [12]

(26)

where

and

One useful quantity to be computed, particularly for the pur-pose of comparison with other theoretical methods is the in-duced current density on the obstacle’s surface. This quantity

in the -field polarization case is given by

(27)

where is the unit vector in the axis direction. For the -fieldpolarization case, the induced surface current density equals themagnitude of the surface magnetic field, but lies in the– planein a direction tangential to the cylinder’s cross sectional profile.

The numerical results presented in this paper are obtained byusing a banded matrix solver that directly inverts the matrix in-volved. Since the resultant banded matrix is sparse, the use ofiterative sparse matrix solvers for nonpositive definite matricessuch as the preconditioned conjugate gradient type GMRES[14], [15] would be more computer memory efficient and wouldallow the solution of larger matrix equations so as to extend the


Fig. 2. A metallic circular cylinder half immersed in an absorbing medium. Radial profiles of generalized scattering amplitudes,E-field polarization,� = 10 .

Fig. 3. A metallic circular cylinder half immersed in an absorbing medium. Radial profiles of generalized scattering amplitudes,H-field polarization,� = 10 .

range of applicability of the present method to higher incidentwave frequencies. However, since the matrix to be inverted de-pends only upon the obstacle’s geometry, the direct matrix in-version methods has the advantage of allowing one to save theelements of the inverse matrix. The saved inverse matrix canthen be used for scatterings with the same geometry but with dif-ferent incident angles that affect only the right hand side of thematrix equation. The advantage is considerable since typicallymore than 90% of the computational time is spent in formingthe matrix elements and finding its inverse. This advantage isunfortunately lost for iterative methods.

IV. RESULTS FOREXAMPLE PROBLEMS AND DISCUSSIONS

Two example problems are solved using the computer codeSHIP2D which is developed and based on the theory and numer-ical method described above. Before applying the code to the

example problems, the code is tested in simplified one-mediumcases where the second medium is assumed to be free space. Thenumerical result [5], [6] previously reported for the scatteringsof circular cylinders and airfoil shaped cylinders are reproduced.The code is also tested for problems involving scatterings of2-D metallic objects partially immersed in a highly conductivedielectric medium to simulate a perfectly conducting groundplane. For this case, the code yields nearly identical results forfar-field cross section and induced surface current density withthose computed by the RAM2D code [16] which handles targetssituated on a perfectly conducting ground plane. The SHIP2Dcode is further tested on nonlossy dielectric media to check itsobedience to the optical theorem.

The first example problem considers a metallic circularcylinder half immersed in a dielectric medium. The permittivityand permeability of the dielectric medium is taken to be thoseof the material used in a recent study [17] of propagation and


Fig. 4. A metallic circular cylinder half immersed in an absorbing medium. Magnitude of total field,� = 90 , � = 10 .

Fig. 5. A metallic circular cylinder half immersed in an absorbing medium. Bistatic radar cross sections,� = 10

excitation of surface waves in an absorbing layer. At 2 GHz,it provides and . Thecylinder’s radius is taken to be the length unitand the com-putation is performed for . The computational resultsare obtained for a 40 (angular) by 52 (radial) grid network.Figs. 2 and 3 show (for -field and -field polarizations,respectively) representative radial profiles of the generalizedscattering amplitudes in three scattering directions for an inci-dent angle of 10above the horizon. For this example problemwe obtain ,for the -field polarization case and, for the -field polarization case.

These radial profiles exhibit the expected behavior that theyrapidly but smoothly vary near the obstacle’s surface, thenquickly approach asymptotic plateaus. This behavior is in sharpcontrast to the radial profiles (Fig. 4) of the total field strengthas computed from (11) along the axis . Theoscillations in the total field are due to the interferences among

its components of incident wave, reflected wave, and scatteredwave. The oscillations continue unabated beyond fifty times thecylinder’s radius at which the generalized scattering amplitudeshave long settled into their asymptotic values (Figs. 2 and 3).These results clearly demonstrate the virtue of solving for thegeneralized scattering amplitudes directly.

For most practical problems, the far-field bistatic cross sec-tion in free-space and the induced surface current density in bothmedia are of much interest. They are shown in Figs. 5–8 for thisexample problem. Fig. 5 shows the qualitative comparison ofthe SHIP2D results between one medium and two media cases.The one medium result can be obtained by SHIP2D assuming

and or using the results of [5]. The differencebetween the two cases is due entirely to the presence of the non-free-space dielectric medium in Region 2. Fig. 6 shows the sameSHIP2D results with those of high frequency approximation[18] for metallic circular cylinders and with the reflected waveexplicitly taken into account. For this comparison, the high-fre-


Fig. 6. A metallic circular cylinder half immersed in an absorbing medium. Bistatic radar cross sections,� = 10 .

Fig. 7. A metallic circular cylinder half immersed in an absorbing medium. Bistatic radar cross sections,� = 10 .

quency scattering amplitude is taken to be

(28)

The first term on the right-hand side accounts for the contribu-tion due to the incident wave while the second term accounts forthat due to the reflected wave with being the amplitude of thereflected wave as defined in (11), (13), or (13)′. The high-fre-quency approximation provides reasonable estimation for bothpolarizations up to , considering is hardly

in the high-frequency regime. Fig. 7 compares the SHIP2D re-sults with those of RAM2D which assumes the second mediumto be a perfectly conducting ground plane. For the-field po-larization case, at is reasonably close to that ofan infinitely conducting ground plane. This explains the closeagreement between SHIP2D results and RAM2D results up toscattering angle at 125. This is not the case in -field polariza-tion as for -field polarization is quite different from thatof an infinitely conducting ground plane. Fig. 8 shows the in-duced current density on the obstacle’s surface for this exampleproblem. The peaks at for both polarizations can beattributed to the interference between the direct and reflectedcreeping waves at the interface in the shadow region. The in-duced surface current density in the dielectric medium falls offsharply as the distance from interface increases due to the ab-sorbent nature of the dielectric.

The second example problem involves a trapezoidal ship-shaped conducting cylinder partially immersed in ocean water(Fig. 1) with and . Representativenear-field and global grid networks for this example problem


Fig. 8. A metallic circular cylinder half immersed in an absorbing medium. Induced surface current density,� = 10 .

Fig. 9. Ship-shaped geometry. Global grid distribution.

Fig. 10. Ship-shaped geometry. Near-field grid distribution.

are shown in Figs. 9 and 10. A much finer grid network of 64(radial) by 96 (angular) grid points are used in the computationof cross section and current density results presented here. Thewaterline width of the ship-shaped cylinder is taken to bewhere is the length unit and the computation is performed

at . The ship’s hull and the ocean surface form aninternal wedge with a wedge angle of .The incident angle is at 10above the horizon. The top deckis at above the water surface while the bottom deckis at below the water surface. The width of the topdeck is while that of the bottom deck is. Under theseconditions, the reflected and transmitted wave amplitudes arefound to be ,

for the -field polarization case and ,for the -field polarization case. The

resultant bistatic cross sections (Fig. 11) for both polarizationsare compared to the RAM2D computations. The RAM2D resultsfor -field polarization are very close to those of SHIP2Dresults throughout the entire scattering angle range due to theamplitude of reflected wave being very close to that of aninfinitely conducting ground plane. As in the first exampleproblem, this is not the case for -field polarization. Thesurface current density (Fig. 12) is smoothly varying alongthe flat contour of the ship’s hull but changes abruptly atthe corners. The peaks at for both polarizationscan be attributed to the interference between the corner edgewaves and their reflections at the interface in the shadowregion. The current density drops more than nine orders ofmagnitude from the top deck to the bottom hull in the-fieldpolarization case and more than ten orders of magnitude in the

-field polarization case as the second medium (ocean water)is highly absorbent. As it is well known [19], in the-fieldpolarization case, the surface current tends to infinity in thevicinity of a sharp corner (wedge) according towhere is the distance from the corner andis the exteriorwedge angle. For the example problem under consideration, thisimplies that the surface current should approachfor the upper corner in the first region (vacuum or air) and

for the lower corner in the second (dielectric)region. However, in the present finite-difference computation,only a finite number of grid points can be employed; andoutside the obstacle, the grid distribution, which transitions thegrids from body-fitting to a large circle, effectively introducesa nonvanishing radius of curvature at the sharp corner’s edge


Fig. 11. Scattering by a ship-shaped conducting cylinder in ocean water. Bistatic radar cross sections,� = 10 above the horizon.

Fig. 12. Scattering by a ship-shaped conducting cylinder in ocean water. Induced surface current density,� = 10 above the horizon.

and rounds it off. Thus, the numerical results do not showthe mathematically rigorous asymptotic behavior.

V. CONCLUSION

A class of scattering problems of practical interest involvesobjects partially immersed in a dielectric medium. This type ofproblem introduces boundary conditions at the interface whichextends from the body’s surface to the far field. To handle theadditional boundary condition, the concept of the generalizedscattering amplitude is applied to the functional form of the scat-tered waves in each medium. This allows the interface boundarycondition to be enforced at a finite number of points on the inter-face without truncating the computational domain. At the inter-face, there exist distinct generalized scattering amplitudes fromboth media but they constitute a common total field value andalso a common field gradient, as scaled by the permittivity andpermeability, across the interface. The theoretical and numer-ical methods and associated results presented above illustrate a

practical solution method for the special class of problems underconsideration.

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[19] P. Ya. Ufimtsev, “Fast convergent integrals for nonuniform currents onwedge faces,”Electromagn., vol. 18, pp. 289–313, 1998.

R. T. Ling (S’00), photograph and biography not available at the time of publi-cation.

P. Y. Ufimtsev (SM’92–F’99), photograph and biography not available at thetime of publication.

Documents

Scattering of electromagnetic waves by a metallic object partially immersed in a semi-infinite dielectric medium